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🧮 Algorithm Title

🎯 Objective

  • Example: "This is a K-Nearest Neighbors (KNN) classifier algorithm used for classifying data points based on their proximity to other points in the dataset."

📚 Prerequisites

  • Linear Algebra Basics
  • Probability and Statistics
  • Libraries: NumPy, TensorFlow, PyTorch (as applicable)

🧩 Inputs

  • Example: The input dataset should be in CSV format with features and labels for supervised learning algorithms.

📤 Outputs

  • Example: The algorithm returns a predicted class label or a regression value for each input sample.

🏛️ Algorithm Architecture

  • Example: "The neural network consists of 3 layers: an input layer, one hidden layer with 128 units, and an output layer with 10 units for classification."

🏋️‍♂️ Training Process

  • Example:
    • The model is trained using the gradient descent optimizer.
    • Learning rate: 0.01
    • Batch size: 32
    • Number of epochs: 50
    • Validation set: 20% of the training data

📊 Evaluation Metrics

  • Example: "Accuracy and F1-Score are used to evaluate the classification performance of the model. Cross-validation is used to reduce overfitting."

💻 Code Implementation

# Example: Bayesian Regression implementation

import numpy as np
from sklearn.linear_model import BayesianRidge
import matplotlib.pyplot as plt

# Generate Synthetic Data
np.random.seed(42)
X = np.random.rand(20, 1) * 10
y = 3 * X.squeeze() + np.random.randn(20) * 2

# Initialize and Train Bayesian Ridge Regression
model = BayesianRidge(alpha_1=1e-6, lambda_1=1e-6, compute_score=True)
model.fit(X, y)

# Make Predictions
X_test = np.linspace(0, 10, 100).reshape(-1, 1)
y_pred, y_std = model.predict(X_test, return_std=True)

# Display Results
print("Coefficients:", model.coef_)
print("Intercept:", model.intercept_)

# Visualization
plt.figure(figsize=(8, 5))
plt.scatter(X, y, color="blue", label="Training Data")
plt.plot(X_test, y_pred, color="red", label="Mean Prediction")
plt.fill_between(
    X_test.squeeze(),
    y_pred - y_std,
    y_pred + y_std,
    color="orange",
    alpha=0.3,
    label="Predictive Uncertainty",
)
plt.title("Bayesian Regression with Predictive Uncertainty")
plt.xlabel("X")
plt.ylabel("y")
plt.legend()
plt.show()

🔍 Scratch Code Explanation

Bayesian Regression is a probabilistic approach to linear regression that incorporates prior beliefs and updates these beliefs based on observed data to form posterior distributions of the model parameters. Below is a breakdown of the implementation, structured for clarity and understanding.

1. Class Constructor: Initialization

class BayesianRegression:
    def __init__(self, alpha=1, beta=1):
        """
        Constructor for the BayesianRegression class.

        Parameters:
        - alpha: Prior precision (controls the weight of the prior belief).
        - beta: Noise precision (inverse of noise variance in the data).
        """
        self.alpha = alpha
        self.beta = beta
        self.w_mean = None
        self.w_precision = None
  • Key Idea
    • The alpha (Prior precision, representing our belief in the model parameters' variability) and beta (Precision of the noise in the data) hyperparameters are crucial to controlling the Bayesian framework. A higher alpha means stronger prior belief in smaller weights, while beta controls the confidence in the noise level of the observations.
    • w_mean - Posterior mean of weights (initialized as None)
    • w_precision - Posterior precision matrix (initialized as None)

2. Fitting the Model: Bayesian Learning

def fit(self, X, y):
    """
    Fit the Bayesian Regression model to the input data.

    Parameters:
    - X: Input features (numpy array of shape [n_samples, n_features]).
    - y: Target values (numpy array of shape [n_samples]).
    """
    # Add a bias term to X for intercept handling.
    X = np.c_[np.ones(X.shape[0]), X]

    # Compute the posterior precision matrix.
    self.w_precision = (
        self.alpha * np.eye(X.shape[1])  # Prior contribution.
        + self.beta * X.T @ X  # Data contribution.
    )

    # Compute the posterior mean of the weights.
    self.w_mean = np.linalg.solve(self.w_precision, self.beta * X.T @ y)

Key Steps in the Fitting Process

  1. Add Bias Term: The bias term (column of ones) is added to X to account for the intercept in the linear model.

  2. Posterior Precision Matrix: $$ \mathbf{S}_w^{-1} = \alpha \mathbf{I} + \beta \mathbf{X}^\top \mathbf{X} $$

    • The prior contributes (\alpha \mathbf{I}), which regularizes the weights.
    • The likelihood contributes (\beta \mathbf{X}^\top \mathbf{X}), based on the observed data.

  3. Posterior Mean of Weights: $$ \mathbf{m}_w = \mathbf{S}_w \beta \mathbf{X}^\top \mathbf{y} $$

    • This reflects the most probable weights under the posterior distribution, balancing prior beliefs and observed data.

3. Making Predictions: Posterior Inference

def predict(self, X):
    """
    Make predictions on new data.

    Parameters:
    - X: Input features for prediction (numpy array of shape [n_samples, n_features]).

    Returns:
    - Predicted values (numpy array of shape [n_samples]).
    """
    # Add a bias term to X for intercept handling.
    X = np.c_[np.ones(X.shape[0]), X]

    # Compute the mean of the predictions using the posterior mean of weights.
    y_pred = X @ self.w_mean

    return y_pred

Key Prediction Details

  1. Adding Bias Term: The bias term ensures that predictions account for the intercept term in the model.
  2. Posterior Predictive Mean: $$ \hat{\mathbf{y}} = \mathbf{X} \mathbf{m}_w $$
    • This computes the expected value of the targets using the posterior mean of the weights.

4. Code Walkthrough

  • Posterior Precision Matrix ((\mathbf{S}_w^{-1})): Balances the prior ((\alpha \mathbf{I})) and the data ((\beta \mathbf{X}^\top \mathbf{X})) to regularize and incorporate observed evidence.
  • Posterior Mean ((\mathbf{m}_w)): Encodes the most likely parameter values given the data and prior.
  • Prediction ((\hat{\mathbf{y}})): Uses the posterior mean to infer new outputs, accounting for both prior knowledge and learned data trends.

🛠️ Example Usage

# Example Data
X = np.array([[1.0], [2.0], [3.0]])  # Features
y = np.array([2.0, 4.0, 6.0])        # Targets

# Initialize and Train Model
model = BayesianRegression(alpha=1.0, beta=1.0)
model.fit(X, y)

# Predict on New Data
X_new = np.array([[4.0], [5.0]])
y_pred = model.predict(X_new)

print(f"Predictions: {y_pred}")
  • Explanation
    • A small dataset is provided where the relationship between (X) and (y) is linear.
    • The model fits this data by learning posterior distributions of the weights.
    • Predictions are made for new inputs using the learned posterior mean.

🌟 Advantages

  • Encodes uncertainty explicitly, providing confidence intervals for predictions.
  • Regularization is naturally incorporated through prior distributions.
  • Handles small datasets effectively by leveraging prior knowledge.

⚠️ Limitations

  • Computationally intensive for high-dimensional data due to matrix inversions.
  • Sensitive to prior hyperparameters ((\alpha, \beta)).

🚀 Application

=== "Application 1" Explain your application

=== "Application 2" Explain your application