Beijing Institute of Mathematical Sciences and Applications

16th September ~ 16th December, 2025

calculus, linear algebra, probability

Introduction to Bayesian Statistics
Instructor: Yongtao Guan
Email: ytguan@bimsa.cn
Term: Fall 2025
Class Meetings: Tuesdays & Thursdays, 16:05–17:40

Course Description
This course provides an introduction to Bayesian statistics, covering foundational concepts,
decision theory, prior and posterior distributions, hypothesis testing, and applications. The course
follows Statistical Decision Theory and Bayesian Analysis by James O. Berger (Chapters 1–7),
supplemented with discussions on Bayesian resolutions to statistical paradoxes and real-world in
terpretations (e.g., medical testing for rare diseases).

Prerequisites
Basic calculus, linear algebra, Probability

Textbook
Primary: Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis.
Supplementary: Leonard J. Savage (1954). The Foundations of Statistics (second revised edition).

Course Schedule (12 Weeks, 24 Lecs)
Week 1: Foundations of Bayesian Inference
1 Frequentist vs. Bayesian; Bayes’ theorem; Basic examples (coin flips, medical tests).
2 Subjective probability and degrees of belief; Likelihood principle (Berger Ch. 1); Conjugate priors (Beta-Binomial model).
Week 2: Decision Theory Basics
1 Framework of statistical decision theory (Ch. 2); Loss functions, risk, and admissibility.
2 Minimax and Bayes rules; Example: Point estimation under squared error loss.
Week 3: Prior Distributions & Elicitation
1 Noninformative priors (Berger Ch. 3); Jeffreys’ prior, reference priors.
2 Informative priors: Elicitation from experts; Hierarchical priors (brief introduction)
Week 4: Bayesian Inference & Posterior Analysis
1 Posterior distributions and conjugacy (Berger Ch. 4); Normal-Normal model.
2 Credible intervals vs. confidence intervals; Predictive distributions.
Week 5: Hypothesis Testing & Model Comparison–
1 Bayesian hypothesis testing (Berger Ch. 4 & 7); Bayes factors.
2 Interpretation of p-values vs. posterior probabilities; Lindley’s paradox.
Week 6: Bayesian Computation (Introduction)
1 Analytical vs. computational methods; Grid approximation, rejection sampling.
2 Introduction to MCMC (Metropolis-Hastings).
Week 7: Bayesian Interpretations of Paradoxes–
1 Monty Hall problem (Bayesian perspective); Simpson’s paradox.
2 Medical testing paradox (false positives in rare diseases); Base rate neglect.
Week 8: Bayesian Linear Models
1 Bayesian linear regression (Berger Ch. 4); Conjugate priors for regression.
2 Comparison with frequentist OLS; Predictive performance.
Week 9: Robustness & Sensitivity
1 Robust Bayesian analysis (Berger Ch. 4); Prior sensitivity checks.
2 Case study: How prior choice affects inference.
Week 10: Empirical Bayes & Shrinkage
1 Introduction to Empirical Bayes (Berger Ch. 4); James-Stein estimator.
2 Applications in A/B testing.
Week 11: Advanced Topics & Applications
1 Bayesian model averaging; Variable selection.
2 Bayesian networks (brief overview); Real-world applications (e.g., spam filtering).
Week 12: Review & Project Presentations
1 Course recap; Q&A on key topics.
2 Student presentations (mini-projects).

Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis (second edition).
Leonard J. Savage (1954). The Foundations of Statistics (second revised edition).

Undergraduate , Advanced Undergraduate , Graduate , Postdoc , Researcher

Yes

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Chinese , English