Informed proposals for local MCMC in discrete spaces

There is a lack of methodological results to design efficient Markov chain Monte Carlo (MCMC) algorithms for statistical models with discrete-valued high-dimensional parameters. Motivated by this consideration, we propose a simple framework for the design of informed MCMC proposals (i.e. Metropolis-Hastings proposal distributions that appropriately incorporate local information about the target) which is naturally applicable to discrete spaces. Using Peskun-type comparisons of Markov kernels, we explicitly characterize the class of asymptotically-optimal proposal distributions under this framework, which we refer to as <i>locally-balanced</i> proposals. The resulting algorithms are straightforward to implement in discrete spaces and provide orders of magnitude improvements in efficiency compared to alternative MCMC schemes, including discrete versions of Hamiltonian Monte Carlo. Simulations are performed with both simulated and real data sets, including a detailed application to Bayesian record linkage. A direct connection with gradient-based MCMC suggests that locally-balanced proposals can be seen as a natural way to extend the latter to discrete spaces.