May 16, 2013

Medallion Lecture: Ya’acov Ritov

Ya’acov Ritov is professor in the Department of Statistics at the Hebrew University of Jerusalem. He received his PhD from the Hebrew University in 1983, and is a fellow of the IMS. Ya’acov’s (statistical) research interests include complex and large dimensional model, empirical Bayes procedures, semi- and non-parametric models. His Medallion Lecture is also at JSM, on Thursday August 8, at 8:30am (see below for the times and locations of other Medallion Lectures, as well as the Wald Lectures, the Rietz Lecture and the Presidential Address.)

A Priori Analysis of Complex Models

We (P.J. Bickel, A.C. Gamst, B.J.K. Kleijn, and Y. Ritov) study a few examples of Bayesian procedures on complex, high-dimensional parameter spaces. The Bayesian procedures we consider are those that adhere to the following paradigm. The prior distribution is announced prior to observing the data. At least we are restricted to priors that do not depend on details of the experimental design or on knowing the specific functions of the parameters that may turn out to be of interest. In this paradigm, it would not, for example, be reasonable for a statistician to use one prior for estimating one function, and another to estimate a different function. We shouldn’t be reminded of Groucho Marx’s quote, “Those are my principles, and if you don’t like them… well, I have others.”

Bayesian procedures can be considered from different points of view. Their closure is the set of admissible procedures, and they are known to generate asymptotic minimax procedures in regular parametric models. However, these claims are valid when the priors are selected to fit frequentist ad-hoc considerations.

Most early discussions of Bayesian analysis presented simple examples, e.g., X ~ N(ϑ, 1). In this case, a statistician might have clear a priori ideas about ϑ, and might well understand the implications of using his prior. Regardless, the data will eventually overwhelm the prior, and typically frequentist and Bayesian inference will coincide. The classical Bernstein–von Mises Theorem encapsulates this observation. Currently, Bayesian procedures are being applied to complex, high-dimensional models, e.g., those used in medical imaging. With a very high-dimensional parameter space (where, for example, laws of large numbers appear in surprising places), it is very difficult to understand the implications of using a particular prior. It is very difficult, if not impossible, to express subjective information about the model in a robust prior, and it is difficult to express this knowledge in a way that would support the data analysis and not dominate it.

We use several examples to illustrate a number of issues. This includes the partial linear model of Engle, Granger, Rice and Weiss (1986) , and different models in the very convenient lab of white noise series. We show that in situations where the nonparametric part of the model is smooth enough, the Bernstein–von Mises phenomenon holds and Bayesian estimators are efficient, but the Bayesian estimator is going to fail in extreme situations where simple frequentist estimation can still work. Then, it may argue that in a given white noise model, the any Bayesian prior would fail in estimation of some linear functional, while trivial frequentist estimator would not.

We also give an example in which Bayesian procedures which ignore the stopping time associated with the data generating process fail, while simple frequentist procedures continue to work. This demonstrates the danger of the classical principle that Bayesians need not pay attention to stopping times.

Share

Leave a comment

*

Share

Welcome!

Welcome to the IMS Bulletin website! We are developing the way we communicate news and information more effectively with members. The print Bulletin is still with us (free with IMS membership), and still available as a PDF to download, but in addition, we are placing some of the news, columns and articles on this blog site, which will allow you the opportunity to interact more. We are always keen to hear from IMS members, and encourage you to write articles and reports that other IMS members would find interesting. Contact the IMS Bulletin at bulletin@imstat.org

What is “Open Forum”?

In the Open Forum, any IMS member can propose a topic for discussion. Email your subject and an opening paragraph (to bulletin@imstat.org) and we'll post it to start off the discussion. Other readers can join in the debate by commenting on the post. Search other Open Forum posts by using the Open Forum category link below. Start a discussion today!

About IMS

The Institute of Mathematical Statistics is an international scholarly society devoted to the development and dissemination of the theory and applications of statistics and probability. We have about 4,500 members around the world. Visit IMS at http://imstat.org
Latest Issue