Course: STAT 25250
Title: Fundamentals of Probability in Statistical Inference
Instructor: Promit Ghosal
Class Schedule: Sec 1: MW 3:00 PM–4:20 PM in TBA
Description: This course provides the necessary probabilistic foundation for tackling statistical problems and explores how these concepts can be applied to real-world data. Beginning with an introduction to probability and statistical inference, the course develops a framework for understanding convergence of probability measures, including convergence in distribution, probability, and almost sure convergence. We examine the Law of Large Numbers and the Central Limit Theorem, emphasizing their importance in statistical inference and real-world applications. Markov Chains are introduced with a focus on their recurrence, transience, and ergodicity, along with their applications in sampling techniques like Markov Chain Monte Carlo and simulated annealing. Methods for parameter estimation are explored in depth. The efficiency and asymptotic properties of estimators are discussed using tools such as the Cramer-Rao bound. Hypothesis testing methods, including Wald tests, t-tests, and goodness-of-fit tests, are studied in detail, with particular emphasis on their asymptotic consistency and power analysis. Bayesian approaches, such as prior and posterior distributions, Bayesian confidence regions, and the Bernstein-von Mises Theorem, are also covered. The course concludes with problems in regression methods, including linear models, least-squares estimation, inference, and a brief exploration of high-dimensional linear regression. Throughout the course, real data examples will be used to demonstrate the practical applications of these probabilistic and statistical concepts. This course is ideal for students seeking a solid probabilistic foundation for statistical problems analysis, with a focus on practical insights and the theoretical rigor of asymptotic properties.
A working knowledge of probability concepts, including expectation, variance, and various distributions, is required. A foundational understanding of linear algebra, including proficiency with vectors, matrices, and eigenvalues, is highly recommended.