Statistics and Information Theory for Machine Learning

This crash course provides the foundational knowledge in statistics and information theory necessary to delve deeper into advanced machine learning topics, particularly those related to optimization, model evaluation, and the geometric interpretation of learning algorithms. It builds upon a foundation of calculus and linear algebra, and familiarity with tensor calculus can be beneficial for multivariate concepts.

This course is structured to first establish the statistical principles needed to work with data and models, and then to introduce information-theoretic measures that quantify uncertainty, similarity between distributions, and the information content relevant for learning and optimization.

Course Structure:

1. Part 1: Statistical Foundations for Machine Learning: This part covers fundamental probability theory, the concept of random variables, essential probability distributions encountered in machine learning, key limit theorems, and the principles of statistical estimation, with a focus on Maximum Likelihood Estimation (MLE).

2. Part 2: Information Theory Essentials for Machine Learning: Building on the statistical foundations, this part introduces core concepts from information theory such as entropy (to quantify uncertainty), mutual information (to measure variable dependence), Kullback-Leibler (KL) divergence (to compare distributions), and cross-entropy (a common loss function). It culminates in an exploration of Fisher Information, a pivotal concept for understanding the geometry of statistical models and its application in advanced optimization algorithms.

These topics are vital for grasping how uncertainty is modeled, how information is quantified, and how these concepts underpin the behavior and design of learning algorithms discussed in the main series on Mathematical Optimization in ML, especially for understanding Information Geometry.