From 2da26769fa92e8790a297c5aa278321e0e1ddef5 Mon Sep 17 00:00:00 2001 From: Lillian Weng Date: Thu, 26 Oct 2023 19:38:52 -0700 Subject: [PATCH] publish note 18 --- _quarto.yml | 2 +- docs/case_study_HCE/case_study_HCE.html | 6 + .../loss_transformations.html | 6 + docs/cv_regularization/cv_reg.html | 6 + docs/eda/eda.html | 6 + .../feature_engineering.html | 6 + docs/gradient_descent/gradient_descent.html | 22 +- docs/index.html | 6 + docs/intro_lec/introduction.html | 6 + docs/intro_to_modeling/intro_to_modeling.html | 6 + .../figure-html/cell-2-output-1.png | Bin 86402 -> 86081 bytes docs/ols/ols.html | 6 + docs/pandas_1/pandas_1.html | 6 + docs/pandas_2/pandas_2.html | 78 +- docs/pandas_3/pandas_3.html | 18 +- docs/probability_1/images/SD_change.png | Bin 28692 -> 0 bytes docs/probability_1/probability_1.html | 188 +-- docs/probability_2/images/CLTdiff.png | Bin 0 -> 10809 bytes docs/probability_2/images/bootstrap.png | Bin 0 -> 380665 bytes .../images/bootstrapped_samples.png | Bin 0 -> 320160 bytes docs/probability_2/images/clt.png | Bin 0 -> 20606 bytes docs/probability_2/images/data.png | Bin 0 -> 86231 bytes .../images/population_samples.png | Bin 0 -> 173371 bytes docs/probability_2/images/y_hat.png | Bin 0 -> 77043 bytes docs/probability_2/probability_2.html | 1164 +++++++++++++++++ docs/regex/regex.html | 10 +- docs/sampling/sampling.html | 16 +- .../figure-html/cell-13-output-1.png | Bin 34559 -> 32008 bytes .../figure-html/cell-15-output-1.png | Bin 58945 -> 57316 bytes docs/visualization_1/visualization_1.html | 6 + docs/visualization_2/visualization_2.html | 6 + .../figure-html/cell-17-output-1.png | Bin 95470 -> 95263 bytes index.log | 2 +- index.pdf | Bin 814941 -> 814952 bytes index.tex | 544 ++++++-- probability_1/probability_1.ipynb | 380 ------ probability_2/probability_2.ipynb | 302 ----- 37 files changed, 1772 insertions(+), 1026 deletions(-) delete mode 100644 docs/probability_1/images/SD_change.png create mode 100644 docs/probability_2/images/CLTdiff.png create mode 100644 docs/probability_2/images/bootstrap.png create mode 100644 docs/probability_2/images/bootstrapped_samples.png create mode 100644 docs/probability_2/images/clt.png create mode 100644 docs/probability_2/images/data.png create mode 100644 docs/probability_2/images/population_samples.png create mode 100644 docs/probability_2/images/y_hat.png create mode 100644 docs/probability_2/probability_2.html delete mode 100644 probability_1/probability_1.ipynb delete mode 100644 probability_2/probability_2.ipynb diff --git a/_quarto.yml b/_quarto.yml index f4b6349c..06c46eac 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -33,7 +33,7 @@ book: - case_study_HCE/case_study_HCE.qmd - cv_regularization/cv_reg.qmd - probability_1/probability_1.qmd - # - probability_2/probability_2.qmd + - probability_2/probability_2.qmd # - case_study_climate/case_study_climate.qmd # - inference_causality/inference_causality.qmd # - sql_I/sql_I.qmd diff --git a/docs/case_study_HCE/case_study_HCE.html b/docs/case_study_HCE/case_study_HCE.html index 2d20476a..00af55c7 100644 --- a/docs/case_study_HCE/case_study_HCE.html +++ b/docs/case_study_HCE/case_study_HCE.html @@ -248,6 +248,12 @@ 17  Random Variables + + diff --git a/docs/constant_model_loss_transformations/loss_transformations.html b/docs/constant_model_loss_transformations/loss_transformations.html index b5cb71ff..6b779453 100644 --- a/docs/constant_model_loss_transformations/loss_transformations.html +++ b/docs/constant_model_loss_transformations/loss_transformations.html @@ -250,6 +250,12 @@ 17  Random Variables + + diff --git a/docs/cv_regularization/cv_reg.html b/docs/cv_regularization/cv_reg.html index ac3047e9..5bae8857 100644 --- a/docs/cv_regularization/cv_reg.html +++ b/docs/cv_regularization/cv_reg.html @@ -253,6 +253,12 @@ 17  Random Variables + + diff --git a/docs/eda/eda.html b/docs/eda/eda.html index a8bd9f81..958b189a 100644 --- a/docs/eda/eda.html +++ b/docs/eda/eda.html @@ -253,6 +253,12 @@ 17  Random Variables + + diff --git a/docs/feature_engineering/feature_engineering.html b/docs/feature_engineering/feature_engineering.html index 14f09cc2..a45c4e3e 100644 --- a/docs/feature_engineering/feature_engineering.html +++ b/docs/feature_engineering/feature_engineering.html @@ -253,6 +253,12 @@ 17  Random Variables + + diff --git a/docs/gradient_descent/gradient_descent.html b/docs/gradient_descent/gradient_descent.html index cf2c6bd5..91a8a3d7 100644 --- a/docs/gradient_descent/gradient_descent.html +++ b/docs/gradient_descent/gradient_descent.html @@ -269,6 +269,12 @@ 17  Random Variables + + @@ -409,9 +415,9 @@

-
+ @@ -249,6 +250,12 @@ 17  Random Variables
+ + @@ -284,13 +291,6 @@

Random Variables

  • 17.3.4 Covariance and Correlation
  • 17.3.5 Summary
  • -
  • 17.4 Common Random Variables
  • -
  • 17.5 Populations and Samples -
  • @@ -752,97 +752,6 @@

    \(\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) 2\text{cov}(X,Y)\) - - -
    -

    17.4 Common Random Variables

    -

    There are several cases of random variables that appear often and have useful properties. Below are the ones we will explore further in this course. The numbers in parentheses are the parameters of a random variable, which are constants. Parameters define a random variable’s shape (i.e., distribution) and its values.

    -
      -
    • Bernoulli(p) -
        -
      • Takes on value 1 with probability p, and 0 with probability 1 - p.
      • -
      • AKA the “indicator” random variable.
      • -
      • Let X be a Bernoulli(p) random variable -
          -
        • \(\mathbb{E}[X] = 1 * p + 0 * (1-p) = p\) -
            -
          • \(\mathbb{E}[X^2] = 1^2 * p + 0 * (1-p) = p\)
          • -
        • -
        • \(\text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2 = p - p^2 = p(1-p)\)
        • -
      • -
    • -
    • Binomial(n, p) -
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      • Number of 1s in \(n\) independent Bernoulli(p) trials.
      • -
      • Let \(Y\) be a Binomial(n, p) random variable. -
          -
        • The distribution of \(Y\) is given by the binomial formula, and we can write \(Y = \sum_{i=1}^n X_i\) where: -
            -
          • \(X_i\) s the indicator of success on trial i. \(X_i = 1\) if trial i is a success, else 0.
          • -
          • All \(X_i\) are i.i.d. and Bernoulli(p).
          • -
        • -
        • \(\mathbb{E}[Y] = \sum_{i=1}^n \mathbb{E}[X_i] = np\)
        • -
        • \(\text{Var}(X) = \sum_{i=1}^n \text{Var}(X_i) = np(1-p)\) -
            -
          • \(X_i\)’s are independent, so \(\text{Cov}(X_i, X_j) = 0\) for all i, j.
          • -
        • -
      • -
    • -
    • Uniform on a finite set of values -
        -
      • Probability of each value is 1 / (number of possible values).
      • -
      • For example, a standard/fair die.
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    • -
    • Uniform on the unit interval (0, 1) -
        -
      • Density is flat at 1 on (0, 1) and 0 elsewhere.
      • -
    • -
    • Normal(\(\mu, \sigma^2\)) -
        -
      • \(f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left( -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{\!2}\,\right)\)
      • -
    • -
    -
    -
    -

    17.5 Populations and Samples

    -

    Today, we’ve talked extensively about populations; if we know the distribution of a random variable, we can reliably compute expectation, variance, functions of the random variable, etc.

    -

    In Data Science, however, we often do not have access to the whole population, so we don’t know its distribution. As such, we need to collect a sample and use its distribution to estimate or infer properties of the population.

    -

    When sampling, we make the (big) assumption that we sample uniformly at random with replacement from the population; each observation in our sample is a random variable drawn i.i.d from our population distribution.

    -
    -

    17.5.1 Sample Mean

    -

    Consider an i.i.d. sample \(X_1, X_2, ..., X_n\) drawn from a population with mean 𝜇 and SD 𝜎. We define the sample mean as \[\bar{X_n} = \frac{1}{n} \sum_{i=1}^n X_i\]

    -

    The expectation of the sample mean is given by: \[\begin{align} - \mathbb{E}[\bar{X_n}] &= \frac{1}{n} \sum_{i=1}^n \mathbb{E}[X_i] \\ - &= \frac{1}{n} (n \mu) \\ - &= \mu -\end{align}\]

    -

    The variance is given by: \[\begin{align} - \text{Var}(\bar{X_n}) &= \frac{1}{n^2} \text{Var}( \sum_{i=1}^n X_i) \\ - &= \frac{1}{n^2} \left( \sum_{i=1}^n \text{Var}(X_i) \right) \\ - &= \frac{1}{n^2} (n \sigma^2) = \frac{\sigma^2}{n} -\end{align}\]

    -

    \(\bar{X_n}\) is normally distributed by the Central Limit Theorem (CLT).

    -
    -
    -

    17.5.2 Central Limit Theorem

    -

    The CLT states that no matter what population you are drawing from, if an i.i.d. sample of size \(n\) is large, the probability distribution of the sample mean is roughly normal with mean 𝜇 and SD \(\sigma/\sqrt{n}\).

    -

    Any theorem that provides the rough distribution of a statistic and doesn’t need the distribution of the population is valuable to data scientists because we rarely know a lot about the population!

    -

    For a more in-depth demo, check out onlinestatbook.

    -

    The CLT applies if the sample size \(n\) is large, but how large does n have to be for the normal approximation to be good? It depends on the shape of the distribution of the population.

    -
      -
    • If the population is roughly symmetric and unimodal/uniform, could need as few as \(n = 20\).
    • -
    • If the population is very skewed, you will need a bigger n.
    • -
    • If in doubt, you can bootstrap the sample mean and see if the bootstrapped distribution is bell-shaped.
    • -
    -
    -
    -

    17.5.3 Using the Sample Mean to Estimate the Population Mean

    -

    Our goal with sampling is often to estimate some characteristic of a population. When we collect a single sample, it has just one average. Since our sample was random, it could have come out differently. The CLT helps us understand this difference. We should consider the average value and spread of all possible sample means, and what this means for how big \(n\) should be.

    -

    For every sample size, the expected value of the sample mean is the population mean. \(\mathbb{E}[\bar{X_n}] = \mu\). We call the sample mean an unbiased estimator of the population mean, and we’ll cover this more in the next lecture.

    -

    The square root law (Data 8) states that if you increase the sample size by a factor, the SD decreases by the square root of the factor. \(\text{SD}(\bar{X_n}) = \frac{\sigma}{\sqrt{n}}\). The sample mean is more likely to be close to the population mean if we have a larger sample size.

    -

    -transformation -

    @@ -1142,6 +1051,9 @@

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    18  Estimators, Bias, and Variance

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    • Explore commonly seen random variables like Bernoulli and Binomial distributions
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    • Apply the Central Limit Theorem to approximate parameters of a population
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    • Use sampled data to model an estimation of and infer the true underlying distribution
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    • Estimate the true population distribution from a sample using the bootstrapping technique
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    Last time, we introduced the idea of random variables: numerical functions of a sample. Most of our work in the last lecture was done to build a background in probability and statistics. Now that we’ve established some key ideas, we’re in a good place to apply what we’ve learned to our original goal – understanding how the randomness of a sample impacts the model design process.

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    In this lecture, we will delve more deeply into the idea of fitting a model to a sample. We’ll explore how to re-express our modeling process in terms of random variables and use this new understanding to steer model complexity.

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    18.1 Common Random Variables

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    There are several cases of random variables that appear often and have useful properties. Below are the ones we will explore further in this course. The numbers in parentheses are the parameters of a random variable, which are constants. Parameters define a random variable’s shape (i.e., distribution) and its values. For this lecture, we’ll focus more heavily on the bolded random variables and their special properties, but you should familiarize yourself with all the ones listed below:

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    • Bernoulli(p) +
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      • Takes on value 1 with probability p, and 0 with probability 1 - p.
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      • AKA the “indicator” random variable.
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      • Let X be a Bernoulli(p) random variable +
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        • \(\mathbb{E}[X] = 1 * p + 0 * (1-p) = p\) +
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          • \(\mathbb{E}[X^2] = 1^2 * p + 0 * (1-p) = p\)
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        • \(\text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2 = p - p^2 = p(1-p)\)
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    • Binomial(n, p) +
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      • Number of 1s in \(n\) independent Bernoulli(p) trials.
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      • Let \(Y\) be a Binomial(n, p) random variable. +
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        • The distribution of \(Y\) is given by the binomial formula, and we can write \(Y = \sum_{i=1}^n X_i\) where: +
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          • \(X_i\) s the indicator of success on trial i. \(X_i = 1\) if trial i is a success, else 0.
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          • All \(X_i\) are i.i.d. and Bernoulli(p).
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        • \(\mathbb{E}[Y] = \sum_{i=1}^n \mathbb{E}[X_i] = np\)
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        • \(\text{Var}(X) = \sum_{i=1}^n \text{Var}(X_i) = np(1-p)\) +
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          • \(X_i\)’s are independent, so \(\text{Cov}(X_i, X_j) = 0\) for all i, j.
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    • Uniform on a finite set of values +
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      • Probability of each value is 1 / (number of possible values).
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      • For example, a standard/fair die.
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      • Density is flat at 1 on (0, 1) and 0 elsewhere.
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    • Normal(\(\mu, \sigma^2\)) +
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      • \(f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left( -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{\!2}\,\right)\)
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    18.1.1 Example

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    Suppose you win cash based on the number of heads you get in a series of 20 coin flips. Let \(X_i = 1\) if the \(i\)-th coin is heads, 0 otherwise. Which payout strategy would you choose?

