New accept/reject methods for independent sampling from posterior probability distributions

Rejection sampling (RS) is a well-known method to generate (pseudo-)random samples from arbitrary probability distributions that enjoys important applications, either by itself or as a tool in more sophisticated Monte Carlo techniques. Unfortunately, the use of RS techniques demands the calculation of tight upper bounds for the ratio of the target probability density function (pdf) over the proposal density from which candidate samples are drawn. Except for the class of log-concave target pdf's, for which an efficient algorithm exists, there are no general methods to analytically determine this bound, which has to be derived from scratch for each specific case. In this paper, we tackle the general problem of applying RS to draw from an arbitrary posterior pdf using the prior density as a proposal function. This is a scenario that appears frequently in Bayesian signal processing methods. We derive a general geometric procedure for the calculation of upper bounds that can be used with a broad class of target pdf's, including scenarios with correlated observations, multimodal and/or mixture measurement noises. We provide some simple numerical examples to illustrate the application of the proposed techniques.