Backward specification of prior in Bayesian inference as an inverse problem

Specification of prior distribution is one of the most important methodological as well practical problems in Bayesian inference. Although a number of approaches have been proposed, none of them is completely satisfactory from both theoretical and practical points of view. We propose a new method to infer prior distribution from a priori information which may be available from observations. The method consists of specifying a predictive distribution of the value of interest and then working backwards towards the prior distribution on the parameters. The method requires the solution of the Fredholm integral equation of the first kind, which can be effectively approximated using Tikhonov regularization. Numerical examples for two cases of Bayesian inference are presented.