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<!doctype html>
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<head>
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<title>Advanced Machine Learning</title>
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<meta name="author" content="Sergey M Plis">
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<section>
<section>
<p>
<h2>Advanced Machine Learning</h2>
<h3>09: Maximum Likelihood Estimation</h3>
<p>
</section>
<section>
<h3>Schedule</h3>
<row>
<col50>
<table style="font-size:14px">
<tr>
<th>#</th>
<th>date</th>
<th>topic</th>
<th>description</th>
</tr>
<tr><td>1</td>
<td> 22-Aug-2022 </td>
<td> Introduction </td>
<td></td>
</tr>
<tr>
<td> 2 </td>
<td> 24-Aug-2022 </td>
<td> Foundations of learning </td>
<td> </td>
</tr>
<tr><td> 3 </td><td> 29-Aug-2022 </td><td> PAC learnability </td><td> </td></tr>
<tr><td> 4 </td><td> 31-Aug-2022 </td><td> Linear algebra (recap) </td><td> hw1 released </td></tr>
<tr style='background-color: #FBEEC2;'><td> </td><td> 05-Sep-2022 </td><td> <em>Holiday</em> </td><td> </td></tr>
<tr style='background-color: #E0E4CC;'><td> 5 </td><td> 07-Sep-2022 </td><td> Linear learning models </td><td> </td></tr>
<tr><td> 6 </td><td> 12-Sep-2022 </td><td> Principal Component Analysis </td><td> project ideas </td></tr>
<tr><td> 7 </td><td> 14-Sep-2022 </td><td> Curse of Dimensionality </td></td></td><td> hw1 due </td></tr>
<tr><td> 8 </td><td> 19-Sep-2022 </td><td> Bayesian Decision Theory </td><td>hw2 release </td></tr>
<tr><td> 9 </td><td> 21-Sep-2022 </td><td> Parameter estimation: MLE </td><td><i class='fa fa-map-marker' style='color: #FA6900;'></i></td></tr>
<tr><td> 10 </td><td> 26-Sep-2022 </td><td> Parameter estimation: MAP & NB</td><td>finalize teams</td></tr>
<tr><td> 11 </td><td> 28-Sep-2022 </td><td> Logistic Regression </td><td> </td></tr>
<tr><td> 12 </td><td> 03-Oct-2022 </td><td> Kernel Density Estimation </td><td> </td></tr>
<tr><td> 13 </td><td> 05-Oct-2022 </td><td> Support Vector Machines </td><td> hw3, hw2 due </td></tr>
<tr style='background-color: #E5DDCB;'><td> </td><td> 10-Oct-2022 </td><td> * Mid-point projects checkpoint </td><td> * </td></tr>
<tr style='background-color: #E5DDCB;'><td> </td><td> 12-Oct-2022 </td><td> * Midterm: Semester Midpoint </td><td> exam </td></tr>
<tr><td> 14 </td><td> 17-Oct-2022 </td><td>Matrix Factorization</td><td> </td></tr>
<tr><td> 15 </td><td> 19-Oct-2022 </td><td>Stochastic Gradient Descent</td><td> </td></tr>
</table>
</col50>
<col50>
<table style="font-size:14px; vertical-align: top;">
<tr>
<th>#</th>
<th>date</th>
<th>topic</th>
<th>description</th>
</tr>
<tr><td> 16 </td><td> 24-Oct-2022 </td><td> k-means clustering </td><td> </td></tr>
<tr><td> 17 </td><td> 26-Oct-2022 </td><td> Expectation Maximization </td><td> hw4, hw3 due </td></tr>
<tr><td> 18 </td><td> 31-Oct-2022 </td><td> Automatic Differentiation </td><td> </td></tr>
<tr><td> 19 </td><td> 02-Nov-2022 </td><td> Nonlinear embedding approaches </td><td> </td></tr>
<tr><td> 20 </td><td> 07-Nov-2022 </td><td> Model comparison I </td><td> </td></tr>
<tr><td> 21 </td><td> 09-Nov-2022 </td><td> Model comparison II </td><td> hw5, hw4 due</td></tr>
<tr><td> 22 </td><td> 14-Nov-2022 </td><td> Model Calibration </td><td> </td></tr>
<tr><td> 23 </td><td> 16-Nov-2022 </td><td> Convolutional Neural Networks </td><td> </td></tr>
<tr style='background-color: #FBEEC2;'><td> </td><td> 21-Nov-2022 </td><td> <em>Fall break</em> </td><td> </td></tr>
<tr style='background-color: #FBEEC2;'><td> </td><td> 23-Nov-2022 </td><td> <em>Fall break</em> </td><td> </td></tr>
<tr><td> 24 </td><td> 28-Nov-2022 </td><td> Word Embedding </td><td> hw5 due </td></tr>
