Data Science: Bayesian Linear Regression in Python

the lecture shows that w_hat is normally distributed with mean w and variance sigma^2 over the sum of x_i^2, making it an unbiased estimator whose uncertainty decreases as data grows.

Perform a handwritten derivation of the linear regression estimator w_hat for y = w x + epsilon, and show its mean is w with variance sigma^2 over sum x_i^2.

Relate the squared error minimization to the maximum likelihood estimate by modeling y_i as normal with mean w x_i and variance sigma^2, showing that maximizing likelihood equals minimizing squared error.

Introduce map estimation by placing a normal prior on W and using Bayes rule to maximize the posterior instead of the likelihood, linking to L2 regularization and bias-variance tradeoff.

Explore how least squares equals maximum likelihood under normal errors, and how Bayesian map estimation with a normal prior yields shrinkage and a bias-variance trade-off.

Provide detailed feedback through the suggestion box form, sharing your background, course, difficulty, missing topics, and concrete examples to guide future course improvements.

Explore the bayesian approach to machine learning as a framework for deriving the bayesian version of linear regression, emphasizing fitting via the posterior of w and predictions as distributions.

Review conjugate priors for normal likelihood in Bayesian linear regression. Use a normal prior for w with mean zero and precision lambda to obtain posterior by proportionality, bypassing the denominator.

Derives the posterior for w in bayesian linear regression by combining normal prior and normal likelihood, yielding mu_w and sigma_w^2 with formulas involving sums of x_i^2 and x_i y_i.

Compute the posterior predictive distribution for y hat by integrating over uncertain weights w, instead of using the posterior mean, in bayesian linear regression.

Complete the square to turn a quadratic into a gaussian form, identify precision and mean, and simplify integrals through variable substitutions in Bayesian linear regression.

Show that the posterior for y hat is normal with mean mu_w x_hat and a variance combining model variance and w uncertainty, and that data reduce this variance toward sigma^2.

Derive the posterior of W in Bayesian linear regression using a Gaussian prior and likelihood, yielding a normal with mu_w and sigma_w^2 via precision updates.

This lecture develops the posterior predictive distribution for Bayesian linear regression on dog food, reducing to a normal with mean mu_w x_hat and variance sigma_w^2 x_hat^2 + sigma^2.

Derive the posterior distribution for the multivariate weight vector in bayesian linear regression, using a gaussian likelihood and a vector normal prior, to obtain the posterior mean and covariance.

Derive the posterior predictive distribution for multivariate Bayesian linear regression, yielding a normal with mean mu_w dot x_hat and variance sigma^2 plus x_hat transpose sigma_w x_hat.

Explore the prediction step in multivariate bayesian linear regression by integrating over the weight posterior to obtain the posterior predictive distribution for y hat given x hat and training data.

Explore how ChatGPT derives the posterior predictive distribution for Bayesian linear regression by treating the model as Gaussian and computing its mean and covariance via the law of total variance.

Prepare code for bayesian linear regression on an advertising dataset and learn to estimate unknown variance with least squares, use the precision matrix, and avoid inversion.

Implement Bayesian linear regression in Python, load and explore data, fit the posterior, and predict with mean and variance, comparing to classical methods using TV, radio, and newspaper ad spend.

Learn how to succeed in this course by asking questions in the q&a, verifying prerequisites, and implementing theory in code to connect background knowledge with practice.

Explore a skill-building roadmap that maps machine learning topics and their dependencies. Discover the progression from linear and logistic regression to deep learning, including neural networks, backpropagation, CNNs, and RNNs.

Discover the machine learning prerequisite roadmap from supervised and unsupervised learning to reinforcement learning, recurrent and convolutional networks, word embeddings, and ensemble methods in nlp paths.

Note that the Python notebooks aren’t on GitHub, log in, and use the official code link; turn off VPN to receive verification emails and resolve access issues.