Stat 180/236

Introduction to Bayesian Statistics

Bayesian statistics enjoys today considerable popularity, as novel powerful computing tools have eased the application of this elegant theory. Indeed, almost any statistician finds himself doing some "Bayesian statistics" occasionally, making it a necessity for anybody seriously interested in the field to acquire first hand knowledge of this approach. The goal of this course is precisely to introduce statistical inference as it can be carried out using the Bayes theorem. We will consider foundational aspects of statistics as well as computational issues encountered in applications in order to provide a sound and complete picture of Bayesian inference. As the number of courses offered at UCLA and covering the Bayesian approach to statistics is increasing, to avoid overlap with others we will pay particular attention to the understanding of the basis of Bayesian inference. The emphasis will be on clear understanding of the concept presented and of their implication for the practice of statistics. Hence, the examples will be chosen so to keep the mathematics simple and to illustrate problems of actual relevance (image reconstruction, protein alignment, etc. ) We will discuss Stein's paradox, non parametric Bayes and statistical learning. This is an upper-level undergraduate course: on the one hand, a previous knowledge of probability and calculus will be assumed; on the other hand, there will not be space for things like a high-powered treatment of decision theory or MCMC techniques and too challenging computations. The course can be taken for graduate credit by complementing the requirements with a data analysis project.

Following is a tentative syllabus. As the course progress, we will precise the topics covered to date, grouping them by week of instruction.

  • The goals of Bayesian statistics. Subjective interpretation of probability. Bayes theorem.
  • The concepts and methods of Bayesian statistics as they appear in Bayes's writings and their generalization with a decision theory framework. Choice of prior distributions and role of predictive distributions.
  • Prior information on parameters (states of nature). Likelihood of the parameter based on data. Posterior distribution of the parameter. Loss and risk functions. Bayesian risk. Estimation, testing, classification and different loss functions.
  • Bayesian calculus. Working out the examples of Normal-normal, gamma-normal and beta-binomial, etc.
  • Choice of the prior distribution and of the statistical model. Exchangeability. Non informative priors. Properties of the posterior distribution.
  • Hierarchical models. Empirical Bayes. Stein's paradox.
  • Model selection. Notions of smoothness, roughness penalty.
  • Non parametric Bayes
  • Bayesian image reconstruction. An example of maximum a-posteriori estimate and the use of the Gibbs sampler.

NOTE there will be no class during the following days: October 6, November 10 and November 17 . Instead, we will meet on the thursday of the respective weeks at 1pm in Boelter 5514.

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