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    A. \(Y_A = 10 * X_1 + 10 * X_2\)

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    B. \(Y_B = \sum_{i=1}^{20} X_i\)

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    C. \(Y_C = 20 * X_1\)

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    Let \(X_1, X_2, ... X_{20}\) be 20 i.i.d Bernoulli(0.5) random variables. Since the \(X_i\)’s are independent, \(\text{Cov}(X_i, X_j) = 0\) for all pairs \(i, j\). Additionally, Since \(X_i\) is Bernoulli(0.5), we know that \(\mathbb{E}[X] = p = 0.5\) and \(\text{Var}(X) = p(1-p) = 0.25\). We can calculate the following for each scenario:

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    A. \(Y_A = 10 * X_1 + 10 * X_2\)B. \(Y_B = \sum_{i=1}^{20} X_i\)C. \(Y_C = 20 * X_1\)
    Expectation\(\mathbb{E}[Y_A] = 10 (0.5) + 10(0.5) = 10\)\(\mathbb{E}[Y_B] = 0.5 + ... + 0.5 = 10\)\(\mathbb{E}[Y_C] = 20(0.5) = 10\)
    Variance\(\text{Var}(Y_A) = 10^2 (0.25) + 10^2 (0.25) = 50\)\(\text{Var}(Y_B) = 0.25 + ... + 0.25 = 5\)\(\text{Var}(Y_C) = 20^2 (0.25) = 100\)
    Standard Deviation\(\text{SD}(Y_A) \approx 7.07\)\(\text{SD}(Y_B) \approx 2.24\)\(\text{SD}(Y_C) = 10\)
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    As we can see, all the scenarios have the same expected value but different variances. The higher the variance, the greater the risk and uncertainty, so the “right” strategy depends on your personal preference. Would you choose the “safest” option B, the most “risky” option C, or somewhere in the middle (option A)?

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    18.2 Sample Statistics

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    Today, we’ve talked extensively about populations; if we know the distribution of a random variable, we can reliably compute expectation, variance, functions of the random variable, etc. Note that:

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    • The distribution of a population describes how a random variable behaves across all individuals of interest.
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    • The distribution of a sample describes how a random variable behaves in a specific sample from the population.
    • +
    +

    In Data Science, however, we often do not have access to the whole population, so we don’t know its distribution. As such, we need to collect a sample and use its distribution to estimate or infer properties of the population. In cases like these, we can take several samples of size \(n\) from the population (an easy way to do this is using df.sample(n, replace=True)), and compute the mean of each sample. When sampling, we make the (big) assumption that we sample uniformly at random with replacement from the population; each observation in our sample is a random variable drawn i.i.d from our population distribution. Remember that our sample mean is a random variable since it depends on our randomly drawn sample! On the other hand, our population mean is simply a number (a fixed value).

    +
    +

    18.2.1 Sample Mean

    +

    Consider an i.i.d. sample \(X_1, X_2, ..., X_n\) drawn from a population with mean 𝜇 and SD 𝜎. We define the sample mean as \[\bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i\]

    +

    The expectation of the sample mean is given by: \[\begin{align} + \mathbb{E}[\bar{X}_n] &= \frac{1}{n} \sum_{i=1}^n \mathbb{E}[X_i] \\ + &= \frac{1}{n} (n \mu) \\ + &= \mu +\end{align}\]

    +

    The variance is given by: \[\begin{align} + \text{Var}(\bar{X}_n) &= \frac{1}{n^2} \text{Var}( \sum_{i=1}^n X_i) \\ + &= \frac{1}{n^2} \left( \sum_{i=1}^n \text{Var}(X_i) \right) \\ + &= \frac{1}{n^2} (n \sigma^2) = \frac{\sigma^2}{n} +\end{align}\]

    +

    \(\bar{X}_n\) is normally distributed by the Central Limit Theorem (CLT).

    +
    +
    +

    18.2.2 Central Limit Theorem

    +

    In Data 8 and in the previous lecture, you encountered the Central Limit Theorem (CLT). This is a powerful theorem for estimating the distribution of a population with mean \(\mu\) and standard deviation \(\sigma\) from a collection of smaller samples. The CLT tells us that if an i.i.d sample of size \(n\) is large, then the probability distribution of the sample mean is roughly normal with mean \(\mu\) and SD of \(\frac{\sigma}{\sqrt{n}}\). More generally, any theorem that provides the rough distribution of a statistic and doesn’t need the distribution of the population is valuable to data scientists! This is because we rarely know a lot about the population.

    +

    +

    clt

    +

    +

    Importantly, the CLT assumes that each observation in our samples is drawn i.i.d from the distribution of the population. In addition, the CLT is accurate only when \(n\) is “large”, but what counts as a “large” sample size depends on the specific distribution. If a population is highly symmetric and unimodal, we could need as few as \(n=20\); if a population is very skewed, we need a larger \(n\). If in doubt, you can bootstrap the sample mean and see if the bootstrapped distribution is bell-shaped. Classes like Data 140 investigate this idea in great detail.

    +

    For a more in-depth demo, check out onlinestatbook.

    +
    +
    +

    18.2.3 Using the Sample Mean to Estimate the Population Mean

    +

    Now let’s say we want to use the sample mean to estimate the population mean, for example, the average height of Cal undergraduates. We can typically collect a single sample, which has just one average. However, what if we happened, by random chance, to draw a sample with a different mean or spread than that of the population? We might get a skewed view of how the population behaves (consider the extreme case where we happen to sample the exact same value \(n\) times!).

    +

    +clt +

    +

    For example, notice the difference in variation between these two distributions that are different in sample size. The distribution with a bigger sample size (\(n=800\)) is tighter around the mean than the distribution with a smaller sample size (\(n=200\)). Try plugging in these values into the standard deviation equation for the normal distribution to make sense of this!

    +

    Applying the CLT allows us to make sense of all of this and resolve this issue. By drawing many samples, we can consider how the sample distribution varies across multiple subsets of the data. This allows us to approximate the properties of the population without the need to survey every single member.

    +

    Given this potential variance, it is also important that we consider the average value and spread of all possible sample means, and what this means for how big \(n\) should be. For every sample size, the expected value of the sample mean is the population mean: \[\mathbb{E}[\bar{X}_n] = \mu\]. We call the sample mean an unbiased estimator of the population mean and will explore this idea more in the next lecture.

    +
    + +
    +
    +

    The square root law (Data 8) states that if you increase the sample size by a factor, the SD decreases by the square root of the factor. \[\text{SD}(\bar{X_n}) = \frac{\sigma}{\sqrt{n}}\] The sample mean is more likely to be close to the population mean if we have a larger sample size.

    +
    +
    +
    +
    +
    +
    +

    18.3 Prediction vs. Inference

    +

    At this point in the course, we’ve spent a great deal of time working with models. When we first introduced the idea of modeling a few weeks ago, we did so in the context of prediction: using models to make accurate predictions about unseen data. Another reason we might build models is to better understand complex phenomena in the world around us. Inference is the task of using a model to infer the true underlying relationships between the feature and response variables. For example, if we are working with a set of housing data, prediction might ask: given the attributes of a house, how much is it worth? Inference might ask: how much does having a local park impact the value of a house?

    +

    A major goal of inference is to draw conclusions about the full population of data given only a random sample. To do this, we aim to estimate the value of a parameter, which is a numerical function of the population (for example, the population mean \(\mu\)). We use a collected sample to construct a statistic, which is a numerical function of the random sample (for example, the sample mean \(\bar{X}_n\)). It’s helpful to think “p” for “parameter” and “population,” and “s” for “sample” and “statistic.”

    +

    Since the sample represents a random subset of the population, any statistic we generate will likely deviate from the true population parameter, and it could have been different. We say that the sample statistic is an estimator of the true population parameter. Notationally, the population parameter is typically called \(\theta\), while its estimator is denoted by \(\hat{\theta}\).

    +

    To address our inference question, we aim to construct estimators that closely estimate the value of the population parameter. We evaluate how “good” an estimator is by answering three questions:

    +
      +
    • Do we get the right answer for the parameter, on average?
    • +
    • How variable is the answer?
    • +
    • How close is our answer to the parameter?
    • +
    +
    +

    18.3.1 Modeling as Estimation

    +

    Now that we’ve established the idea of an estimator, let’s see how we can apply this learning to the modeling process. To do so, we’ll take a moment to formalize our data collection and models in the language of random variables.

    +

    Say we are working with an input variable, \(x\), and a response variable, \(Y\). We assume that \(Y\) and \(x\) are linked by some relationship \(g\); in other words, \(Y = g(x)\). \(g\) represents some “universal truth” or “law of nature” that defines the underlying relationship between \(x\) and \(Y\). In the image below, \(g\) is denoted by the red line.

    +

    As data scientists, however, we have no way of directly “seeing” the underlying relationship \(g\). The best we can do is collect observed data out in the real world to try to understand this relationship. Unfortunately, the data collection process will always have some inherent error (think of the randomness you might encounter when taking measurements in a scientific experiment). We say that each observation comes with some random error or noise term, \(\epsilon\). This error is assumed to be a random variable with expectation \(\mathbb{E}(\epsilon)=0\), variance \(\text{Var}(\epsilon) = \sigma^2\), and be i.i.d. across each observation. The existence of this random noise means that our observations, \(Y(x)\), are random variables.

    +

    +data +

    +

    We can only observe our random sample of data, represented by the blue points above. From this sample, we want to estimate the true relationship \(g\). We do this by constructing the model \(\hat{Y}(x)\) to estimate \(g\).

    +

    \[\text{True relationship: }Y = g(x)\]

    +

    \[\text{Observed relationship: }Y = g(x) + \epsilon\]

    +

    \[\text{Prediction: }\hat{Y}(x)\]

    +

    +y_hat +

    +
    +

    18.3.1.1 Estimating a Linear Relationship

    +

    If we assume that the true relationship \(g\) is linear, we can express the response as \(Y = f_{\theta}(x)\), where our true relationship is modeled by \[Y = g(x) + \epsilon\] \[ f_{\theta}(x) = Y = \theta_0 + \sum_{j=1}^p \theta_j x_j + \epsilon\]

    +
    +
    +
    + +
    +
    +Which Expressions are random? +
    +
    +
    +

    In our two equations above, the true relationship \(g(x) = \theta_0 + \sum_{j=1}^p \theta_j x_j\) is not random, but \(\epsilon\) is random. Hence, \(Y = f_{\theta}(x)\) is also random.

    +
    +
    +

    This true relationship has true, unobservable parameters \(\theta\), and it has random noise \(\epsilon\), so we can never observe the true relationship. Instead, the next best thing we can do is obtain a sample \(\Bbb{X}\), \(\Bbb{Y}\) of \(n\) observed relationships, \((x, Y)\) and use it to train a model and obtain an estimate of \(\hat{\theta}\) \[\hat{Y}(x) = f_{\hat{\theta}}(x) = \hat{\theta_0} + \sum_{j=1}^p \hat{\theta_j} x_j\]

    +
    +
    +
    + +
    +
    +Which Expressions are random? +
    +
    +
    +

    In our estimating equation above, our sample \(\Bbb{X}\), \(\Bbb{Y}\) are random. Hence, the estimates we calculate from our samples \(\hat{\theta}\) are also random, so our predictor \(\hat{Y}(x)\) is also random.

    +
    +
    +

    Now taking a look at our original equations, we can see that they both have differing sources of randomness. For our observed relationship, \(Y = g(x) + \epsilon\), \(\epsilon\) represents measurement errors and reflects randomness from the future. For the estimation model, the data we have is a random sample collected from the population, so the randomness from the past.

    +
    +
    +
    +
    +

    18.4 Bootstrap Resampling (Review)

    +

    To determine properties of the sampling distribution of an estimator like variance, we’d need to have access to the population so that we can consider all possible samples and compute an estimate for each sample.

    +

    +y_hat +

    +

    However, we don’t have access to the population; we only have one random sample from the population. How can we consider all possible samples if we only have one?

    +The idea of bootstrapping is to treat our random sample as a “population” and resample from it with replacement. Intuitively, a random sample resembles the population, so a random resample also resamples a random sample. +

    +y_hat +

    +

    Bootstrap resampling is a technique for estimating the sampling distribution of an estimator. To execute it, we can follow the pseudocode below:

    +
    collect a random sample of size n (called the bootstrap population)
    +
    +initiate list of estimates
    +
    +repeat 10,000 times:
    +    resample with replacement n times from bootstrap population
    +
    +apply estimator f to resample
    +
    +store in list
    +
    +list of estimates is the bootstrapped sampling distribution of f
    +
    + +
    +
    +

    Given an original sample of size \(n\), we want a resample that has the same size \(n\) as the original. Sampling without replacement will give us the original sample with shuffled rows. Hence, when we calculate summary statistics like the average, our sample without replacement will always have the same average as the original sample, defeating the purpose of a bootstrap.