<tr style='background-color: #FBEEC2;'><td> </td><td> 30-Nov-2022 </td><td> Presentation and exam prep day </td><td> </td></tr>
<tr style='background-color: #E5DDCB;'><td> </td><td> 02-Dec-2022 </td><td> * Project Final Presentations </td><td> * </td></tr>
<tr style='background-color: #E5DDCB;'><td> </td><td> 07-Dec-2022 </td><td> * Project Final Presentations </td><td> * </td></tr>
<tr style='background-color: #E5DDCB;'><td> </td><td> 12-Dec-2022 </td><td> * Final Exam </td><td> * </td></tr>
<tr><td> </td><td> 15-Dec-2022 </td><td> Grades due </td><td> </td></tr>
</table>
</col50>
</row>
</section>
<section>
<h3>Outline for the lecture</h3>
<ul>
<li class="fragment roll-in"> Independence
<li class="fragment roll-in"> Parameter estimation: MLE
<li class="fragment roll-in"> MLE and KL-divergence
</ul>
</section>
</section>
<!-- ------------------------------------------------------------------------- -->
<section>
<section>
<h2>Independence</h2>
<div class='slide-footer'>
based on Barnabas Poczos' slides
</div>
<aside class="notes">
<ul>
<li> just a template
</ul>
</aside>
</section>
<section>
<h2>Independence</h2>
<blockquote>
<b>Independent random variables:</b>
\begin{align}
\prob{P}{X,Y} &= \prob{P}{X}\prob{P}{Y}\\
\prob{P}{X|Y} &= \prob{P}{X}
\end{align}
</blockquote>
<ul style="list-style-type: none;">
<li class="fragment roll-in"> $Y$ and $X$ don't contain information about each other.
<li class="fragment roll-in"> Observing $Y$ does not help predicting $X$.
<li class="fragment roll-in"> Observing $X$ does not help predicting $Y$.
</ul>
<ul style="list-style-type: none; font-size: 32px;">
<li class="fragment roll-in"> <b>Examples:</b>
<li class="fragment roll-in"> <b>Independent:</b> winning on roulette this week and next week
<li class="fragment roll-in"> <b>Dependent:</b> Russian roulette
</ul>
</section>
<section>
<h2>inependent/dependent</h2>
<div class="row">
<div class="col_left5">
<img style="border:0; box-shadow: 0px 0px 0px rgba(255, 255, 255, 255);" width="1200"
src="figures/independent_samples.png" alt="independent">
</div>
<div class="col_right">
<img style="border:0; box-shadow: 0px 0px 0px rgba(255, 255, 255, 255);" width="450"
src="figures/dependent_samples.png" alt="dependent">
</div>
</div>
</section>
<section>
<h2>Conditionally Independent</h2>
<div class="fragment" data-fragment-index="0">
<blockquote>
<b>Conditionally independent:</b><br>
$$\prob{P}{X,Y|Z} = \prob{P}{X|Z}\prob{P}{Y|Z}$$
Knowing $Z$ makes $X$ and $Y$ independent
</blockquote>
</div>
<div class="fragment" data-fragment-index="1">
<blockquote style="font-size:28px">
<ul style="list-style-type: none;">
<li class="fragment roll-in" data-fragment-index="1"> <b>Examples:</b>
<li class="fragment roll-in" data-fragment-index="2"> <b>Dependent:</b> shoe size and reading skills in kids
<li class="fragment roll-in" data-fragment-index="3"> <b>Conditionally Independent:</b> shoe size and readnig skills given <b>age</b>
</ul>
</blockquote>
</div>
<div class="fragment" data-fragment-index="4" >
<row>
<col70>
<blockquote style="font-size:30px; width: 100%; text-align: left;">
<b>Storks deliver babies:</b>
Highly statistically significant correlation ($p=0.008$) exists between stork populations and human birth rates across Europe
</blockquote>
</col70>
<col30>
<img style="width: 100%; margin-top: -7%;" src="figures/stork_baby.png" alt="stork">
</col30>
</row>
</div>
</section>
<section>
<h2>Conditionally Independent</h2>
<blockquote style="font-size:32px; width: 100%; text-align: left;">
<b>London taxi drivers:</b>
A survey has pointed out a positive and significant correlation between the number of accidents and wearing coats. They concluded that coats could hinder movements of drivers and be the cause of accidents. A new law was prepared to prohibit drivers from wearing coats when driving.
