Optimal Thinning of MCMC Output

Abstract

The use of heuristics to assess the convergence and compress the output of Markov chain Monte Carlo can be sub-optimal in terms of the empirical approximations that are produced. Typically a number of the initial states are attributed to "burn in" and removed, whilst the remainder of the chain is "thinned" if compression is also required. In this paper we consider the problem of retrospectively selecting a subset of states, of fixed cardinality, from the sample path such that the approximation provided by their empirical distribution is close to optimal. A novel method is proposed, based on greedy minimisation of a kernel Stein discrepancy, that is suitable for problems where heavy compression is required. Theoretical results guarantee consistency of the method and its effectiveness is demonstrated in the challenging context of parameter inference for ordinary differential equations. Software is available in the Stein Thinning package in both Python and MATLAB.

Online, Informative MCMC Thinning with Kernelized Stein Discrepancy

“…Several MCMC thinning procedures have been developed based upon RKHS embedding: Offline methods such as Stein Thinning andd MMD thinning (Riabiz et al, 2020;Teymur et al, 2020) take a full chain S of MCMC samples as input and iteratively build a subset D by greedy KSD/MMD minimization. The online method Stein Point MCMC (SPMCMC) (Chen et al, 2019a) selects the optimal sample from a batch of m samples during MCMC sampling: at each step it adds the best of m points to a D. Doing so mitigates both the aforementioned redundancy and representational complexity issues; however (Chen et al, 2019a) only append new points to the existing empirical measure estimates, which may still retain too many redundant points.…”

“…Method Online Informative Discard Past Samples Model Order Growth MCMC Thinning n Stein Thinning (Riabiz et al, 2020) NA SPMCMC (Chen et al, 2019a) n…”

“…Related work by (Riabiz et al, 2020;Teymur et al, 2020) present constructive greedy approaches to subset selection during post-hoc thinning. The challenge with online constructive approaches is time complexity, as the inner loop requires | Dt | point selections in the worst case (no points are thinned).…”

“…Classically, to deal with the redundancy issue, one may employ online or posthoc "thinning," which discards all but a random subset of MCMC samples (Raftery & Lewis, 1996;Link & Eaton, 2012). More recently this task is sometimes called "quantizing" a posterior distribution (Riabiz et al, 2020), especially when the work studies Bayesian cubature (Teymur et al, 2020). Existing approaches generate a large set of samples (usually via MCMC) and then "thin" the sample set.…”

“…Existing approaches generate a large set of samples (usually via MCMC) and then "thin" the sample set. Doing so may require storing a large number of samples before the final post-processing stage (Riabiz et al, 2020;Teymur et al, 2020) and does not allow the sampler to target a compressed representation during the MCMC iterations. Traditional online thinning is done without any goodness-of-fit metric on the samples, which may ignore gradient information generated by modern stochastic gradient MCMC methods (Welling & Teh, 2011;Ma et al, 2015).…”