    +
    +
    +
    +

    How well does bootstrapping actually represent our population? The bootstrapped sampling distribution of an estimator does not exactly match the sampling distribution of that estimator, but it is often close. Similarly, the variance of the bootstrapped distribution is often close to the true variance of the estimator. The example below displays the results from different bootstraps from a known population using a sample size of \(n=50\).

    +

    +y_hat +

    +

    In the real world, we don’t know the population distribution. The center of the boostrapped distribution is the estimator applied to our original sample, so we have no way of recovering the estimator’s true expected value. The quality of our bootstrapped distribution depends on the quality of our original sample; if our original sample was not representative of the population, bootstrap is next to useless.

    +

    One thing to note is that the bootstrap often does not work well for some statistics, like the median or other quantile-based statistics, that depend heavily on a small number of observations out of a larger sample. Bootstrapping does not overcome the weakness of small samples as a basis for inference. Indeed, for the very smallest samples, it may be better to make additional assumptions such as a parametric family.

    + + + + +
    + +
    + + +
    + + + + \ No newline at end of file diff --git a/docs/regex/regex.html b/docs/regex/regex.html index 4a384825..12a61ba4 100644 --- a/docs/regex/regex.html +++ b/docs/regex/regex.html @@ -251,6 +251,12 @@ 17  Random Variables + + @@ -670,11 +676,11 @@

    county_and_state['clean_county_pandas'] = canonicalize_county_series(county_and_state['County']) display(county_and_pop), display(county_and_state);
    -
    /var/folders/sy/b85yc0p951zdr__z5hvdmbjm0000gn/T/ipykernel_19948/2523629438.py:7: FutureWarning:
    +
    /var/folders/sy/b85yc0p951zdr__z5hvdmbjm0000gn/T/ipykernel_60573/2523629438.py:7: FutureWarning:
     
     The default value of regex will change from True to False in a future version. In addition, single character regular expressions will *not* be treated as literal strings when regex=True.
     
    -/var/folders/sy/b85yc0p951zdr__z5hvdmbjm0000gn/T/ipykernel_19948/2523629438.py:7: FutureWarning:
    +/var/folders/sy/b85yc0p951zdr__z5hvdmbjm0000gn/T/ipykernel_60573/2523629438.py:7: FutureWarning:
     
     The default value of regex will change from True to False in a future version. In addition, single character regular expressions will *not* be treated as literal strings when regex=True.
     
    diff --git a/docs/sampling/sampling.html b/docs/sampling/sampling.html index aa3c5bb2..933c4b56 100644 --- a/docs/sampling/sampling.html +++ b/docs/sampling/sampling.html @@ -251,6 +251,12 @@ 17  Random Variables
    + + @@ -680,7 +686,7 @@

    random_sample = movie.sample(n, replace = False) ## By default, replace = False np.mean(random_sample["barbie"])
    -
    0.5302674487195183
    +
    0.5308934990929226

    This is very close to the actual vote of 0.5302792307692308!

    @@ -698,7 +704,7 @@

    Markdown(f"**Actual** = {actual_barbie:.4f}, **Sample** = {sample_barbie:.4f}, " f"**Err** = {100*err:.2f}%.")
    -

    Actual = 0.5303, Sample = 0.5787, Err = 9.14%.

    +

    Actual = 0.5303, Sample = 0.5413, Err = 2.07%.

    We’ll learn how to choose this number when we (re)learn the Central Limit Theorem later in the semester.

    @@ -723,7 +729,7 @@

    ax.axvline(actual_barbie, color="orange", lw=4);
    -

    +

    What fraction of these simulated samples would have predicted Barbie?

    @@ -731,7 +737,7 @@

    poll_result = pd.Series(poll_result)
     np.sum(poll_result > 0.5)/1000
    -
    0.939
    +
    0.949

    You can see the curve looks roughly Gaussian/normal. Using KDE:

    @@ -741,7 +747,7 @@

    sns.histplot(poll_result, stat='density', kde=True);
    -

    +

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Calling @@ -45939,7 +45939,7 @@ \section{Parting Note}\label{parting-note-1}} \hypertarget{pandas-iii}{% \chapter{Pandas III}\label{pandas-iii}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -61137,7 +61137,7 @@ \chapter{Data Cleaning and EDA}\label{data-cleaning-and-eda}} \end{Highlighting} \end{Shaded} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -63674,7 +63674,7 @@ \section{EDA and Data Wrangling}\label{eda-and-data-wrangling}} \hypertarget{regular-expressions}{% \chapter{Regular Expressions}\label{regular-expressions}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -64014,11 +64014,11 @@ \subsubsection{Canonicalization with Pandas Series \end{Shaded} \begin{verbatim} -/var/folders/sy/b85yc0p951zdr__z5hvdmbjm0000gn/T/ipykernel_19979/2523629438.py:7: FutureWarning: +/var/folders/sy/b85yc0p951zdr__z5hvdmbjm0000gn/T/ipykernel_60601/2523629438.py:7: FutureWarning: The default value of regex will change from True to False in a future version. In addition, single character regular expressions will *not* be treated as literal strings when regex=True. -/var/folders/sy/b85yc0p951zdr__z5hvdmbjm0000gn/T/ipykernel_19979/2523629438.py:7: FutureWarning: +/var/folders/sy/b85yc0p951zdr__z5hvdmbjm0000gn/T/ipykernel_60601/2523629438.py:7: FutureWarning: The default value of regex will change from True to False in a future version. In addition, single character regular expressions will *not* be treated as literal strings when regex=True. @@ -64965,7 +64965,7 @@ \section{Limitations of Regular \hypertarget{visualization-i}{% \chapter{Visualization I}\label{visualization-i}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -65784,7 +65784,7 @@ \subsection{Modes}\label{modes}} \hypertarget{visualization-ii}{% \chapter{Visualization II}\label{visualization-ii}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -67242,7 +67242,7 @@ \subsection{Harnessing Context}\label{harnessing-context}} \hypertarget{sampling}{% \chapter{Sampling}\label{sampling}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -67823,7 +67823,7 @@ \subsubsection{Simple Random Sample}\label{simple-random-sample}} \end{Shaded} \begin{verbatim} -0.5298333871272913 +0.5285117252278824 \end{verbatim} This is very close to the actual vote of 0.5302792307692308! @@ -67847,8 +67847,8 @@ \subsubsection{Simple Random Sample}\label{simple-random-sample}} \end{Highlighting} \end{Shaded} -\textbf{Actual} = 0.5303, \textbf{Sample} = 0.5350, \textbf{Err} = -0.89\%. +\textbf{Actual} = 0.5303, \textbf{Sample} = 0.5300, \textbf{Err} = +0.05\%. We'll learn how to choose this number when we (re)learn the Central Limit Theorem later in the semester. @@ -67898,7 +67898,7 @@ \subsubsection{Quantifying Chance \end{Shaded} \begin{verbatim} -0.948 +0.963 \end{verbatim} You can see the curve looks roughly Gaussian/normal. Using KDE: @@ -67931,7 +67931,7 @@ \section{Summary}\label{summary-1}} \hypertarget{introduction-to-modeling}{% \chapter{Introduction to Modeling}\label{introduction-to-modeling}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -68294,7 +68294,7 @@ \subsection{Derivation}\label{derivation}} \(\hat{a} = \text{average of }y - \text{slope}\cdot\text{average of }x\) \end{itemize} -\begin{tcolorbox}[enhanced jigsaw, opacityback=0, left=2mm, rightrule=.15mm, arc=.35mm, breakable, bottomrule=.15mm, colback=white, leftrule=.75mm, toprule=.15mm] +\begin{tcolorbox}[enhanced jigsaw, colback=white, arc=.35mm, breakable, left=2mm, opacityback=0, bottomrule=.15mm, rightrule=.15mm, toprule=.15mm, leftrule=.75mm] Proof: @@ -68646,7 +68646,7 @@ \section{Fitting the Model}\label{fitting-the-model}} \chapter{Constant Model, Loss, and Transformations}\label{constant-model-loss-and-transformations}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -69840,7 +69840,7 @@ \section{Transformations to fit Linear \hypertarget{ordinary-least-squares}{% \chapter{Ordinary Least Squares}\label{ordinary-least-squares}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -70470,7 +70470,7 @@ \section{OLS Properties}\label{ols-properties}} \[\mathbb{X}^Te = 0 \] -\begin{tcolorbox}[enhanced jigsaw, opacityback=0, left=2mm, rightrule=.15mm, arc=.35mm, breakable, bottomrule=.15mm, colback=white, leftrule=.75mm, toprule=.15mm] +\begin{tcolorbox}[enhanced jigsaw, colback=white, arc=.35mm, breakable, left=2mm, opacityback=0, bottomrule=.15mm, rightrule=.15mm, toprule=.15mm, leftrule=.75mm] Proof: @@ -70508,7 +70508,7 @@ \section{OLS Properties}\label{ols-properties}} \[\sum_i^n e_i = 0\] -\begin{tcolorbox}[enhanced jigsaw, opacityback=0, left=2mm, rightrule=.15mm, arc=.35mm, breakable, bottomrule=.15mm, colback=white, leftrule=.75mm, toprule=.15mm] +\begin{tcolorbox}[enhanced jigsaw, colback=white, arc=.35mm, breakable, left=2mm, opacityback=0, bottomrule=.15mm, rightrule=.15mm, toprule=.15mm, leftrule=.75mm] Proof: @@ -70537,7 +70537,7 @@ \section{OLS Properties}\label{ols-properties}} only if \(\mathbb{X}\) is \textbf{full column rank}. \end{enumerate} -\begin{tcolorbox}[enhanced jigsaw, opacityback=0, left=2mm, rightrule=.15mm, arc=.35mm, breakable, bottomrule=.15mm, colback=white, leftrule=.75mm, toprule=.15mm] +\begin{tcolorbox}[enhanced jigsaw, colback=white, arc=.35mm, breakable, left=2mm, opacityback=0, bottomrule=.15mm, rightrule=.15mm, toprule=.15mm, leftrule=.75mm] Proof: @@ -70619,7 +70619,7 @@ \section{OLS Properties}\label{ols-properties}} \hypertarget{gradient-descent}{% \chapter{Gradient Descent}\label{gradient-descent}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -71351,7 +71351,7 @@ \section{Batch, Mini-Batch Gradient Descent and Stochastic Gradient \chapter{Sklearn and Feature Engineering}\label{sklearn-and-feature-engineering}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -72163,7 +72163,7 @@ \section{Complexity and Overfitting}\label{complexity-and-overfitting}} \chapter{Case Study in Human Contexts and Ethics}\label{case-study-in-human-contexts-and-ethics}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -72354,7 +72354,7 @@ \section{The Response: Cook County Open Data \subsection{Question/Problem Formulation}\label{questionproblem-formulation}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Driving Questions}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Driving Questions}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -72400,7 +72400,7 @@ \subsection{Question/Problem \end{itemize} \end{enumerate} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Definitions: Fairness and Transparency}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-tip-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-tip-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-tip-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Definitions: Fairness and Transparency}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-tip-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] The definitions, as given by the Cook County Assessor's Office, are given below: @@ -72451,7 +72451,7 @@ \subsection{Question/Problem \subsection{Data Acquisition and Cleaning}\label{data-acquisition-and-cleaning}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Driving Questions}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Driving Questions}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -72498,7 +72498,7 @@ \subsection{Data Acquisition and \hypertarget{exploratory-data-analysis}{% \subsection{Exploratory Data Analysis}\label{exploratory-data-analysis}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Driving Questions}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Driving Questions}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -72543,7 +72543,7 @@ \subsection{Exploratory Data Analysis}\label{exploratory-data-analysis}} \hypertarget{prediction-and-inference}{% \subsection{Prediction and Inference}\label{prediction-and-inference}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Driving Questions}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Driving Questions}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -72584,7 +72584,7 @@ \subsection{Prediction and Inference}\label{prediction-and-inference}} \subsection{Reports Decisions, and Conclusions}\label{reports-decisions-and-conclusions}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Driving Questions}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Driving Questions}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -72724,7 +72724,7 @@ \section{Lessons for Data Science \chapter{Cross Validation and Regularization}\label{cross-validation-and-regularization}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -73478,7 +73478,7 @@ \section{Regression Summary}\label{regression-summary}} \hypertarget{random-variables}{% \chapter{Random Variables}\label{random-variables}} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \begin{itemize} \tightlist @@ -73521,7 +73521,7 @@ \chapter{Random Variables}\label{random-variables}} perspective to investigate our choice of model complexity \end{enumerate} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Data 8 Recap}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-tip-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-tip-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-tip-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Data 8 Recap}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-tip-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] Recall the following concepts from Data 8: @@ -73763,7 +73763,7 @@ \subsection{Variance}\label{variance}} \[\text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2\] -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Proof}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-tip-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-tip-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-tip-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Proof}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-tip-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \[\begin{align} \text{Var}(X) &= \mathbb{E}[(X-\mathbb{E}[X])^2] \\ @@ -73791,7 +73791,7 @@ \subsection{Example: Dice}\label{example-dice}} 0, \text{otherwise} \end{cases}\] -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-caution-color}{\faFire}\hspace{0.5em}{What's the expectation \(\mathbb{E}[X]?\)}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-caution-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-caution-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-caution-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-caution-color}{\faFire}\hspace{0.5em}{What's the expectation \(\mathbb{E}[X]?\)}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-caution-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \[ \begin{align} \mathbb{E}[X] &= 1(\frac{1}{6}) + 2(\frac{1}{6}) + 3(\frac{1}{6}) + 4(\frac{1}{6}) + 5(\frac{1}{6}) + 6(\frac{1}{6}) \\ @@ -73801,7 +73801,7 @@ \subsection{Example: Dice}\label{example-dice}} \end{tcolorbox} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-caution-color}{\faFire}\hspace{0.5em}{What's the variance \(\text{Var}(X)?\)}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-caution-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-caution-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-caution-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-caution-color}{\faFire}\hspace{0.5em}{What's the variance \(\text{Var}(X)?\)}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-caution-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] Using approach 1: \[\begin{align} \text{Var}(X) &= (\frac{1}{6})((1 - \frac{7}{2})^2 + (2 - \frac{7}{2})^2 + (3 - \frac{7}{2})^2 + (4 - \frac{7}{2})^2 + (5 - \frac{7}{2})^2 + (6 - \frac{7}{2})^2) \\ @@ -73900,7 +73900,7 @@ \subsection{Properties of Expectation}\label{properties-of-expectation}} \[\mathbb{E}[aX+b] = aE[\mathbb{X}] + b\] -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Proof}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-tip-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-tip-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-tip-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Proof}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-tip-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \[\begin{align} \mathbb{E}[aX+b] &= \sum_{x} (ax + b) P(X=x) \\ @@ -73921,7 +73921,7 @@ \subsection{Properties of Expectation}\label{properties-of-expectation}} \[\mathbb{E}[X+Y] = \mathbb{E}[X] + \mathbb{E}[Y]\] -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Proof}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-tip-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-tip-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-tip-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Proof}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-tip-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] \[\begin{align} \mathbb{E}[X+Y] &= \sum_{s} (X+Y)(s) P(s) \\ @@ -73971,7 +73971,7 @@ \subsection{Properties of Variance}\label{properties-of-variance}} \(X\) by \(b\) units. \end{itemize} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Proof}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-tip-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-tip-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-tip-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Proof}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-tip-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] We know that \[\mathbb{E}[aX+b] = aE[\mathbb{X}] + b\] @@ -74009,7 +74009,7 @@ \subsection{Properties of Variance}\label{properties-of-variance}} \[\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) \qquad \text{if } X, Y \text{ independent}\] \end{enumerate} -\begin{tcolorbox}[enhanced jigsaw, left=2mm, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Proof}, opacitybacktitle=0.6, toptitle=1mm, toprule=.15mm, opacityback=0, bottomtitle=1mm, titlerule=0mm, coltitle=black, arc=.35mm, rightrule=.15mm, colbacktitle=quarto-callout-tip-color!10!white, breakable, bottomrule=.15mm, leftrule=.75mm, colback=white, colframe=quarto-callout-tip-color-frame] +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-tip-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Proof}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-tip-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] The variance of a sum is affected by the dependence between the two random variables that are being added. Let's expand out the definition @@ -74088,6 +74088,44 @@ \subsection{Summary}\label{summary-2}} \end{itemize} \end{itemize} +\bookmarksetup{startatroot} + +\hypertarget{estimators-bias-and-variance}{% +\chapter{Estimators, Bias, and +Variance}\label{estimators-bias-and-variance}} + +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-note-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Learning Outcomes}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] + +\begin{itemize} +\tightlist +\item + Explore commonly seen random variables like Bernoulli and Binomial + distributions +\item + Apply the Central Limit Theorem to approximate parameters of a + population +\item + Use sampled data to model an estimation of and infer the true + underlying distribution +\item + Estimate the true population distribution from a sample using the + bootstrapping technique +\end{itemize} + +\end{tcolorbox} + +Last time, we introduced the idea of random variables: numerical +functions of a sample. Most of our work in the last lecture was done to +build a background in probability and statistics. Now that we've +established some key ideas, we're in a good place to apply what we've +learned to our original goal -- understanding how the randomness of a +sample impacts the model design process. + +In this lecture, we will delve more deeply into the idea of fitting a +model to a sample. We'll explore how to re-express our modeling process +in terms of random variables and use this new understanding to steer +model complexity. + \hypertarget{common-random-variables}{% \section{Common Random Variables}\label{common-random-variables}} @@ -74095,12 +74133,15 @@ \section{Common Random Variables}\label{common-random-variables}} useful properties. Below are the ones we will explore further in this course. The numbers in parentheses are the parameters of a random variable, which are constants. Parameters define a random variable's -shape (i.e., distribution) and its values. +shape (i.e., distribution) and its values. For this lecture, we'll focus +more heavily on the bolded random variables and their special +properties, but you should familiarize yourself with all the ones listed +below: \begin{itemize} \tightlist \item - Bernoulli(p) + \textbf{Bernoulli(p)} \begin{itemize} \tightlist @@ -74126,7 +74167,7 @@ \section{Common Random Variables}\label{common-random-variables}} \end{itemize} \end{itemize} \item - Binomial(n, p) + \textbf{Binomial(n, p)} \begin{itemize} \tightlist @@ -74190,100 +74231,387 @@ \section{Common Random Variables}\label{common-random-variables}} \end{itemize} \end{itemize} -\hypertarget{populations-and-samples}{% -\section{Populations and Samples}\label{populations-and-samples}} +\hypertarget{example}{% +\subsection{Example}\label{example}} + +Suppose you win cash based on the number of heads you get in a series of +20 coin flips. Let \(X_i = 1\) if the \(i\)-th coin is heads, 0 +otherwise. Which payout strategy would you choose? + +A. \(Y_A = 10 * X_1 + 10 * X_2\) + +B. \(Y_B = \sum_{i=1}^{20} X_i\) + +C. \(Y_C = 20 * X_1\) + +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-caution-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-caution-color}{\faFire}\hspace{0.5em}{Solution}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-caution-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] + +Let \(X_1, X_2, ... X_{20}\) be 20 i.i.d Bernoulli(0.5) random +variables. Since the \(X_i\)'s are independent, +\(\text{Cov}(X_i, X_j) = 0\) for all pairs \(i, j\). Additionally, Since +\(X_i\) is Bernoulli(0.5), we know that \(\mathbb{E}[X] = p = 0.5\) and +\(\text{Var}(X) = p(1-p) = 0.25\). We can calculate the following for +each scenario: + +\begin{longtable}[]{@{} + >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}} + >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}} + >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}} + >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}}@{}} +\toprule\noalign{} +\begin{minipage}[b]{\linewidth}\raggedright +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +A. \(Y_A = 10 * X_1 + 10 * X_2\) +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +B. \(Y_B = \sum_{i=1}^{20} X_i\) +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +C. \(Y_C = 20 * X_1\) +\end{minipage} \\ +\midrule\noalign{} +\endhead +\bottomrule\noalign{} +\endlastfoot +Expectation & \(\mathbb{E}[Y_A] = 10 (0.5) + 10(0.5) = 10\) & +\(\mathbb{E}[Y_B] = 0.5 + ... + 0.5 = 10\) & +\(\mathbb{E}[Y_C] = 20(0.5) = 10\) \\ +Variance & \(\text{Var}(Y_A) = 10^2 (0.25) + 10^2 (0.25) = 50\) & +\(\text{Var}(Y_B) = 0.25 + ... + 0.25 = 5\) & +\(\text{Var}(Y_C) = 20^2 (0.25) = 100\) \\ +Standard Deviation & \(\text{SD}(Y_A) \approx 7.07\) & +\(\text{SD}(Y_B) \approx 2.24\) & \(\text{SD}(Y_C) = 10\) \\ +\end{longtable} + +As we can see, all the scenarios have the same expected value but +different variances. The higher the variance, the greater the risk and +uncertainty, so the ``right'' strategy depends on your personal +preference. Would you choose the ``safest'' option B, the most ``risky'' +option C, or somewhere in the middle (option A)? + +\end{tcolorbox} + +\hypertarget{sample-statistics}{% +\section{Sample Statistics}\label{sample-statistics}} Today, we've talked extensively about populations; if we know the distribution of a random variable, we can reliably compute expectation, -variance, functions of the random variable, etc. +variance, functions of the random variable, etc. Note that: + +\begin{itemize} +\tightlist +\item + The distribution of a \emph{population} describes how a random + variable behaves across \emph{all} individuals of interest. +\item + The distribution of a \emph{sample} describes how a random variable + behaves in a \emph{specific sample} from the population. +\end{itemize} In Data Science, however, we often do not have access to the whole population, so we don't know its distribution. As such, we need to collect a sample and use its distribution to estimate or infer -properties of the population. - -When sampling, we make the (big) assumption that we sample uniformly at -random with replacement from the population; each observation in our -sample is a random variable drawn i.i.d from our population -distribution. +properties of the population. In cases like these, we can take several +samples of size \(n\) from the population (an easy way to do this is +using \texttt{df.sample(n,\ replace=True)}), and compute the mean of +each \emph{sample}. When sampling, we make the (big) assumption that we +sample uniformly at random with replacement from the population; each +observation in our sample is a random variable drawn i.i.d from our +population distribution. Remember that our sample mean is a random +variable since it depends on our randomly drawn sample! On the other +hand, our population mean is simply a number (a fixed value). \hypertarget{sample-mean}{% \subsection{Sample Mean}\label{sample-mean}} Consider an i.i.d. sample \(X_1, X_2, ..., X_n\) drawn from a population with mean 𝜇 and SD 𝜎. We define the sample mean as -\[\bar{X_n} = \frac{1}{n} \sum_{i=1}^n X_i\] +\[\bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i\] The expectation of the sample mean is given by: \[\begin{align} - \mathbb{E}[\bar{X_n}] &= \frac{1}{n} \sum_{i=1}^n \mathbb{E}[X_i] \\ + \mathbb{E}[\bar{X}_n] &= \frac{1}{n} \sum_{i=1}^n \mathbb{E}[X_i] \\ &= \frac{1}{n} (n \mu) \\ &= \mu \end{align}\] The variance is given by: \[\begin{align} - \text{Var}(\bar{X_n}) &= \frac{1}{n^2} \text{Var}( \sum_{i=1}^n X_i) \\ + \text{Var}(\bar{X}_n) &= \frac{1}{n^2} \text{Var}( \sum_{i=1}^n X_i) \\ &= \frac{1}{n^2} \left( \sum_{i=1}^n \text{Var}(X_i) \right) \\ &= \frac{1}{n^2} (n \sigma^2) = \frac{\sigma^2}{n} \end{align}\] -\(\bar{X_n}\) is normally distributed by the Central Limit Theorem +\(\bar{X}_n\) is normally distributed by the Central Limit Theorem (CLT). \hypertarget{central-limit-theorem}{% \subsection{Central Limit Theorem}\label{central-limit-theorem}} -The CLT states that no matter what population you are drawing from, if -an i.i.d. sample of size \(n\) is large, the probability distribution of -the sample mean is roughly normal with mean 𝜇 and SD -\(\sigma/\sqrt{n}\). - -Any theorem that provides the rough distribution of a statistic and -doesn't need the distribution of the population is valuable to data -scientists because we rarely know a lot about the population! +In +\href{https://inferentialthinking.com/chapters/14/4/Central_Limit_Theorem.html?