</blockquote>
<div class="fragment" data-fragment-index="1" style="font-size:32px">
<em>Finally another study pointed out that people wear coats when it rains...</em>
</div>
</section>
<section>
<h2>Correlation $\ne$ Causation</h2>
<img style="border:0; box-shadow: 0px 0px 0px rgba(255, 255, 255, 255);" width="450" class="reveal"
src="figures/causation_correlation.png" alt="correlation is not causation">
<div class='slide-footer'>
xkcd.com
</div>
</section>
</section>
<!-- ------------------------------------------------------------------------- -->
<section>
<section>
<h2>Parameter estimation: MLE</h2>
<h3>a machine learning problem</h3>
<div class="row">
<div class="col_left5">
Estimating probabilities
</div>
<div class="col_right">
<img style="border:0; box-shadow: 0px 0px 0px rgba(255, 255, 255, 255);" width="250"
src="figures/hand_flipping.png" alt="flipping">
</div>
</div>
</section>
<section>
<h2>Flipping a coin</h2>
<blockquote style="font-size:36px">
I have a coin, if I flip it, what's the probability it will fall with head up?
</blockquote>
<div class="fragment" data-fragment-index="0">
Let us flip it a few times to estimate the probability:
<img style="border:0; box-shadow: 0px 0px 0px rgba(255, 255, 255, 255);" width="600"
src="figures/coin_row.png" alt="a flip">
</div>
<div class="fragment" data-fragment-index="1">
<blockquote>
The estimated probability is $\frac{3}{5}$. "Frequency of heads"
</blockquote>
</div>
<aside class="notes">
<ul>
<li> Don't tell me the class is difficult, we're only working with pocket change problems here. Easy
</ul>
</aside>
</section>
<section>
<h2>Flipping a coin</h2>
<img style="border:0; box-shadow: 0px 0px 0px rgba(255, 255, 255, 255);" width="600"
src="figures/coin_row.png" alt="a flip">
<blockquote>
The estimated probability is $\frac{3}{5}$. "Frequency of heads"
</blockquote>
<ol>
<li class="fragment roll-in" data-fragment-index="0"> Why frequency of heads???
<li class="fragment roll-in" data-fragment-index="1"> How good is this estimation???
<li class="fragment roll-in" data-fragment-index="2"> Why is this a machine learning problem???
