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

## TL;DR

This paper introduces a method to find the optimal joint distribution that best fits incompatible conditional distributions, enabling practical Bayesian Kriging analysis with improved computational tractability and reliable prediction intervals.

## Contribution

It proposes a novel approach to approximate incompatible conditionals with an optimal compromise distribution, facilitating objective Bayesian inference in Kriging models.

## Key findings

- The pseudo-Gibbs sampler converges to the optimal compromise distribution.
- The method achieves near-optimal frequentist coverage in prediction intervals.
- It simplifies Bayesian correlation parameter estimation in Kriging models.

## 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.