Perfect Simulation for Mixtures with Known and Unknown Number of Components

Abstract

We propose and develop a novel and effective perfect sampling methodology for simulating from posteriors corresponding to mixtures with either known (fixed) or unknown number of components. For the latter we consider the Dirichlet process-based mixture model developed by these authors, and show that our ideas are applicable to conjugate, and importantly, to non-conjugate cases. As to be expected, and as we show, perfect sampling for mixtures with known number of components can be achieved with much less effort with a simplified version of our general methodology, whether or not conjugate or non-conjugate priors are used. While no special assumption is necessary in the conjugate set-up for our theory to work, we require the assumption of compact parameter space in the non-conjugate set-up. However, we argue, with appropriate analytical, simulation, and real data studies as support, that such compactness assumption is not unrealistic and is not an impediment in practice. Not only do we validate our ideas theoretically and with simulation studies, but we also consider application of our proposal to three real data sets used by several authors in the past in connection with mixture models. The results we achieved in each of our experiments with either simulation study or real data application, are quite encouraging. However, the computation can be extremely burdensome in the case of large number of mixture components and in massive data sets. We discuss the role of parallel processing in mitigating the extreme computational burden.