MCMC convergence diagnosis via multivariate bounds on log-concave densities

We begin by showing how piecewise linear bounds may be devised, which bound both above and below any concave log-density in general dimensions. We then show how these bounds may be used to gain an upper bound to the volume in the tails outside the convex hull of the sample path in order to assess how well the sampler has explored the target distribution. This method can be used as a stand-alone diagnostic to determine when the sampler output provides a reliable basis for inference on the stationary density, or in conjunction with existing convergence diagnostics to ensure that they are based upon good sampler output. We provide an example and briefly discuss possible extensions to the method and alternative applications of the bounds. Ž. 1. Introduction. The use of Markov chain Monte Carlo MCMC methods has increased dramatically within the statistical community since their Ž.