Annals of Mathematical Sciences and Applications

Volume 3 (2018)

Number 1

Special issue in honor of Professor David Mumford, dedicated to the memory of Jennifer Mumford

Guest Editors: Stuart Geman, David Gu, Stanley Osher, Chi-Wang Shu, Yang Wang, and Shing-Tung Yau

Bayesian goodness of fit tests: a conversation for David Mumford

Pages: 287 – 308



Persi Diaconis (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)

Guanyang Wang (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)


The problem of making practical, useful goodness of fit tests in the Bayesian paradigm is largely open. We introduce a class of special cases (testing for uniformity: have the cards been shuffled enough; does my random generator work) and a class of sensible Bayes tests inspired by Mumford, Wu and Zhu. Calculating these tests presents the challenge of ‘doubly intractable distributions’. In present circumstances, modern MCMC techniques are up to the challenge. But many other problems remain. Our paper is didactic, we hope to induce the reader to help take it further.


Bayes test, MCMC, doubly intractable

The research of Persi Diaconis was partially supported by NSF Grant DMS-1208775.

Received 14 November 2017

Published 27 March 2018