Informed Proposals for Local MCMC in Discrete Spaces

Abstract

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 both discrete and continuous spaces. We explicitly characterize the class of optimal proposal distributions under this framework, which we refer to as locallybalanced proposals, and prove their Peskun-optimality in high-dimensional regimes. 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 datasets, including a detailed application to Bayesian record linkage. A direct connection with gradient-based MCMC suggests that locally-balanced proposals may be seen as a natural way to extend the latter to discrete spaces. arXiv:1711.07424v1 [stat.CO]

Scalable Importance Tempering and Bayesian Variable Selection

“…The choice of the number of variables to select would then be related to a cost per iteration versus mixing trade‐off. See section 6.4 of Zanella () for a discussion of similar blockwise implementations. Also, computing p i ( x ) exactly may be infeasible in some contexts, and thus it would be interesting to design a version of TGS where the terms p i ( x ) are replaced by unbiased estimators while preserving the correct invariant distribution.…”

Informed reversible jump algorithms

“…The resulting proposal distributions are called globally-balanced in Zanella (2020). To understand why, consider the situation where the size of K is small and it is feasible to switch from any model to any other one, and thus we can set N(k) = K for all k.…”

“…We recommend to set h to the identity only when it is to feasible to set N(k) = K for all k. Otherwise, it seems better to use locallybalanced proposal distributions; this is the case for instance in Zanella (2020) and our moderate-size (both in n and dimension) variable-selection example. Two choices of locally-balanced functions h are considered in Zanella (2020): h(x) = √ x and h(x) = x/(1 + x).…”

“…The empirical results of Zanella (2020) suggest that this latter choice is superior because h is bounded, which stabilizes the normalizing constants c(k) and c(k ) and thus the acceptance probabilities. Livingstone and Zanella (2019) reached the same conclusion using locally-balanced proposal distributions for continuous random variables.…”

Uncertainty modelling and computational aspects of data association

“…This result is achieved by leveraging the efficiency of an approximate multi-object filtering method. The use of MCMC in discrete spaces is discussed more generally in Zanella (2019). MCMC has also been used in conjunction with, or as a replacement of, sequential Monte Carlo in the context of filtering for single-object systems (Gilks and Berzuini 2001) and multi-object systems (Khan et al 2005;Septier et al 2009;Carmi et al 2012;Maroulas and Stinis 2012;Bao and Maroulas 2017); however, this type of approach is less directly related to the method proposed in this article.…”