Geometrically Aware Dynamic Markov Bases for Statistical Linear Inverse Problems

For statistical linear inverse problems involving count data, inference typically requires sampling a latent variable with conditional support comprising the lattice points in a convex polytope. Irreducibility of random walk samplers is guaranteed only if a sufficiently rich array of sampling directions is available. In principle this can be achieved by finding a Markov basis of moves ab initio, but in practice doing so may be computationally infeasible. What is more, the use of a full Markov basis can lead to very poor mixing. It is far simpler to find a lattice basis of moves, which can be tailored to the overall geometry of the polytope. However, a single lattice basis generally does not connect all points in the polytope. In response, we propose a dynamic lattice basis sampler. This sampler can access a sufficient variety of sampling directions to guarantee irreducibility, but also privileges moves that are well aligned to the polytope geometry, hence promoting good mixing. The probability with which the sampler selects different bases can be tuned. We present an efficient algorithm for updating the lattice basis, obviating the need for repeated matrix inversion.