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In order to estimate model parameters and circumvent possible difficultiesencountered with the likelihood function, we propose to replace the likelihoodin the formula of the posterior distribution by a function depending on acontrast. The properties of the contrast-based (CB) posterior distribution andMAP estimator are studied to understand what the consequences of incorporatinga contrast in the Bayesian formula are. We show that the proposed method can beused to make frequentist inference and allows the reduction of analyticalcalculations to get the limit variance matrix of the estimator. For specificcontrasts, the CB--posterior distribution directly approximates the limitdistribution of the estimator; the calculation of the limit variance matrix isthen avoided. Moreover, for these contrasts, the CB--posterior distribution canalso be used to make inference in the Bayesian way. The method is applied tothree spatial data sets.

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