</ol>
<div class="fragment" data-fragment-index="3 ">
Let's go ahead and answer these questions
</div>
</section>
<section>
<h3>QUESTION 1: Why frequency of heads???</h3>
<ul class="fa-ul">
<li class="fragment roll-in"><i class="fa-li fa fa-thumbs-o-up"></i> Frequency of heads is exactly the <b>maximum likelihood estimator</b> (MLE) for this problem
<li class="fragment roll-in"><i class="fa-li fa fa-thumbs-o-up"></i> MLE has nice properties
<li class="fragment roll-in"><i class="fa-li fa fa-thumbs-o-down"></i> and bad ones too, but that's another story
</ul>
</section>
<section>
<h2>Maximum Likelihood Estimation</h2>
</section>
<section data-fullscreen>
<h2>MLE for Bernoulli distribution</h2>
<blockquote>
Data $D = $ <img style="vertical-align: middle;" height="100" src="figures/coin_row.png" alt="a flip">
$D = \{x_i\}_{i=1}^n, x_i \in \{\text{H}, \text{T}\}$
</blockquote>
<span style="font-size: 32px;">
$\prob{P}{\text{Heads}} = \theta, \prob{P}{\text{Tails}} = 1-\theta$
</span>
<div class="fragment" data-fragment-index="0">
<blockquote style="background-color: #eee8d5; font-size: 34px;">
Flips are <b>i.i.d.</b>:
<ul>
<li><b>Independent</b> events
<li><b>Identically distributed</b> according to Bernoulli distribution
</ul>
</blockquote>
</div>
<div class="fragment" data-fragment-index="1">
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 34px;">
MLE: Choose $\theta$ that maximizes the probability of observed data
</blockquote>
</div>
<aside class="notes">
<ul>
<li> discuss the likelihood on the blackboard: function of theta
</ul>
</aside>
</section>
<section>
<h2>Maximum Likelihood Estimation</h2>
<blockquote style="background-color: #93a1a1; width: 100%; color: #fdf6e3; font-size: 34px;">
MLE: Choose $\theta$ that maximizes the probability of observed data
</blockquote>
<div class="fragment" data-fragment-index="0" style="font-size: 34px;">
\begin{align}
\hat{\theta}_{MLE}
&\fragment{1}{ = \underset{\theta}{\argmax} \prob{P}{D|\theta}}\\
&\fragment{2}{ = \underset{\theta}{\argmax} \displaystyle{\prod_{i=1}^n}\prob{P}{x_i|\theta} \color{#dc322f}{\text{ independent draws}}}\\
&\fragment{3}{ = \underset{\theta}{\argmax} \displaystyle{\prod_{i:x_i=H}^{\alpha_H}}\theta \displaystyle{\prod_{j:x_j=T}^{\alpha_T}}(1-\theta) \color{#dc322f}{\stackrel{\text{identically}}{\text{distributed}}}}\\
&\fragment{4}{ = \underset{\theta}{\argmax} \theta^{\alpha_H} (1-\theta)^{\alpha_T}}\\
\end{align}
</div>
<blockquote style="background-color: #93a1a1; width: 100%; color: #fdf6e3; font-size: 34px;">
$J(\theta) = \theta^{\alpha_H} (1-\theta)^{\alpha_T}$
</blockquote>
</section>
<section>
<!-- <h2>Maximum Likelihood Estimation</h2> -->
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 38px;">
MLE: Choose $\theta$ that maximizes the probability of observed data
</blockquote>
<div class="fragment" data-fragment-index="0" style="font-size: 30px;">
\begin{align}
\hat{\theta}_{MLE} & = \underset{\theta}{\argmax} \prob{P}{D|\theta}\\
J(\theta) & = \theta^{\alpha_H} (1-\theta)^{\alpha_T}\\
\frac{\partial J(\theta)}{\partial \theta} &= \alpha_H \theta^{\alpha_H-1} (1-\theta)^{\alpha_T} - \alpha_T \theta^{\alpha_H} (1-\theta)^{\alpha_T-1} \stackrel{\text{set}}{=} 0
\end{align}
\begin{align}
(\alpha_H(1 - \theta) - \alpha_T\theta)\theta^{\alpha_h-1}(1-\theta)^{\alpha_T-1} &= 0\\
\alpha_H(1 - \theta) - \alpha_T\theta &= 0\\
\hat{\theta}_{MLE} &= \frac{\alpha_H}{\alpha_H + \alpha_T}\\
\end{align}
</div>
<div class="fragment" data-fragment-index="1">
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 38px;">
That's exactly "Frequency of heads" <i class="far fa-hand-point-up"></i>
</blockquote>
</div>
</section>
<section>
<h2>Flipping a coin</h2>
<img style="border:0; box-shadow: 0px 0px 0px rgba(255, 255, 255, 255);" width="600"
src="figures/coin_row.png" alt="a flip">
<blockquote>
The estimated probability is $\frac{3}{5}$. "Frequency of heads"
</blockquote>
<ol>
<li> <i class="far fa-hand-point-up"></i> Why frequency of heads???
<li> <b>How good is this estimation???</b>
<li> Why is this a machine learning problem???
