Some rapidly mixing hit-and-run samplers for latent counts in linear inverse problems

Abstract

Linear inverse problems for count data arise in a myriad of settings. The latent counts lie on a fibre that is too large to enumerate in most practical problems, but inference can proceed by sampling the fibre. We examine the mixing properties of hit-and-run samplers in this context. In general convergence can be arbitrarily slow. However, there is a class of linear inverse problems for which rapid mixing for uniform fibre sampling is possible, using Markov sub-bases that are of minimum size but yet provide a sufficiently rich range of sampling directions to avoid the need for zig-zagging walks to ensure connectivity. Focussing on such problems, we study a particular class of bases that enjoy these properties under certain easily checkable conditions on the configuration matrix. We also examine the mixing properties of these bases when employing commonly used Poisson models. Our theoretical results provide practical guidance on optimizing these Markov sub-bases.