**COURSE DESCRIPTION **

The course covers Bayesian statistical methods for inference and prediction including: estimation; model selection and prediction; exchangeability; prior, likelihood, posterior and predictive distributions; coherence and calibration; conjugate analysis; Markov Chain Monte Carlo methods for simulation-based computation; hierarchical modeling; Bayesian modeldiagnostics, model selection and sensitivity analysis.

**PREREQUISITES**

AMS-203. Enrollment restricted to graduate students.

**CLASS SCHEDULE **

Below is the tentative class schedule. Please check the online schedule as the list of topics and/or the dates when such topics will be covered may change.

DATE TOPICS

01/08 Introduction.

01/10 Review of distributions. Exponential families. Conjugate analysis.

01/15 Conjugate analysis.

01/17 Foundations: A decision theoretic approach to statistical inference.

01/22 Foundations: Priors and the subjective interpretation of probability. Exchangeability.

01/24 Bayesian point and interval estimation. Loss functions. Admissibility of

Bayes estimators. Shrikage Priors. Bias and MSE of Bayes estimators.

01/29 More foundational issues. The likelihood principle. Sufficiency.

Exchangeability revisited. Sequential updating.

01/31 More general priors and asymptotic properties of Bayesian estimates. Discrete case.

Continuous case: Laplace expansions & “Bayesian CLT”. Approx inference.

02/05 EXAM 1

02/07 More general priors: Simulation-based inference. MC integration.

Generating pseudo-random variables (inversion, transformation, rejection).

02/12 More on random variate generation.

02/14 Simulation-based inference: Importance sampling, Metropolis Hastings.

02/19 Simulation-based inference: Gibbs sampling.

02/21 Objective Bayes and estimation problems.

02/26 Bayesian hypothesis testing and model comparison.

02/28 Simulation-based methods for model comparison.

03/05 Objective Bayes and hypothesis testing.

03/07 EXAM 2

03/12 Hierarchical modeling and mixture representations. Data augmentation.

03/14 Complementary topics.

03/22 FINAL EXAM 12-3pm

**BIBLIOGRAPHY**

- Robert, C. (2007) The Bayesian Choice. Second Edition, Springer Verlag: New York. Available electronically, UCSC library
- Hoff, P. (2009) A First Course in Bayesian Statistical Methods. Springer: New York. Available electronically, UCSC library.
- Berger J.O. (1984) Statistical Decision Theory and Bayesian Analysis. Springer.
- Bernardo, J.M. and Smith, A.F.M. (2000) Bayesian Theory.

Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B. (2003). Bayesian Data Analysis. Second Edition. Chapman & Hall.

**EVALUATION**

* Homework.* There will be about 3-5 homework assignments. Some of the problems in the homework assignments will be graded. In addition, the exams will be based on the homework assignments so it is very important that you solve

provided by the instructor.

__ Exams.__ There will be two exams and a final exam. All the exams will have an in class part and some may have a take home part (probably only the final).

__ Your Grade.__ Your course grade will be based on the exams and homework assignments as follows: (a) Homework (25\%); (b) Exam 1 (20\%); (c) Exam 2 (25\%); (d) Final (30\%).