</ol>
</section>
<section>
<h3>Question2: How good is this estimation???</h3>
$$
\hat{\theta}_{MLE} = \frac{\alpha_H}{\alpha_H + \alpha_T}
$$
</section>
<section>
<h2>How many flips do I need ?</h2>
<ul style="list-style-type: none;">
<li class="fragment roll-in" data-fragment-index="0">I flipped the coins 5 times: 3 heads, 2 tails
$$
\hat{\theta}_{MLE} = \frac{3}{5}
$$
<li class="fragment roll-in" data-fragment-index="1"> What if I flipped 26 heads and 24 tails?
$$
\hat{\theta}_{MLE} = \frac{26}{50}
$$
</ul>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 38px;" class="fragment" data-fragment-index="2">
Which estimator should we trust more? <i class="fas fa-meh-rolling-eyes"></i>
</blockquote>
</section>
<section>
<h2>Simple bound</h2>
Let $\theta^*$ be the true parameter.
<div class="fragment" data-fragment-index="0">
For $n = \alpha_H + \alpha_T$, and $\hat{\theta}_{MLE} = \frac{\alpha_H}{\alpha_H + \alpha_T}$
</div>
<div class="fragment" data-fragment-index="1">
For any $\epsilon \gt 0$:
</div>
<div class="fragment" data-fragment-index="2">
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 38px;">
Hoeffding's inequality:
</blockquote>
<blockquote>
\begin{align}
\prob{P}{|\hat{\theta} - \theta^*| \ge \epsilon} \le 2e^{-2n\epsilon^2}
\end{align}
</blockquote>
</div>
<aside class="notes">
<ul>
<li> In Hoeffding's inequality the bound is divided by $(a-b)^2$ for the interval where the parameter is defined
</ul>
</aside>
</section>
<section>
<h2>PAC learning</h2>
I want to know the coin parameter $\theta$, within $\epsilon = 0.1$ error with probability at least $1-\delta = 0.95$
<ul style="list-style-type: none;">
<li class="fragment roll-in"> How many flips do I need?
<li class="fragment roll-in">
\begin{align}
\prob{P}{|\hat{\theta} - \theta^*| \ge \epsilon} & \le 2e^{-2n\epsilon^2} \le \delta
\end{align}
<li class="fragment roll-in"> How many samples do I need?
<li class="fragment roll-in">
\begin{align}
n & \ge \frac{\ln (2/\delta)}{2\epsilon^2} \approx 185
\end{align}
</ul>
<aside class="notes">
<ul>
<li> Derive the number of flips as
<li> $e^{-2n\epsilon^2} \le \delta/2$
<li> $-2n\epsilon^2 \le \ln(\delta/2)$ take a log
<li> divide by $-2\epsilon^2$ flipping the inequality because of the negative sign
<li> $n \le \frac{\ln(\delta/2)}{-2\epsilon^2}$ pull the negative into the logarithm
<li> arrive at the value of bound and plug in 0.05 for delta and 0.1 for epsilon
</ul>
</aside>
</section>
<section>
<h2>Flipping a coin</h2>
<img style="border:0; box-shadow: 0px 0px 0px rgba(255, 255, 255, 255);" width="600"
src="figures/coin_row.png" alt="a flip">
<blockquote>
The estimated probability is $\frac{3}{5}$. "Frequency of heads"
</blockquote>
<ol>
<li> <i class="far fa-hand-point-up"></i> Why frequency of heads???
<li> <i class="far fa-hand-point-up"></i> How good is this estimation???
<li> <b>Why is this a machine learning problem???</b>
</ol>
</section>
<section>
<h3>Question2: Why is this an ML problem???</h3>
<blockquote style="width: 100%; background-color: #93a1a1; color: #fdf6e3; font-size: 38px;" class="fragment" data-fragment-index="1">
Machine Learning is the study of algorithms that
<ul>
<li> improve their performance
<li> at some task
<li> with experience
</ul>
</blockquote>
<ul>
<li class="fragment roll-in"> improves: accuracy of the predicted probability
<li class="fragment roll-in"> task: predicting the probability of heads
<li class="fragment roll-in"> experience: the more flips the better the estimate
</ul>
</section>
<section>
<h2>What about continuous features?</h2>
<img style="border:0; box-shadow: 0px 0px 0px rgba(150, 150, 255, 1);" width="600"
src="figures/gaussian_pink.svg" alt="Gaussian samples">
<div class="fragment" data-fragment-index="0">
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 38px;">
Let us try Gaussians...
