GSO Spring Speaker 2017

What Bayes Did, and What Bayes Did Not Do

Arthur P. Dempster
Emeritus Professor of Theoretical Statistics
Department of Statistics, Harvard University

Venue: LWSN 1142

Abstract:

The basic probe model used for illustration in Thomas Bayes’s 1763 posthumous paper can be weakened by removing the controversial prior distribution. The example then becomes a useful illustration of basic features of the Dempster-Shafer (DS) theory of statistical inference. DS provides modeling and reasoning tools for advising users about whether assertions of fact are true or false. The weakening leads directly to relaxing the strict Bayesian requirement of “probabilities of everything”, while inferential probabilities p and q that quantify “surely true” and “surely false” are complemented by a new species of inferential probability r that quantifies “neither” option, signaling inadequacy of underlying evidence. Whereas the familiar or ordinary calculus of probability (OCP) has p + q = 1, DS theory has p + q + r = 1 creating an extended calculus of probability (ECP).

Scientific applications depend on basic choices among competing assumed models. Since these choices are proposed, discussed, and accepted, by analysts, they are personal in nature, so lead to personal probability inferences. Familiar inferential outputs such as significance tests and interval estimates are restated in DS terms. While mathematical and computational aspects of DS often require heavy lifting, much of the abstract framework of the ECP is essentially the same as that of the OCP, the difference being that the former distributes probability components called masses over subsets of a state space, while the latter distributes probabilities directly over singleton elements of the state space. Although unfamiliar to most statisticians, ECP mathematics is more streamlined than OCP mathematics, with only two basic operators, namely, projection and combination. Modern MCMC techniques developed for the OCP can be adapted for use with the ECP.