Posterior sampling in the presence of unknown normalising constants: An adaptive pseudo-marginal approach.

We consider a Bayesian setup where the data distribution is specified by an unnormalised density with an intractable normalising
constant. This precludes the possibility of using a simple Metropolis-Algorithm for sampling of the posterior. Assuming that it is possible to draw samples from the data distribution we device a
auxiliary variable technique for sampling the posterior(asymptotically). This technique eliminates the need for knowing the normalising constant of the data distribution. Our auxiliary variable technique is a special case of the pseudo marginal algorithm (Andrieu and Roberts, 2007) while extending the auxiliary variable method of Møller et al. (2006). The efficiency depends on the choice of the auxiliary variable. We avoid making this choice by applying a variation of the adaptive Markov chain Monte Carlo approach of Roberts and Rosenthal (2007).