Optimal compromise between incompatible conditional probability distributions, with application to Objective Bayesian Kriging

¹ EDF R&D, Dpt. PRISME, 6 quai Watier, 78401 Chatou, France.
² Université Paris Diderot, Laboratoire de Probabilités, Statistique et Modélisation, France.

^* Corresponding author: joseph.mure@edf.fr

Received: 11 October 2017
Accepted: 31 October 2018

Abstract

Models are often defined through conditional rather than joint distributions, but it can be difficult to check whether the conditional distributions are compatible, i.e. whether there exists a joint probability distribution which generates them. When they are compatible, a Gibbs sampler can be used to sample from this joint distribution. When they are not, the Gibbs sampling algorithm may still be applied, resulting in a “pseudo-Gibbs sampler”. We show its stationary probability distribution to be the optimal compromise between the conditional distributions, in the sense that it minimizes a mean squared misfit between them and its own conditional distributions. This allows us to perform Objective Bayesian analysis of correlation parameters in Kriging models by using univariate conditional Jeffreys-rule posterior distributions instead of the widely used multivariate Jeffreys-rule posterior. This strategy makes the full-Bayesian procedure tractable. Numerical examples show it has near-optimal frequentist performance in terms of prediction interval coverage.