DEPARTMENT OF
APPLIED MATHEMATICS & STATISTICS
(AMS)
Bayesian Model Specification and Bayesian Hierarchical Modeling
A foundational theorem, proved independently by the statistician and actuary Bruno de Finetti (1937) and the physicist Richard Cox (1946), says that if You wish to summarize Your information about θ, in light of D and B, in a logically-consistent manner, then You need to specify two ingredients, for inference about θ and prediction of future data D^{*} -- namely, p ( θ | B ) (Your prior distribution for θ) and p ( D | θ B ) (Your sampling distribution for D) -- and two more ingredients for decision-making: a set A of possible actions and a utility function U ( a, θ ) quantifying Your judgments about the rewards (monetary or otherwise) that would ensue if You chose action a and the unknown actually took the value θ. This theorem then goes on, through a series of corollaries, to tell You how these ingredients should be combined for logically-consistent inference, prediction and decision-making, using the basic rules we examined in the first short course in Oct 2010 (for example, Bayes's Theorem for inference).
The de Finetti-Cox theorem is great as far as it goes -- it tells You which four ingredients You have to specify -- but it's almost entirely silent on how to specify those ingredients. The first half of this short course will be about how to perform Bayesian model specification: how to go from {the context of the problem and the design of the data-gathering process} to good choices for the ingredients mentioned above.
Let's agree to call { p ( θ | B ), p ( D | θ B ) } Your model M for Your uncertainty about θ -- such a model is needed for inference and prediction. In this short course I'll concentrate mainly on specifying M, not because decision-theory is unimportant but for lack of time to do justice to both inference/prediction and decision-making. Topics to be covered on model specification will include the following:
Another example involves the discipline of meta-analysis, in which there are a number of studies of the same basic phenomenon and Your goal is to combine information across studies to produce a more accurate answer than that obtainable from any single study.
Even when Your data set is not hierarchical, there are many situations in which You'll find it useful to employ Bayesian hierarchical modeling, especially when You want to build a mixture model. Mixture models are helpful in settings (which occur frequently) in which You discover that Your data values exhibit unexplained heterogeneity and You want to realistically describe this heterogeneity to produce well-calibrated inferences and predictions.
As usual I'll motivate each unit of methodology in the context of a case study. The case studies in this short course will be drawn from health policy, medicine and the physical sciences, but the methods illustrated will apply to a broad range of subject areas in the natural and social sciences, business (including topics of direct relevance to pharmaceutical companies), and public policy.
The course will liberally illustrate user-friendly implementations of the methods covered via the freeware Bayesian analysis program WinBUGS and the freeware statistical computing package R.
The course is intended mainly for people who often use statistics in their research; some graduate coursework in statistics will provide sufficient mathematical background for participants, and -- since this is a follow-on course from the one a year ago -- I'll assume some familiarity with Bayesian reasoning and methods. To get the most out of the course, participants should be comfortable with hearing the course presenter mention (at least briefly) (a) differentiation and integration of functions of several variables and (b) discrete and continuous probability distributions (joint, marginal, and conditional) for several variables at a time, but all necessary concepts will be approached in a sufficiently intuitive manner that rustiness on these topics will not prevent understanding of the key ideas.
He is a Fellow of the American Association for the Advancement of Science, the American Statistical Association (ASA), the Institute of Mathematical Statistics, and the Royal Statistical Society; from 2001 to 2003 he served as the President-Elect, President, and Past President of the International Society for Bayesian Analysis (ISBA).
He is the author or co-author of more than 100 contributions to the methodological and applied statistical literature, including articles in the Journal of the Royal Statistical Society (Series A, B and C), the Journal of the American Statistical Association, the Annals of Applied Statistics, Bayesian Analysis, Statistical Science, the New England Journal of Medicine, and the Journal of the American Medical Association; his 1995 JRSS-B article on assessment and propagation of model uncertainty has been cited more than 900 times.
His research is in the areas of Bayesian inference and prediction, model uncertainty and empirical model-building, hierarchical modeling, Markov Chain Monte Carlo methods, and Bayesian nonparametric methods, with applications mainly in medicine, health policy, education, and environmental risk assessment.
When he gave an earlier version of the Oct 2010 short course at the Anaheim Joint Statistical Meetings (JSM) in 1997 it received the 1998 ASA Excellence in Continuing Education Award, and a short course he gave on intermediate and advanced-level topics in Bayesian hierarchical modeling at the San Francisco JSM in 2003 received the 2004 ASA Excellence in Continuing Education Award.
He has won or been nominated for major teaching awards everywhere he has taught (the University of Chicago; the RAND Graduate School of Public Policy Studies; the University of California, Los Angeles; the University of Bath (UK); and the University of California, Santa Cruz).
He has a particular interest in the exposition of complex statistical methods and ideas in the context of real-world applications.
(last modified: 6 November 2011)