</blockquote>
\begin{align}
\prob{p}{x|\mu,\sigma} &= \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} = {\cal N}_x(\mu, \sigma)
\end{align}
</div>
</section>
<section>
<h2>MLE for Gaussian $\mu$ and $\sigma^2$</h2>
$\theta = (\mu, \sigma^2)$ that maximizes the probability of observed data
<span style="font-size: 32px;">
\begin{align}
\hat{\theta}_{MLE} & = \underset{\theta}{\argmax} \prob{P}{D|\theta}\\
& = \underset{\theta}{\argmax} \displaystyle{\prod_{i=1}^n}\prob{P}{x_i|\theta} \color{#dc322f}{\text{ independent draws}}\\
& = \underset{\theta}{\argmax} \displaystyle{\prod_{i=1}^n} \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x_i-\mu)^2}{2\sigma^2}} \color{#dc322f}{\text{ i.i.d}}\\
& = \underset{\theta}{\argmax} \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{\sum_{i=1}^n(x_i-\mu)^2}{2\sigma^2}}\\
\end{align}
</span>
</section>
<section>
<h2>Derive $\hat{\mu}_{MLE}$ </h2>
<aside class="notes">
<ul>
<li> Derive the mu in one way
<li> Talk about log likelihood and derive in this way as well
<li> Mention how I consistently made mistakes and got lucky in the video and importantly, I was not corrected by any of my students :(
</ul>
</aside>
</section>
<section>
<h3>MLE for Gaussian $\mu$ and $\sigma^2$</h3>
<blockquote style="font-size: 30px;">
\begin{align}
\hat{\mu}_{MLE} &= \frac{1}{n} \displaystyle\sum_{i=1}^n x_i\\
\hat{\sigma}^2_{MLE} &= \frac{1}{n} \displaystyle\sum_{i=1}^n (x_i - \hat{\mu}_{MLE})^2\\
\end{align}
</blockquote>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 30px;" class="fragment" data-fragment-index="0">
MLE for $\sigma^2$ of a Gaussian is <b>biased</b>: expected result of estimation is <b>not</b> the true parameter!
$$\hat{\sigma}^2_{unbiased} = \frac{1}{n-1} \displaystyle\sum_{i=1}^n (x_i - \hat{\mu}_{MLE})^2$$
</blockquote>
<aside class="notes">
<ul>
<li> Because sample mean needs to be used in the variance estimator the estimator converges to $\frac{n-1}{n} \sigma^2$
</ul>
</aside>
</section>
<section>
<h2>Refresher: Exponential Family</h2>
<img style="border:0; box-shadow: 0px 0px 0px rgba(150, 150, 255, 1);" width="100%"
src="figures/Gaussian_ductape.png" alt="Gaussian duct tape">
<div class="slide-footer">
<a href="https://arxiv.org/abs/0911.4863" target="_blank">Statistical exponential families: A digest with flash cards</a>
</div>
<aside class="notes">
<ul>
<li> The complete family can be expressed as $\prob{x|\eta} = h(x) g(\eta) \exp{\eta^Tu(x)}$
<li> $\eta$ - "natural parameters" of a distribution
<li> $g(\eta)$ is a normalization constant that ensures that $\int h(x)\exp{\eta^Tu(x)}dx = 1$
<li> Express Bernoulli as an exponential family distribution $\theta^H(1-\theta)^H = \exp{H\ln{\theta} + (1-H)\ln{1-\theta)}} = (1-\theta) \exp{\ln{\frac{\theta}{1-\theta}}H}$
</ul>
</aside>
</section>
</section>
<section>
<section>
<h2>MLE and KL-divergence</h2>
</section>
<section>
<h2>How to measure Information</h2>
<ul style="font-size: 36px;">
<li class="fragment roll-in"> Messages are strings of characters from a fixed alphabet.
<li class="fragment roll-in"> The amount of information contained in a message should be a
function of the total number of possible messages.
<li class="fragment roll-in"> If you have an alphabet with $s$ symbols, then there are
$s^\ell$ messages of length, $\ell$.
<li class="fragment roll-in"> The amount of information contained in two messages should be
the sum of the information contained in the individual messages.
