ims.09

IMS

Session Slot: 10:30-12:20 Wednesday

Estimated Audience Size:

AudioVisual Request: Two Overheads

Session Title: Wald Lecture III

Theme Session: No

Applied Session: No

Session Organizer: Lindsay, Bruce The Pennsylvania State University

Address: 422 Thomas Building, Department of Statistics, University Park, PA16802

Phone: (814)865-1220

Fax: (814) 863-7114

Email: bgl@psu.edu

Session Timing: 110 minutes total (Sorry about format):

Opening Remarks by Chair - 5 First Speaker - 90 minutes

Floor Discussion - 10 minutes

Session Chair: Owen, Art Stanford University

Address: Department of Statistics Stanford University Stanford, CA 94305

Phone:

Fax:

Email: art@stat.Stanford.EDU

1. De Finetti's Theorem

Freedman, David,   University of California at Berkeley

Address: Department of Statistics University of California, Berkeley, CA 94720-4735

Phone: 510-642-2781

Fax:

Email: freedman@stat.berkeley.edu

Abstract: Consider an infinite sequence of independent, identically distributed (IID) random variables; for simplicity, suppose each variable takes only finitely many values. Plainly, the variables are ``exchangeable,'' in the sense that permuting them does not affect their joint distribution. (Multiplication is, after all, commutative.) Consider next a Bayesian statistician whose predictive distribution--in advance of data collection--can be described as follows: (i) a marginal distribution is chosen at random, (ii) a sequence of independent random draws are made from this marginal. Such a predictive distribution is called a ``mixture of IID processes.'' And any such mixture must also be exchangeable. (The expected value of a constant is constant.) A celebrated theorem of de Finetti's asserts the converse: any exchangeable distribution is a mixture of IID processes.

De Finetti's theorem has been generalized in many different directions. For example, the state space can be the real line (or any complete, separable metric space). Mixtures of specific parametric families of distributions can be characterized. So can mixtures of Markov chains. Bounds can be given for finite sequences of random variables. I will review some of the work on de Finetti's theorem.

List of speakers who are nonmembers: None