}{Data +8} and in the previous lecture, you encountered the \textbf{Central +Limit Theorem (CLT)}. This is a powerful theorem for estimating the +distribution of a population with mean \(\mu\) and standard deviation +\(\sigma\) from a collection of smaller samples. The CLT tells us that +if an i.i.d sample of size \(n\) is large, then the probability +distribution of the \textbf{sample mean} is \textbf{roughly normal} with +mean \(\mu\) and SD of \(\frac{\sigma}{\sqrt{n}}\). More generally, any +theorem that provides the rough distribution of a statistic and +\textbf{doesn't need the distribution of the population} is valuable to +data scientists! This is because we rarely know a lot about the +population. + +Importantly, the CLT assumes that each observation in our samples is +drawn i.i.d from the distribution of the population. In addition, the +CLT is accurate only when \(n\) is ``large'', but what counts as a +``large'' sample size depends on the specific distribution. If a +population is highly symmetric and unimodal, we could need as few as +\(n=20\); if a population is very skewed, we need a larger \(n\). If in +doubt, you can bootstrap the sample mean and see if the bootstrapped +distribution is bell-shaped. Classes like Data 140 investigate this idea +in great detail. For a more in-depth demo, check out \href{https://onlinestatbook.com/stat_sim/sampling_dist/}{onlinestatbook}. -The CLT applies if the sample size \(n\) is large, but how large does n -have to be for the normal approximation to be good? It depends on the -shape of the distribution of the population. - -\begin{itemize} -\tightlist -\item - If the population is roughly symmetric and unimodal/uniform, could - need as few as \(n = 20\). -\item - If the population is very skewed, you will need a bigger n. -\item - If in doubt, you can bootstrap the sample mean and see if the - bootstrapped distribution is bell-shaped. -\end{itemize} - \hypertarget{using-the-sample-mean-to-estimate-the-population-mean}{% \subsection{Using the Sample Mean to Estimate the Population Mean}\label{using-the-sample-mean-to-estimate-the-population-mean}} -Our goal with sampling is often to estimate some characteristic of a -population. When we collect a single sample, it has just one average. -Since our sample was random, it \emph{could} have come out differently. -The CLT helps us understand this difference. We should consider the -average value and spread of all possible sample means, and what this -means for how big \(n\) should be. - -For every sample size, the expected value of the sample mean is the -population mean. \(\mathbb{E}[\bar{X_n}] = \mu\). We call the sample -mean an unbiased estimator of the population mean, and we'll cover this -more in the next lecture. +Now let's say we want to use the sample mean to \textbf{estimate} the +population mean, for example, the average height of Cal undergraduates. +We can typically collect a \textbf{single sample}, which has just one +average. However, what if we happened, by random chance, to draw a +sample with a different mean or spread than that of the population? We +might get a skewed view of how the population behaves (consider the +extreme case where we happen to sample the exact same value \(n\) +times!). + +For example, notice the difference in variation between these two +distributions that are different in sample size. The distribution with a +bigger sample size (\(n=800\)) is tighter around the mean than the +distribution with a smaller sample size (\(n=200\)). Try plugging in +these values into the standard deviation equation for the normal +distribution to make sense of this! + +Applying the CLT allows us to make sense of all of this and resolve this +issue. By drawing many samples, we can consider how the sample +distribution varies across multiple subsets of the data. This allows us +to approximate the properties of the population without the need to +survey every single member. + +Given this potential variance, it is also important that we consider the +\textbf{average value and spread} of all possible sample means, and what +this means for how big \(n\) should be. For every sample size, the +expected value of the sample mean is the population mean: +\[\mathbb{E}[\bar{X}_n] = \mu\]. We call the sample mean an +\textbf{unbiased estimator} of the population mean and will explore this +idea more in the next lecture. + +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-tip-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-tip-color}{\faLightbulb}\hspace{0.5em}{Data 8 Recap: Square Root Law}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-tip-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] The square root law (\href{https://inferentialthinking.com/chapters/14/5/Variability_of_the_Sample_Mean.html\#the-square-root-law}{Data 8}) states that if you increase the sample size by a factor, the SD decreases by the square root of the factor. -\(\text{SD}(\bar{X_n}) = \frac{\sigma}{\sqrt{n}}\). The sample mean is +\[\text{SD}(\bar{X_n}) = \frac{\sigma}{\sqrt{n}}\] The sample mean is more likely to be close to the population mean if we have a larger sample size. +\end{tcolorbox} + +\hypertarget{prediction-vs.-inference}{% +\section{Prediction vs.~Inference}\label{prediction-vs.-inference}} + +At this point in the course, we've spent a great deal of time working +with models. When we first introduced the idea of modeling a few weeks +ago, we did so in the context of \textbf{prediction}: using models to +make \emph{accurate predictions} about unseen data. Another reason we +might build models is to better understand complex phenomena in the +world around us. \textbf{Inference} is the task of using a model to +infer the true underlying relationships between the feature and response +variables. For example, if we are working with a set of housing data, +\emph{prediction} might ask: given the attributes of a house, how much +is it worth? \emph{Inference} might ask: how much does having a local +park impact the value of a house? + +A major goal of inference is to draw conclusions about the full +population of data given only a random sample. To do this, we aim to +estimate the value of a \textbf{parameter}, which is a numerical +function of the \emph{population} (for example, the population mean +\(\mu\)). We use a collected sample to construct a \textbf{statistic}, +which is a numerical function of the random \emph{sample} (for example, +the sample mean \(\bar{X}_n\)). It's helpful to think ``p'' for +``parameter'' and ``population,'' and ``s'' for ``sample'' and +``statistic.'' + +Since the sample represents a \emph{random} subset of the population, +any statistic we generate will likely deviate from the true population +parameter, and it \emph{could have been different}. We say that the +sample statistic is an \textbf{estimator} of the true population +parameter. Notationally, the population parameter is typically called +\(\theta\), while its estimator is denoted by \(\hat{\theta}\). + +To address our inference question, we aim to construct estimators that +closely estimate the value of the population parameter. We evaluate how +``good'' an estimator is by answering three questions: + +\begin{itemize} +\tightlist +\item + Do we get the right answer for the parameter, on average? +\item + How variable is the answer? +\item + How close is our answer to the parameter? +\end{itemize} + +\hypertarget{modeling-as-estimation}{% +\subsection{Modeling as Estimation}\label{modeling-as-estimation}} + +Now that we've established the idea of an estimator, let's see how we +can apply this learning to the modeling process. To do so, we'll take a +moment to formalize our data collection and models in the language of +random variables. + +Say we are working with an input variable, \(x\), and a response +variable, \(Y\). We assume that \(Y\) and \(x\) are linked by some +relationship \(g\); in other words, \(Y = g(x)\). \(g\) represents some +``universal truth'' or ``law of nature'' that defines the underlying +relationship between \(x\) and \(Y\). In the image below, \(g\) is +denoted by the red line. + +As data scientists, however, we have no way of directly ``seeing'' the +underlying relationship \(g\). The best we can do is collect observed +data out in the real world to try to understand this relationship. +Unfortunately, the data collection process will always have some +inherent error (think of the randomness you might encounter when taking +measurements in a scientific experiment). We say that each observation +comes with some random error or \textbf{noise} term, \(\epsilon\). This +error is assumed to be a random variable with expectation +\(\mathbb{E}(\epsilon)=0\), variance +\(\text{Var}(\epsilon) = \sigma^2\), and be i.i.d. across each +observation. The existence of this random noise means that our +observations, \(Y(x)\), are \emph{random variables}. + +We can only observe our random sample of data, represented by the blue +points above. From this sample, we want to estimate the true +relationship \(g\). We do this by constructing the model \(\hat{Y}(x)\) +to estimate \(g\). + +\[\text{True relationship: }Y = g(x)\] + +\[\text{Observed relationship: }Y = g(x) + \epsilon\] + +\[\text{Prediction: }\hat{Y}(x)\] + +\hypertarget{estimating-a-linear-relationship}{% +\subsubsection{Estimating a Linear +Relationship}\label{estimating-a-linear-relationship}} + +If we assume that the true relationship \(g\) is linear, we can express +the response as \(Y = f_{\theta}(x)\), where our true relationship is +modeled by \[Y = g(x) + \epsilon\] +\[ f_{\theta}(x) = Y = \theta_0 + \sum_{j=1}^p \theta_j x_j + \epsilon\] + +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-warning-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-warning-color}{\faExclamationTriangle}\hspace{0.5em}{Which Expressions are random?}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-warning-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] + +In our two equations above, the true relationship +\(g(x) = \theta_0 + \sum_{j=1}^p \theta_j x_j\) is not random, but +\(\epsilon\) is random. Hence, \(Y = f_{\theta}(x)\) is also random. + +\end{tcolorbox} + +This true relationship has true, unobservable parameters \(\theta\), and +it has random noise \(\epsilon\), so we can never observe the true +relationship. Instead, the next best thing we can do is obtain a sample +\(\Bbb{X}\), \(\Bbb{Y}\) of \(n\) observed relationships, \((x, Y)\) and +use it to train a model and obtain an estimate of \(\hat{\theta}\) +\[\hat{Y}(x) = f_{\hat{\theta}}(x) = \hat{\theta_0} + \sum_{j=1}^p \hat{\theta_j} x_j\] + +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-warning-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-warning-color}{\faExclamationTriangle}\hspace{0.5em}{Which Expressions are random?}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-warning-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] + +In our estimating equation above, our sample \(\Bbb{X}\), \(\Bbb{Y}\) +are random. Hence, the estimates we calculate from our samples +\(\hat{\theta}\) are also random, so our predictor \(\hat{Y}(x)\) is +also random. + +\end{tcolorbox} + +Now taking a look at our original equations, we can see that they both +have differing sources of randomness. For our observed relationship, +\(Y = g(x) + \epsilon\), \(\epsilon\) represents measurement errors and +reflects randomness from the future. For the estimation model, the data +we have is a random sample collected from the population, so the +randomness from the past. + +\hypertarget{bootstrap-resampling-review}{% +\section{Bootstrap Resampling +(Review)}\label{bootstrap-resampling-review}} + +To determine properties of the sampling distribution of an estimator +like variance, we'd need to have access to the population so that we can +consider all possible samples and compute an estimate for each sample. + +However, we don't have access to the population; we only have \emph{one} +random sample from the population. How can we consider all possible +samples if we only have one? + +The idea of bootstrapping is to treat our random sample as a +``population'' and resample from it \emph{with replacement}. +Intuitively, a random sample resembles the population, so a random +\emph{resample} also resamples a random sample. + +Bootstrap resampling is a technique for estimating the sampling +distribution of an estimator. To execute it, we can follow the +pseudocode below: + +\begin{verbatim} +collect a random sample of size n (called the bootstrap population) + +initiate list of estimates + +repeat 10,000 times: + resample with replacement n times from bootstrap population + +apply estimator f to resample + +store in list + +list of estimates is the bootstrapped sampling distribution of f +\end{verbatim} + +\begin{tcolorbox}[enhanced jigsaw, toptitle=1mm, colbacktitle=quarto-callout-warning-color!10!white, opacityback=0, left=2mm, rightrule=.15mm, colback=white, coltitle=black, title=\textcolor{quarto-callout-warning-color}{\faExclamationTriangle}\hspace{0.5em}{Why must we resample \emph{with replacement}?}, bottomtitle=1mm, titlerule=0mm, breakable, arc=.35mm, colframe=quarto-callout-warning-color-frame, bottomrule=.15mm, opacitybacktitle=0.6, toprule=.15mm, leftrule=.75mm] + +Given an original sample of size \(n\), we want a resample that has the +same size \(n\) as the original. Sampling \emph{without} replacement +will give us the original sample with shuffled rows. Hence, when we +calculate summary statistics like the average, our sample \emph{without} +replacement will always have the same average as the original sample, +defeating the purpose of a bootstrap. + +\end{tcolorbox} + +How well does bootstrapping actually represent our population? The +bootstrapped sampling distribution of an estimator does not exactly +match the sampling distribution of that estimator, but it is often +close. Similarly, the variance of the bootstrapped distribution is often +close to the true variance of the estimator. The example below displays +the results from different bootstraps from a \emph{known} population +using a sample size of \(n=50\). + +In the real world, we don't know the population distribution. The center +of the boostrapped distribution is the estimator applied to our original +sample, so we have no way of recovering the estimator's true expected +value. The quality of our bootstrapped distribution depends on the +quality of our original sample; if our original sample was not +representative of the population, bootstrap is next to useless. + +One thing to note is that the bootstrap often does not work well for +some statistics, like the median or other quantile-based statistics, +that depend heavily on a small number of observations out of a larger +sample. \textbf{Bootstrapping does not overcome the weakness of small +samples as a basis for inference}. Indeed, for the very smallest +samples, it may be better to make additional assumptions such as a +parametric family. + \end{document} diff --git a/probability_1/probability_1.ipynb b/probability_1/probability_1.ipynb deleted file mode 100644 index 298d75e7..00000000 --- a/probability_1/probability_1.ipynb +++ /dev/null @@ -1,380 +0,0 @@ -{ - "cells": [ - { - "cell_type": "raw", - "metadata": {}, - "source": [ - "---\n", - "title: Random Variables\n", - "execute:\n", - " echo: true\n", - "format:\n", - " html:\n", - " code-fold: true\n", - " code-tools: true\n", - " toc: true\n", - " toc-title: Random Variables\n", - " page-layout: full\n", - " theme:\n", - " - cosmo\n", - " - cerulean\n", - " callout-icon: false\n", - "---" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "::: {.callout-note collapse=\"false\"}\n", - "## Learning Outcomes\n", - "* Define a random variable in terms of its distribution\n", - "* Compute the expectation and variance of a random variable\n", - "* Gain familiarity with the Bernoulli and binomial random variables\n", - ":::\n", - "\n", - "In the past few lectures, we've examined the role of complexity in influencing model performance. We've considered model complexity in the context of a tradeoff between two competing factors: model variance and training error. \n", - "\n", - "Thus far, our analysis has been mostly qualitative. We've acknowledged that our choice of model complexity needs to strike a balance between model variance and training error, but we haven't yet discussed *why* exactly this tradeoff exists.\n", - "\n", - "To better understand the origin of this tradeoff, we will need to introduce the language of **random variables**. The next two lectures on probability will be a brief digression from our work on modeling so we can build up the concepts needed to understand this so-called **bias-variance tradeoff**. Our roadmap for the next few lectures will be:\n", - "\n", - "1. Random Variables Estimators: introduce random variables, considering the concepts of expectation, variance, and covariance\n", - "2. Estimators, Bias, and Variance: re-express the ideas of model variance and training error in terms of random variables and use this new perspective to investigate our choice of model complexity\n", - "\n", - "::: {.callout-tip collapse=\"true\"}\n", - "## Data 8 Recap\n", - "Recall the following concepts from Data 8: \n", - "\n", - "1. Sample mean: the mean of your random sample\n", - "2. Central Limit Theorem: If you draw a large random sample with replacement, then, regardless of the population distribution, the probability distribution of the sample mean\n", - "\n", - " a. is roughly normal\n", - "\n", - " b. is centered at the population mean\n", - " \n", - " c. has an $SD = \\frac{\\text{population SD}}{\\sqrt{\\text{sample size}}}$\n", - "\n", - ":::\n", - "\n", - "## Random Variables and Distributions\n", - "\n", - "Suppose we generate a set of random data, like a random sample from some population. A **random variable** is a *numerical function* of the randomness in the data. It is *random* since our sample was drawn at random; it is *variable* because its exact value depends on how this random sample came out. As such, the domain or input of our random variable is all possible (random) outcomes in a *sample space*, and its range or output is the number line. We typically denote random variables with uppercase letters, such as $X$ or $Y$. \n", - "\n", - "### Distribution\n", - "For any random variable $X$, we need to be able to specify 2 things: \n", - "\n", - "1. Possible values: the set of values the random variable can take on.\n", - "2. Probabilities: the set of probabilities describing how the total probability of 100% is split over the possible values.\n", - "\n", - "If $X$ is discrete (has a finite number of possible values), the probability that a random variable $X$ takes on the value $x$ is given by $P(X=x)$, and probabilities must sum to 1: $\\sum_{\\text{all} x} P(X=x) = 1$,\n", - "\n", - "We can often display this using a **probability distribution table**, which you will see in the coin toss example below.\n", - "\n", - "The **distribution** of a random variable $X$ is a description of how the total probability of 100% is split over all the possible values of $X$, and it fully defines a random variable. The distribution of a discrete random variable can also be represented using a histogram. If a variable is **continuous** – it can take on infinitely many values – we can illustrate its distribution using a density curve. \n", - "\n", - "

