Bayesian evaluation of non-admissible conditioning: the case of Fisher test

We first analyse the general problem of admissible conditioning and next consider the evaluation of the loss of information when a non-admissible conditioning is used as an approximaton of the exact posterior distribution. Considering the case of Fisher test, we evaluate from a Bayesian point of view how much information is lost when the sampling process for a 2x2 contingency table is analysed conditionally on the two margins. This loss of information due to non-admissible conditioning is evaluated for different sampling models and with respect to the entropy divergence and to the Hellinger distance between the exact and the approximate posterior distributions and with respect to relative risks based on a quadratic loss function. The numerical results obtained through simulation indicate that for a specific range of parameters the loss of information increases with the sample size and decreases with the precision of the a priori distribution. Hence such an approximation is shown to be a non-asymptotic one