2 00 8 Incorporating a contrast in the Bayesian formula : What 1 consequences for the MAP estimator and the posterior 2 distribution ? Applications in spatial statistics

In order to estimate model parameters and circumvent possible dif-6 ficulties encountered with the likelihood function, we propose to replace the like-7 lihood in the formula of the posterior distribution by a function depending on a 8 contrast. The properties of the contrast-based (CB) posterior distribution and 9 MAP estimator are studied to understand what the consequences of incorporat-10 ing a contrast in the Bayesian formula are. We show that the proposed method 11 can be used to make frequentist inference and allows the reduction of analytical 12 calculations to get the limit variance matrix of the estimator. For specific con-13 trasts, the CB–posterior distribution directly approximates the limit distribution 14 of the estimator; the calculation of the limit variance matrix is then avoided. 1 make inference in the Bayesian way. The method is applied to three spatial data 1 sets.