    \n", - "discrete_continuous\n", - "

    \n", - "\n", - "Probabilities are areas. For discrete random variables, the *area of the red bars* represent the probability that a discrete random variable $X$ falls within those values. For continuous random variables, the *area under the curve* represents the probability that a discrete random variable $Y$ falls within those values.\n", - "\n", - "

    \n", - "discrete_continuous\n", - "

    \n", - "\n", - "If we sum up the total area of the bars/under the density curve, we should get 100%, or 1.\n", - "\n", - "\n", - "### Example: Tossing a Coin\n", - "To give a concrete example, let's formally define a fair coin toss. A fair coin can land on heads ($H$) or tails ($T$), each with a probability of 0.5. With these possible outcomes, we can define a random variable $X$ as: \n", - "$$X = \\begin{cases} \n", - " 1, \\text{if the coin lands heads} \\\\\n", - " 0, \\text{if the coin lands tails} \n", - " \\end{cases}$$\n", - "\n", - "$X$ is a function with a domain, or input, of $\\{H, T\\}$ and a range, or output, of $\\{1, 0\\}$. We can write this in function notation as \n", - "$$\\begin{cases} X(H) = 1 \\\\ X(T) = 0 \\end{cases}$$\n", - "The probability distribution table of $X$ is given by.\n", - "\n", - "| $x$ | $P(X=x)$ | \n", - "| --- | -------- |\n", - "| 0 | $\\frac{1}{2}$ | \n", - "| 1 | $\\frac{1}{2}$ |\n", - "\n", - "Suppose we draw a random sample $s$ of size 3 from all students enrolled in Data 100. \n", - "We can define $Y$ as the number of data science students in our sample. Its domain is all possible samples of size 3, and its range is $\\{0, 1, 2, 3\\}$.\n", - "\n", - "

    \n", - " rv\n", - "

    \n", - "\n", - "We can show the distribution of $Y$ in the following tables. The table on the left lists all possible samples of $s$ and the number of times they can appear ($Y(s)$). We can use this to calculate the values for the table on the right, a **probability distribution table**. \n", - "\n", - "

    \n", - "distribution\n", - "

    \n", - "\n", - "### Simulation\n", - "Given a random variable $X$’s distribution, how could we **generate/simulate** a population? To do so, we can randomly pick values of $X$ according to its distribution using `np.random.choice` or `df.sample`. \n", - "\n", - "## Expectation and Variance\n", - "There are several ways to describe a random variable. The methods shown above -- a table of all samples $s, X(s)$, distribution table $P(X=x)$, and histograms -- are all definitions that *fully describe* a random variable. Often, it is easier to describe a random variable using some *numerical summary* rather than fully defining its distribution. These numerical summaries are numbers that characterize some properties of the random variable. Because they give a \"summary\" of how the variable tends to behave, they are *not* random – think of them as a static number that describes a certain property of the random variable. In Data 100, we will focus our attention on the expectation and variance of a random variable.\n", - "\n", - "### Expectation\n", - "The **expectation** of a random variable $X$ is the weighted average of the values of $X$, where the weights are the probabilities of each value occurring. There are two equivalent ways to compute the expectation: \n", - "\n", - "1. Apply the weights one *sample* at a time: $$\\mathbb{E}[X] = \\sum_{\\text{all possible } s} X(s) P(s)$$.\n", - "2. Apply the weights one possible *value* at a time: $$\\mathbb{E}[X] = \\sum_{\\text{all possible } x} x P(X=x)$$\n", - "\n", - "We want to emphasize that the expectation is a *number*, not a random variable. Expectation is a generalization of the average, and it has the same units as the random variable. It is also the center of gravity of the probability distribution histogram, meaning if we simulate the variable many times, it is the long-run average of the random variable.\n", - "\n", - "#### Example 1: Coin Toss\n", - "Going back to our coin toss example, we define a random variable $X$ as: \n", - "$$X = \\begin{cases} \n", - " 1, \\text{if the coin lands heads} \\\\\n", - " 0, \\text{if the coin lands tails} \n", - " \\end{cases}$$\n", - "We can calculate its expectation $\\mathbb{E}[X]$ using the second method of applying the weights one possible value at a time: \n", - "$$\\begin{align}\n", - " \\mathbb{E}[X] &= \\sum_{x} x P(X=x) \\\\\n", - " &= 1 * 0.5 + 0 * 0.5 \\\\\n", - " &= 0.5\n", - "\\end{align}$$\n", - "Note that $\\mathbb{E}[X] = 0.5$ is not a possible value of $X$; it's an average. **The expectation of X does not need to be a possible value of X**.\n", - "\n", - "#### Example 2\n", - "Consider the random variable $X$: \n", - "\n", - "| $x$ | $P(X=x)$ | \n", - "| --- | -------- |\n", - "| 3 | 0.1 | \n", - "| 4 | 0.2 |\n", - "| 6 | 0.4 | \n", - "| 8 | 0.3 |\n", - "\n", - "To calculate it's expectation, \n", - "$$\\begin{align}\n", - " \\mathbb{E}[X] &= \\sum_{x} x P(X=x) \\\\\n", - " &= 3 * 0.1 + 4 * 0.2 + 6 * 0.4 + 8 * 0.3 \\\\\n", - " &= 0.3 + 0.8 + 2.4 + 2.4 \\\\\n", - " &= 5.9\n", - "\\end{align}$$\n", - "Again, note that $\\mathbb{E}[X] = 5.9$ is not a possible value of $X$; it's an average. **The expectation of X does not need to be a possible value of X**.\n", - "\n", - "### Variance\n", - "The **variance** of a random variable is a measure of its chance error. It is defined as the expected squared deviation from the expectation of $X$. Put more simply, variance asks: how far does $X$ typically vary from its average value, just by chance? What is the spread of $X$'s distribution?\n", - "\n", - "$$\\text{Var}(X) = \\mathbb{E}[(X-\\mathbb{E}[X])^2]$$\n", - "\n", - "The units of variance are the square of the units of $X$. To get it back to the right scale, use the standard deviation of $X$: $$\\text{SD}(X) = \\sqrt{\\text{Var}(X)}$$\n", - "\n", - "Like with expectation, **variance is a number, not a random variable**! Its main use is to quantify chance error.\n", - "\n", - "By [Chebyshev’s inequality](https://www.inferentialthinking.com/chapters/14/2/Variability.html#Chebychev's-Bounds), which you saw in Data 8, no matter what the shape of the distribution of X is, the vast majority of the probability lies in the interval “expectation plus or minus a few SDs.”\n", - "\n", - "If we expand the square and use properties of expectation, we can re-express variance as the **computational formula for variance**. This form is often more convenient to use when computing the variance of a variable by hand, and it is also useful in Mean Squared Error calculations, as $\\mathbb{E}[X^2] = \\text{Var}(X)$ if $X$ is centered and $E(X)=0$.\n", - "\n", - "$$\\text{Var}(X) = \\mathbb{E}[X^2] - (\\mathbb{E}[X])^2$$\n", - "\n", - "::: {.callout-tip collapse=\"true\"}\n", - "## Proof\n", - "$$\\begin{align}\n", - " \\text{Var}(X) &= \\mathbb{E}[(X-\\mathbb{E}[X])^2] \\\\\n", - " &= \\mathbb{E}(X^2 - 2X\\mathbb{E}(X) + (\\mathbb{E}(X))^2) \\\\\n", - " &= \\mathbb{E}(X^2) - 2 \\mathbb{E}(X)\\mathbb{E}(X) +( \\mathbb{E}(X))^2\\\\\n", - " &= \\mathbb{E}[X^2] - (\\mathbb{E}[X])^2\n", - "\\end{align}$$\n", - ":::\n", - "\n", - "How do we compute $\\mathbb{E}[X^2]$? Any function of a random variable is *also* a random variable – that means that by squaring $X$, we've created a new random variable. To compute $\\mathbb{E}[X^2]$, we can simply apply our definition of expectation to the random variable $X^2$.\n", - "\n", - "$$\\mathbb{E}[X^2] = \\sum_{x} x^2 P(X = x)$$ \n", - "\n", - "### Example: Dice\n", - "Let $X$ be the outcome of a single fair dice roll. $X$ is a random variable defined as \n", - "$$X = \\begin{cases} \n", - " \\frac{1}{6}, \\text{if } x \\in \\{1,2,3,4,5,6\\} \\\\\n", - " 0, \\text{otherwise} \n", - " \\end{cases}$$\n", - "\n", - "::: {.callout-caution collapse=\"true\"}\n", - "## What's the expectation $\\mathbb{E}[X]?$\n", - "\n", - "$$ \\begin{align} \n", - " \\mathbb{E}[X] &= 1(\\frac{1}{6}) + 2(\\frac{1}{6}) + 3(\\frac{1}{6}) + 4(\\frac{1}{6}) + 5(\\frac{1}{6}) + 6(\\frac{1}{6}) \\\\\n", - " &= (\\frac{1}{6}) ( 1 + 2 + 3 + 4 + 5 + 6) \\\\\n", - " &= \\frac{7}{2}\n", - " \\end{align}$$\n", - ":::\n", - "\n", - "::: {.callout-caution collapse=\"true\"}\n", - "## What's the variance $\\text{Var}(X)?$\n", - "\n", - "Using approach 1: \n", - " $$\\begin{align} \n", - " \\text{Var}(X) &= (\\frac{1}{6})((1 - \\frac{7}{2})^2 + (2 - \\frac{7}{2})^2 + (3 - \\frac{7}{2})^2 + (4 - \\frac{7}{2})^2 + (5 - \\frac{7}{2})^2 + (6 - \\frac{7}{2})^2) \\\\\n", - " &= \\frac{35}{12}\n", - " \\end{align}$$\n", - "\n", - "Using approach 2: \n", - "$$\\mathbb{E}[X^2] = \\sum_{x} x^2 P(X = x) = \\frac{91}{6}$$\n", - "$$\\text{Var}(X) = \\frac{91}{6} - (\\frac{7}{2})^2 = \\frac{35}{12}$$\n", - ":::\n", - "\n", - "## Sums of Random Variables\n", - "Often, we will work with multiple random variables at the same time. A function of a random variable is also a random variable; if you create multiple random variables based on your sample, then functions of those random variables are also random variables.\n", - "\n", - "For example, if $X_1, X_2, ..., X_n$ are random variables, then so are all of these: \n", - "\n", - "* $X_n^2$\n", - "* $\\#\\{i : X_i > 10\\}$\n", - "* $\\text{max}(X_1, X_2, ..., X_n)$\n", - "* $\\frac{1}{n} \\sum_{i=1}^n (X_i - c)^2$\n", - "* $\\frac{1}{n} \\sum_{i=1}^n X_i$\n", - "\n", - "\n", - "### Equal vs. Identically Distributed vs. i.i.d\n", - "Suppose that we have two random variables $X$ and $Y$:\n", - "\n", - "* $X$ and $Y$ are **equal** if $X(s) = Y(s)$ for every sample $s$. Regardless of the exact sample drawn, $X$ is always equal to $Y$.\n", - "* $X$ and $Y$ are **identically distributed** if the distribution of $X$ is equal to the distribution of $Y$. We say “X and Y are equal in distribution.” That is, $X$ and $Y$ take on the same set of possible values, and each of these possible values is taken with the same probability. On any specific sample $s$, identically distributed variables do *not* necessarily share the same value. If X = Y, then X and Y are identically distributed; however, the converse is not true (ex: Y = 7-X, X is a die)\n", - "* $X$ and $Y$ are **independent and identically distributed (i.i.d)** if \n", - " 1. The variables are identically distributed. \n", - " 2. Knowing the outcome of one variable does not influence our belief of the outcome of the other.\n", - "\n", - "For example, let $X_1$ and $X_2$ be numbers on rolls of two fair die. $X_1$ and $X_2$ are i.i.d, so $X_1$ and $X_2$ have the same distribution. However, the sums $Y = X_1 + X_1 = 2X_1$ and $Z=X_1+X_2$ have different distributions but the same expectation.\n", - "\n", - "

    \n", - "distribution\n", - "

    \n", - "\n", - "However, $Y = X_1$ has a larger variance\n", - "\n", - "

    \n", - "distribution\n", - "