<li class="fragment roll-in"> The amount of information in $\ell$ messages of length one
should equal the amount of information in one message of length $\ell$.
</ul>
<aside class="notes">
<ul>
<li> In communication we exchange signals to convey messages
<li> Intuitively, a message is not always equal to another message
<li> Colloquially we speak of the messages conveying some information, but what is information and how can we measure it?
<li> If you think about it for some time, you'll arrive at the following requirements on this hypothetical information measure under the following assumptions.
</ul>
</aside>
</section>
<section>
<h2>Hartley's Information (1928)</h2>
<blockquote style="width: 100%">
The only function which satisfies these requirements:
\[
\ell \log(s) = \log(s^\ell)
\]
</blockquote>
</section>
<section>
<h2>Shannon's entropy (1948)</h2>
<blockquote style="background-color: #93a1a1; color: #fdf6e3; font-size: 38px; width: 100%;" class="fragment" data-fragment-index="0">
Let $X$ be a discrete random variable with $n$ outcomes,
$\{x_1,...,x_n\}$. The probability that the outcome will be $x_i$ is
$p(x_i)$. The <emph>average information</emph> (or <emph>entropy</emph>)
contained in a message about the outcome of $X$ is:
</blockquote>
<blockquote class="fragment" data-fragment-index="1">
\[
H_p = -\sum_{i=1}^n p_X(x_i) \log p_X(x_i)
\]
</blockquote>
<aside class="notes">
von Neumann told him firmly: "You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one really knows what entropy really is, so in a debate you will always have the advantage."
</aside>
</section>
<section>
<h2>Cross Entropy</h2>
<blockquote>
\[
H_{p,q} = -\sum_{i=1}^n p_X(x_i) \log q_X(x_i)
\]
</blockquote>
<aside class="notes">
Number of bits needed to send a message containing symbols drawn from probability distribution p, when we use a code that was designed to minimize the length of the messages drawn from probability distribution q
</aside>
</section>
<section>
<h2>Kullback-Leibler (KL) divergence</h2>
<blockquote class="fragment" data-fragment-index="0">
\[
D_{\rm KL} (P\|Q) = \int P(x) \log \frac{P(x)}{Q(x)}
\]
</blockquote>
<blockquote class="fragment" data-fragment-index="1">
\[
D_{\rm KL} (P\|Q) = \EE_{X\sim P} \left[ \log \frac{P(x)}{Q(x)} \right]
\]
</blockquote>
<div class="fragment" data-fragment-index="2">
\[
D_{\rm KL} (P\|Q) = \EE_{X\sim P} \log P(x) - \EE_{X\sim P} \log Q(x)
\]
</blockquote>
</div>
<aside class="notes">
Is an extra Number of bits needed to send a message containing symbols drawn from probability distribution p, when we use a code that was designed to minimize the length of the messages drawn from probability distribution q <br>
Cross entropy
</aside>
</section>
<section>
<h3>KL divergence is not symmetric</h3>
<img style="border:0; box-shadow: 0px 0px 0px rgba(150, 150, 150, 1); " width="100%"
src="figures/bimodal_KL.png" alt="KLD">
https://www.cs.toronto.edu/~duvenaud/distill_bayes_net/public/
</section>
<section>
<h3>MLE is KL-divergence minimization</h3>
<ul style="list-style-type: none;">
<li class="fragment roll-in">
$
\hat{\theta}_{MLE} = \underset{\theta}{\argmax} \prob{Q}{D|\theta}
$
<li class="fragment roll-in">
$
\hat{\theta}_{MLE} = \underset{\theta}{\argmax} \prod_{i=1}^{n} \prob{Q}{x_i|\theta}
$
<li class="fragment roll-in">
$
\hat{\theta}_{MLE} = \underset{\theta}{\argmax} \sum_{i=1}^{n} \log \prob{Q}{x_i|\theta}
$
<li class="fragment roll-in">
$
\hat{\theta}_{MLE} = \underset{\theta}{\argmax} \EE_{X\sim P} \log \prob{Q}{X|\theta}
$
<li class="fragment roll-in">
$
D_{\rm KL} (P\|Q) = \EE_{X\sim P} \log P(x) - \EE_{X\sim P} \log Q(x)
$
</ul>
</section>
</section>
</div>
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