Incorporating a contrast in the Bayesian formula: What consequences for the MAP estimator and the posterior distribution? Applications in spatial statistics

TL;DR

This paper introduces a contrast-based modification to the Bayesian posterior, enabling both Bayesian and frequentist inference, simplifying calculations, and demonstrating applications in spatial statistics.

Contribution

It proposes replacing the likelihood with a contrast function in Bayesian inference, bridging Bayesian and frequentist methods, and simplifying variance estimation.

Findings

The contrast-based posterior can approximate the limit distribution of estimators.

The method facilitates Bayesian and frequentist inference using the same framework.

Applications to spatial data demonstrate practical utility.

Abstract

In order to estimate model parameters and circumvent possible difficulties encountered with the likelihood function, we propose to replace the likelihood in the formula of the posterior distribution by a function depending on a contrast. The properties of the contrast-based (CB) posterior distribution and MAP estimator are studied to understand what the consequences of incorporating a contrast in the Bayesian formula are. We show that the proposed method can be used to make frequentist inference and allows the reduction of analytical calculations to get the limit variance matrix of the estimator. For specific contrasts, the CB--posterior distribution directly approximates the limit distribution of the estimator; the calculation of the limit variance matrix is then avoided. Moreover, for these contrasts, the CB--posterior distribution can also be used to make inference in the Bayesian way. The method is applied to three spatial data sets.