    \n", - "\n", - "### Properties of Expectation \n", - "Instead of simulating full distributions, we often just compute expectation and variance directly. Recall the definition of expectation: $$\\mathbb{E}[X] = \\sum_{x} x P(X=x)$$\n", - "From it, we can derive some useful properties of expectation: \n", - "\n", - "1. **Linearity of expectation**. The expectation of the linear transformation $aX+b$, where $a$ and $b$ are constants, is:\n", - "\n", - "$$\\mathbb{E}[aX+b] = aE[\\mathbb{X}] + b$$\n", - "\n", - "::: {.callout-tip collapse=\"true\"}\n", - "## Proof\n", - "$$\\begin{align}\n", - " \\mathbb{E}[aX+b] &= \\sum_{x} (ax + b) P(X=x) \\\\\n", - " &= \\sum_{x} (ax P(X=x) + bP(X=x)) \\\\\n", - " &= a\\sum_{x}P(X=x) + b\\sum_{x}P(X=x)\\\\\n", - " &= a\\mathbb{E}(X) + b * 1\n", - " \\end{align}$$\n", - ":::\n", - "\n", - "2. Expectation is also linear in *sums* of random variables. \n", - "\n", - "$$\\mathbb{E}[X+Y] = \\mathbb{E}[X] + \\mathbb{E}[Y]$$\n", - "\n", - "::: {.callout-tip collapse=\"true\"}\n", - "## Proof\n", - "$$\\begin{align}\n", - " \\mathbb{E}[X+Y] &= \\sum_{s} (X+Y)(s) P(s) \\\\\n", - " &= \\sum_{s} (X(s)P(s) + Y(s)P(s)) \\\\\n", - " &= \\sum_{s} X(s)P(s) + \\sum_{s} Y(s)P(s)\\\\\n", - " &= \\mathbb{E}[X] + \\mathbb{E}[Y]\n", - "\\end{align}$$\n", - ":::\n", - "\n", - "3. If $g$ is a non-linear function, then in general, \n", - "$$\\mathbb{E}[g(X)] \\neq g(\\mathbb{E}[X])$$ For example, if $X$ is -1 or 1 with equal probability, then $\\mathbb{E}[X] = 0$, but $\\mathbb{E}[X^2] = 1 \\neq 0$.\n", - "\n", - "### Properties of Variance\n", - "Recall the definition of variance: \n", - "$$\\text{Var}(X) = \\mathbb{E}[(X-\\mathbb{E}[X])^2]$$\n", - "Combining it with the properties of expectation, we can derive some useful properties of variance: \n", - "\n", - "1. Unlike expectation, variance is *non-linear*. The variance of the linear transformation $aX+b$ is:\n", - "$$\\text{Var}(aX+b) = a^2 \\text{Var}(X)$$\n", - "\n", - "* Subsequently, $$\\text{SD}(aX+b) = |a| \\text{SD}(X)$$\n", - "* The full proof of this fact can be found using the definition of variance. As general intuition, consider that $aX+b$ scales the variable $X$ by a factor of $a$, then shifts the distribution of $X$ by $b$ units. \n", - "\n", - "::: {.callout-tip collapse=\"true\"}\n", - "## Proof\n", - "We know that $$\\mathbb{E}[aX+b] = aE[\\mathbb{X}] + b$$\n", - "\n", - "In order to compute $\\text{Var}(aX+b)$, consider that a shift by b units does not affect spread, so $\\text{Var}(aX+b) = \\text{Var}(aX)$.\n", - "\n", - "Then, \n", - "$$\\begin{align}\n", - " \\text{Var}(aX+b) &= \\text{Var}(aX) \\\\\n", - " &= E((aX)^2) - (E(aX))^2 \\\\\n", - " &= E(a^2 X^2) - (aE(X))^2\\\\\n", - " &= a^2 (E(X^2) - (E(X))^2) \\\\\n", - " &= a^2 \\text{Var}(X)\n", - "\\end{align}$$\n", - ":::\n", - "\n", - "* Shifting the distribution by $b$ *does not* impact the *spread* of the distribution. Thus, $\\text{Var}(aX+b) = \\text{Var}(aX)$.\n", - "* Scaling the distribution by $a$ *does* impact the spread of the distribution.\n", - "\n", - "

    \n", - "transformation\n", - "

    \n", - "\n", - "2. Variance of sums of RVs is affected by the (in)dependence of the RVs.\n", - "$$\\text{Var}(X + Y) = \\text{Var}(X) + \\text{Var}(Y) 2\\text{cov}(X,Y)$$\n", - "$$\\text{Var}(X + Y) = \\text{Var}(X) + \\text{Var}(Y) \\qquad \\text{if } X, Y \\text{ independent}$$\n", - "\n", - "\n", - "::: {.callout-tip collapse=\"true\"}\n", - "## Proof\n", - "The variance of a sum is affected by the dependence between the two random variables that are being added. Let’s expand out the definition of $\\text{Var}(X + Y)$ to see what’s going on.\n", - "\n", - "To simplify the math, let $\\mu_x = \\mathbb{E}[X]$ and $\\mu_y = \\mathbb{E}[Y]$.\n", - "\n", - "$$ \\begin{align}\n", - "\\text{Var}(X + Y) &= \\mathbb{E}[(X+Y- \\mathbb{E}(X+Y))^2] \\\\\n", - "&= \\mathbb{E}[((X - \\mu_x) + (Y - \\mu_y))^2] \\\\\n", - "&= \\mathbb{E}[(X - \\mu_x)^2 + 2(X - \\mu_x)(Y - \\mu_y) + (Y - \\mu_y)^2] \\\\\n", - "&= \\mathbb{E}[(X - \\mu_x)^2] + \\mathbb{E}[(Y - \\mu_y)^2] + \\mathbb{E}[(X - \\mu_x)(Y - \\mu_y)] \\\\\n", - "&= \\text{Var}(X) + \\text{Var}(Y) + \\mathbb{E}[(X - \\mu_x)(Y - \\mu_y)] \n", - "\\end{align}$$\n", - ":::\n", - "\n", - "### Covariance and Correlation\n", - "We define the **covariance** of two random variables as the expected product of deviations from expectation. Put more simply, covariance is a generalization of variance to *two* random variables: $\\text{Cov}(X, X) = \\mathbb{E}[(X - \\mathbb{E}[X])^2] = \\text{Var}(X)$\n", - "\n", - "$$\\text{Cov}(X, Y) = \\mathbb{E}[(X - \\mathbb{E}[X])(Y - \\mathbb{E}[Y])]$$\n", - "\n", - "We can treat the covariance as a measure of association. Remember the definition of correlation given when we first established SLR?\n", - "\n", - "$$r(X, Y) = \\mathbb{E}\\left[\\left(\\frac{X-\\mathbb{E}[X]}{\\text{SD}(X)}\\right)\\left(\\frac{Y-\\mathbb{E}[Y]}{\\text{SD}(Y)}\\right)\\right] = \\frac{\\text{Cov}(X, Y)}{\\text{SD}(X)\\text{SD}(Y)}$$\n", - "\n", - "It turns out we've been quietly using covariance for some time now! If $X$ and $Y$ are independent, then $\\text{Cov}(X, Y) =0$ and $r(X, Y) = 0$. Note, however, that the converse is not always true: $X$ and $Y$ could have $\\text{Cov}(X, Y) = r(X, Y) = 0$ but not be independent. \n", - "\n", - "### Summary \n", - "* Let $X$ be a random variable with distribution $P(X=x)$. \n", - " * $\\mathbb{E}[X] = \\sum_{x} x P(X=x)$\n", - " * $\\text{Var}(X) = \\mathbb{E}[(X-\\mathbb{E}[X])^2] = \\mathbb{E}[X^2] - (\\mathbb{E}[X])^2$\n", - "* Let $a$ and $b$ be scalar values. \n", - " * $\\mathbb{E}[aX+b] = aE[\\mathbb{X}] + b$\n", - " * $\\text{Var}(aX+b) = a^2 \\text{Var}(X)$\n", - "* Let $Y$ be another random variable. \n", - " * $\\mathbb{E}[X+Y] = \\mathbb{E}[X] + \\mathbb{E}[Y]$\n", - " * $\\text{Var}(X + Y) = \\text{Var}(X) + \\text{Var}(Y) 2\\text{cov}(X,Y)$" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3 (ipykernel)", - "language": "python", - "name": "python3" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/probability_2/probability_2.ipynb b/probability_2/probability_2.ipynb deleted file mode 100644 index df66e5b4..00000000 --- a/probability_2/probability_2.ipynb +++ /dev/null @@ -1,302 +0,0 @@ -{ - "cells": [ - { - "cell_type": "raw", - "metadata": {}, - "source": [ - "---\n", - "title: 'Estimators, Bias, and Variance'\n", - "execute:\n", - " echo: true\n", - "format:\n", - " html:\n", - " code-fold: true\n", - " code-tools: true\n", - " toc: true\n", - " toc-title: 'Estimators, Bias, and Variance'\n", - " page-layout: full\n", - " theme:\n", - " - cosmo\n", - " - cerulean\n", - " callout-icon: false\n", - "---" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "::: {.callout-note collapse=\"false\"}\n", - "## Learning Outcomes\n", - "* Explore commonly seen random variables like Bernoulli and Binomial distributions\n", - "* Apply the Central Limit Theorem to approximate parameters of a population\n", - "* Use sampled data to model an estimation of and infer the true underlying distribution\n", - "* Estimate the true population distribution from a sample using the bootstrapping technique\n", - ":::\n", - "\n", - "Last time, we introduced the idea of random variables: numerical functions of a sample. Most of our work in the last lecture was done to build a background in probability and statistics. Now that we've established some key ideas, we're in a good place to apply what we've learned to our original goal -- understanding how the randomness of a sample impacts the model design process. \n", - "\n", - "In this lecture, we will delve more deeply into the idea of fitting a model to a sample. We'll explore how to re-express our modeling process in terms of random variables and use this new understanding to steer model complexity. \n", - "\n", - "\n", - "## Common Random Variables\n", - "There are several cases of random variables that appear often and have useful properties. Below are the ones we will explore further in this course. The numbers in parentheses are the parameters of a random variable, which are constants. Parameters define a random variable’s shape (i.e., distribution) and its values. For this lecture, we'll focus more heavily on the bolded random variables and their special properties, but you should familiarize yourself with all the ones listed below: \n", - "\n", - "* **Bernoulli(p)**\n", - " * Takes on value 1 with probability p, and 0 with probability 1 - p.\n", - " * AKA the “indicator” random variable.\n", - " * Let X be a Bernoulli(p) random variable\n", - " * $\\mathbb{E}[X] = 1 * p + 0 * (1-p) = p$\n", - " * $\\mathbb{E}[X^2] = 1^2 * p + 0 * (1-p) = p$\n", - " * $\\text{Var}(X) = \\mathbb{E}[X^2] - (\\mathbb{E}[X])^2 = p - p^2 = p(1-p)$\n", - "* **Binomial(n, p)**\n", - " * Number of 1s in $n$ independent Bernoulli(p) trials.\n", - " * Let $Y$ be a Binomial(n, p) random variable.\n", - " * The distribution of $Y$ is given by the binomial formula, and we can write $Y = \\sum_{i=1}^n X_i$ where:\n", - " * $X_i$ s the indicator of success on trial i. $X_i = 1$ if trial i is a success, else 0.\n", - " * All $X_i$ are i.i.d. and Bernoulli(p).\n", - " * $\\mathbb{E}[Y] = \\sum_{i=1}^n \\mathbb{E}[X_i] = np$\n", - " * $\\text{Var}(X) = \\sum_{i=1}^n \\text{Var}(X_i) = np(1-p)$ \n", - " * $X_i$'s are independent, so $\\text{Cov}(X_i, X_j) = 0$ for all i, j.\n", - "* Uniform on a finite set of values\n", - " * Probability of each value is 1 / (number of possible values).\n", - " * For example, a standard/fair die.\n", - "* Uniform on the unit interval (0, 1)\n", - " * Density is flat at 1 on (0, 1) and 0 elsewhere.\n", - "* Normal($\\mu, \\sigma^2$)\n", - " * $f(x) = \\frac{1}{\\sigma\\sqrt{2\\pi}} \\exp\\left( -\\frac{1}{2}\\left(\\frac{x-\\mu}{\\sigma}\\right)^{\\!2}\\,\\right)$\n", - "\n", - "### Example\n", - "Suppose you win cash based on the number of heads you get in a series of 20 coin flips.\n", - "Let $X_i = 1$ if the $i$-th coin is heads, 0 otherwise. Which payout strategy would you choose?\n", - "\n", - "A. $Y_A = 10 * X_1 + 10 * X_2$\n", - "\n", - "B. $Y_B = \\sum_{i=1}^{20} X_i$\n", - "\n", - "C. $Y_C = 20 * X_1$\n", - "\n", - "::: {.callout-caution collapse=\"true\"}\n", - "## Solution\n", - "\n", - "Let $X_1, X_2, ... X_{20}$ be 20 i.i.d Bernoulli(0.5) random variables. Since the $X_i$'s are independent, $\\text{Cov}(X_i, X_j) = 0$ for all pairs $i, j$. Additionally, Since $X_i$ is Bernoulli(0.5), we know that $\\mathbb{E}[X] = p = 0.5$ and $\\text{Var}(X) = p(1-p) = 0.25$. We can calculate the following for each scenario: \n", - "\n", - "| | A. $Y_A = 10 * X_1 + 10 * X_2$ | B. $Y_B = \\sum_{i=1}^{20} X_i$ | C. $Y_C = 20 * X_1$ | \n", - "| --- | --- | --- | --- |\n", - "| Expectation | $\\mathbb{E}[Y_A] = 10 (0.5) + 10(0.5) = 10$ | $\\mathbb{E}[Y_B] = 0.5 + ... + 0.5 = 10$ | $\\mathbb{E}[Y_C] = 20(0.5) = 10$ | \n", - "| Variance | $\\text{Var}(Y_A) = 10^2 (0.25) + 10^2 (0.25) = 50$ | $\\text{Var}(Y_B) = 0.25 + ... + 0.25 = 5$ | $\\text{Var}(Y_C) = 20^2 (0.25) = 100$ | \n", - "| Standard Deviation | $\\text{SD}(Y_A) \\approx 7.07$ | $\\text{SD}(Y_B) \\approx 2.24$ | $\\text{SD}(Y_C) = 10$ | \n", - "\n", - "As we can see, all the scenarios have the same expected value but different variances. The higher the variance, the greater the risk and uncertainty, so the \"right\" strategy depends on your personal preference. Would you choose the \"safest\" option B, the most \"risky\" option C, or somewhere in the middle (option A)?\n", - ":::\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Sample Statistics \n", - "Today, we've talked extensively about populations; if we know the distribution of a random variable, we can reliably compute expectation, variance, functions of the random variable, etc. \n", - "Note that:\n", - "\n", - "* The distribution of a *population* describes how a random variable behaves across *all* individuals of interest. \n", - "* The distribution of a *sample* describes how a random variable behaves in a *specific sample* from the population. \n", - "\n", - "In Data Science, however, we often do not have access to the whole population, so we don’t know its distribution. As such, we need to collect a sample and use its distribution to estimate or infer properties of the population. In cases like these, we can take several samples of size $n$ from the population (an easy way to do this is using `df.sample(n, replace=True)`), and compute the mean of each *sample*. When sampling, we make the (big) assumption that we sample uniformly at random with replacement from the population; each observation in our sample is a random variable drawn i.i.d from our population distribution. Remember that our sample mean is a random variable since it depends on our randomly drawn sample! On the other hand, our population mean is simply a number (a fixed value).\n", - "\n", - "### Sample Mean \n", - "Consider an i.i.d. sample $X_1, X_2, ..., X_n$ drawn from a population with mean 𝜇 and SD 𝜎.\n", - "We define the sample mean as $$\\bar{X}_n = \\frac{1}{n} \\sum_{i=1}^n X_i$$\n", - "\n", - "The expectation of the sample mean is given by: \n", - "$$\\begin{align} \n", - " \\mathbb{E}[\\bar{X}_n] &= \\frac{1}{n} \\sum_{i=1}^n \\mathbb{E}[X_i] \\\\\n", - " &= \\frac{1}{n} (n \\mu) \\\\\n", - " &= \\mu \n", - "\\end{align}$$\n", - "\n", - "The variance is given by: \n", - "$$\\begin{align} \n", - " \\text{Var}(\\bar{X}_n) &= \\frac{1}{n^2} \\text{Var}( \\sum_{i=1}^n X_i) \\\\\n", - " &= \\frac{1}{n^2} \\left( \\sum_{i=1}^n \\text{Var}(X_i) \\right) \\\\\n", - " &= \\frac{1}{n^2} (n \\sigma^2) = \\frac{\\sigma^2}{n}\n", - "\\end{align}$$\n", - " \n", - "$\\bar{X}_n$ is normally distributed by the Central Limit Theorem (CLT).\n", - "\n", - "### Central Limit Theorem\n", - "In [Data 8](https://inferentialthinking.com/chapters/14/4/Central_Limit_Theorem.html?) and in the previous lecture, you encountered the **Central Limit Theorem (CLT)**. \n", - "This is a powerful theorem for estimating the distribution of a population with mean $\\mu$ and standard deviation $\\sigma$ from a collection of smaller samples. The CLT tells us that if an i.i.d sample of size $n$ is large, then the probability distribution of the **sample mean** is **roughly normal** with mean $\\mu$ and SD of $\\frac{\\sigma}{\\sqrt{n}}$. More generally, any theorem that provides the rough distribution of a statistic and **doesn't need the distribution of the population** is valuable to data scientists! This is because we rarely know a lot about the population. \n", - "\n", - "\n", - "

    \n", - "clt\n", - "\n", - "

    \n", - "\n", - "Importantly, the CLT assumes that each observation in our samples is drawn i.i.d from the distribution of the population. In addition, the CLT is accurate only when $n$ is \"large\", but what counts as a \"large\" sample size depends on the specific distribution. If a population is highly symmetric and unimodal, we could need as few as $n=20$; if a population is very skewed, we need a larger $n$. If in doubt, you can bootstrap the sample mean and see if the bootstrapped distribution is bell-shaped. Classes like Data 140 investigate this idea in great detail.\n", - "\n", - "\n", - "For a more in-depth demo, check out [onlinestatbook](https://onlinestatbook.com/stat_sim/sampling_dist/). \n", - "\n", - "### Using the Sample Mean to Estimate the Population Mean\n", - "Now let's say we want to use the sample mean to **estimate** the population mean, for example, the average height of Cal undergraduates. We can typically collect a **single sample**, which has just one average. However, what if we happened, by random chance, to draw a sample with a different mean or spread than that of the population? We might get a skewed view of how the population behaves (consider the extreme case where we happen to sample the exact same value $n$ times!). \n", - "\n", - "\n", - "

    \n", - "clt\n", - "

    \n", - "\n", - "\n", - "For example, notice the difference in variation between these two distributions that are different in sample size. The distribution with a bigger sample size ($n=800$) is tighter around the mean than the distribution with a smaller sample size ($n=200$). Try plugging in these values into the standard deviation equation for the normal distribution to make sense of this! \n", - "\n", - "Applying the CLT allows us to make sense of all of this and resolve this issue. By drawing many samples, we can consider how the sample distribution varies across multiple subsets of the data. This allows us to approximate the properties of the population without the need to survey every single member. \n", - "\n", - "Given this potential variance, it is also important that we consider the **average value and spread** of all possible sample means, and what this means for how big $n$ should be. For every sample size, the expected value of the sample mean is the population mean: $$\\mathbb{E}[\\bar{X}_n] = \\mu$$. We call the sample mean an **unbiased estimator** of the population mean and will explore this idea more in the next lecture. \n", - "\n", - "\n", - "::: {.callout-tip collapse=\"true\"}\n", - "## Data 8 Recap: Square Root Law\n", - "\n", - "The square root law ([Data 8](https://inferentialthinking.com/chapters/14/5/Variability_of_the_Sample_Mean.html#the-square-root-law)) states that if you increase the sample size by a factor, the SD decreases by the square root of the factor. $$\\text{SD}(\\bar{X_n}) = \\frac{\\sigma}{\\sqrt{n}}$$ The sample mean is more likely to be close to the population mean if we have a larger sample size.\n", - ":::" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Prediction vs. Inference\n", - "\n", - "At this point in the course, we've spent a great deal of time working with models. When we first introduced the idea of modeling a few weeks ago, we did so in the context of **prediction**: using models to make *accurate predictions* about unseen data. Another reason we might build models is to better understand complex phenomena in the world around us. **Inference** is the task of using a model to infer the true underlying relationships between the feature and response variables. For example, if we are working with a set of housing data, *prediction* might ask: given the attributes of a house, how much is it worth? *Inference* might ask: how much does having a local park impact the value of a house?\n", - "\n", - "A major goal of inference is to draw conclusions about the full population of data given only a random sample. To do this, we aim to estimate the value of a **parameter**, which is a numerical function of the *population* (for example, the population mean $\\mu$). We use a collected sample to construct a **statistic**, which is a numerical function of the random *sample* (for example, the sample mean $\\bar{X}_n$). It's helpful to think \"p\" for \"parameter\" and \"population,\" and \"s\" for \"sample\" and \"statistic.\"\n", - "\n", - "Since the sample represents a *random* subset of the population, any statistic we generate will likely deviate from the true population parameter, and it *could have been different*. We say that the sample statistic is an **estimator** of the true population parameter. Notationally, the population parameter is typically called $\\theta$, while its estimator is denoted by $\\hat{\\theta}$.\n", - "\n", - "To address our inference question, we aim to construct estimators that closely estimate the value of the population parameter. We evaluate how \"good\" an estimator is by answering three questions:\n", - "\n", - "* Do we get the right answer for the parameter, on average?\n", - "* How variable is the answer?\n", - "* How close is our answer to the parameter?\n", - "\n", - "### Modeling as Estimation\n", - "\n", - "Now that we've established the idea of an estimator, let's see how we can apply this learning to the modeling process. To do so, we'll take a moment to formalize our data collection and models in the language of random variables.\n", - "\n", - "Say we are working with an input variable, $x$, and a response variable, $Y$. We assume that $Y$ and $x$ are linked by some relationship $g$; in other words, $Y = g(x)$. $g$ represents some \"universal truth\" or \"law of nature\" that defines the underlying relationship between $x$ and $Y$. In the image below, $g$ is denoted by the red line.\n", - "\n", - "As data scientists, however, we have no way of directly \"seeing\" the underlying relationship $g$. The best we can do is collect observed data out in the real world to try to understand this relationship. Unfortunately, the data collection process will always have some inherent error (think of the randomness you might encounter when taking measurements in a scientific experiment). We say that each observation comes with some random error or **noise** term, $\\epsilon$. This error is assumed to be a random variable with expectation $\\mathbb{E}(\\epsilon)=0$, variance $\\text{Var}(\\epsilon) = \\sigma^2$, and be i.i.d. across each observation. The existence of this random noise means that our observations, $Y(x)$, are *random variables*.\n", - "\n", - "

    \n", - "data\n", - "

    \n", - "\n", - "We can only observe our random sample of data, represented by the blue points above. From this sample, we want to estimate the true relationship $g$. We do this by constructing the model $\\hat{Y}(x)$ to estimate $g$. \n", - "\n", - "$$\\text{True relationship: }Y = g(x)$$\n", - "\n", - "$$\\text{Observed relationship: }Y = g(x) + \\epsilon$$\n", - "\n", - "$$\\text{Prediction: }\\hat{Y}(x)$$\n", - "\n", - "

    \n", - "y_hat\n", - "

    \n", - "\n", - "#### Estimating a Linear Relationship\n", - "If we assume that the true relationship $g$ is linear, we can express the response as $Y = f_{\\theta}(x)$, where our true relationship is modeled by \n", - "$$Y = g(x) + \\epsilon$$\n", - "$$ f_{\\theta}(x) = Y = \\theta_0 + \\sum_{j=1}^p \\theta_j x_j + \\epsilon$$\n", - "\n", - "::: {.callout-warning}\n", - "##### Which Expressions are random? \n", - "In our two equations above, the true relationship $g(x) = \\theta_0 + \\sum_{j=1}^p \\theta_j x_j$ is not random, but $\\epsilon$ is random. Hence, $Y = f_{\\theta}(x)$ is also random. \n", - ":::\n", - "\n", - "This true relationship has true, unobservable parameters $\\theta$, and it has random noise $\\epsilon$, so we can never observe the true relationship. Instead, the next best thing we can do is obtain a sample $\\Bbb{X}$, $\\Bbb{Y}$ of $n$ observed relationships, $(x, Y)$ and use it to train a model and obtain an estimate of $\\hat{\\theta}$\n", - "$$\\hat{Y}(x) = f_{\\hat{\\theta}}(x) = \\hat{\\theta_0} + \\sum_{j=1}^p \\hat{\\theta_j} x_j$$\n", - "\n", - "::: {.callout-warning}\n", - "##### Which Expressions are random? \n", - "In our estimating equation above, our sample $\\Bbb{X}$, $\\Bbb{Y}$ are random. Hence, the estimates we calculate from our samples $\\hat{\\theta}$ are also random, so our predictor $\\hat{Y}(x)$ is also random. \n", - ":::\n", - "\n", - "\n", - "Now taking a look at our original equations, we can see that they both have differing sources of randomness. For our observed relationship, $Y = g(x) + \\epsilon$, $\\epsilon$ represents measurement errors and reflects randomness from the future. For the estimation model, the data we have is a random sample collected from the population, so the randomness from the past." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Bootstrap Resampling (Review)\n", - "\n", - "To determine properties of the sampling distribution of an estimator like variance, we’d need to have access to the population so that we can consider all possible samples and compute an estimate for each sample. \n", - "\n", - "

    \n", - "y_hat\n", - "

    \n", - "\n", - "However, we don’t have access to the population; we only have *one* random sample from the population. How can we consider all possible samples if we only have one? \n", - "\n", - "The idea of bootstrapping is to treat our random sample as a \"population\" and resample from it *with replacement*. Intuitively, a random sample resembles the population, so a random *resample* also resamples a random sample. \n", - "

    \n", - "y_hat\n", - "

    \n", - "\n", - "Bootstrap resampling is a technique for estimating the sampling distribution of an estimator. To execute it, we can follow the pseudocode below: \n", - "```\n", - "collect a random sample of size n (called the bootstrap population)\n", - "\n", - "initiate list of estimates\n", - "\n", - "repeat 10,000 times:\n", - "\tresample with replacement n times from bootstrap population\n", - "\n", - "apply estimator f to resample\n", - "\n", - "store in list\n", - "\n", - "list of estimates is the bootstrapped sampling distribution of f\n", - "```\n", - "\n", - "::: {.callout-warning collapse=\"true\"}\n", - "### Why must we resample *with replacement*? \n", - "Given an original sample of size $n$, we want a resample that has the same size $n$ as the original. Sampling *without* replacement will give us the original sample with shuffled rows. Hence, when we calculate summary statistics like the average, our sample *without* replacement will always have the same average as the original sample, defeating the purpose of a bootstrap. \n", - ":::\n", - "\n", - "How well does bootstrapping actually represent our population? The bootstrapped sampling distribution of an estimator does not exactly match the sampling distribution of that estimator, but it is often close. Similarly, the variance of the bootstrapped distribution is often close to the true variance of the estimator. The example below displays the results from different bootstraps from a *known* population using a sample size of $n=50$.\n", - "\n", - "

    \n", - "y_hat\n", - "

    \n", - "\n", - "In the real world, we don't know the population distribution. The center of the boostrapped distribution is the estimator applied to our original sample, so we have no way of recovering the estimator's true expected value. The quality of our bootstrapped distribution depends on the quality of our original sample; if our original sample was not representative of the population, bootstrap is next to useless.\n", - "\n", - "One thing to note is that the bootstrap often does not work well for some statistics, like the median or other quantile-based statistics, that depend heavily on a small number of observations out of a larger sample. **Bootstrapping does not overcome the weakness of small samples as a basis for inference**. Indeed, for the very smallest samples, it may be better to make additional assumptions such as a parametric family. " - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3 (ipykernel)", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.9.16" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -}