Stein Point Markov Chain Monte Carlo

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. A similar line of research in Gaussian Processes (Williams & Rasmussen, 2006) and kernel regression (Hofmann et al, 2008) reduces the complexity of a nonparametric distributional representation through offline point selection rules such as Nyström sampling (Williams & Seeger, 2001), greedy forward selection (Seeger et al, 2003;Wang et al, 2012), or inducing inputs (Snelson & Ghahramani, 2005).…”

“…Most similar to our work is a variant of Stein Point MCMC proposed in (Chen et al, 2019a) [Appendix A.6.5] which develops a non-adaptive add/drop criterion where a pre-fixed number of points are dropped at each stage and the dictionary size grows linearly with the time step. By contrast, our work develops a flexible and adaptive scheme that automatically determines the number of points to drop and the dictionary size grows sub-linearly with the time step.…”

“…We assume that the unnormalized density p and its score function ∇ log p may be evaluated in a computationally affordable manner. This set of assumptions is standard in many Bayesian inference problems that arise in machine learning and uncertainty quantification (Welling & Teh, 2011;Chen et al, 2019a;Constantine et al, 2016;Peherstorfer et al, 2018).…”

“…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…”

“…We discuss practical selection of and the resultant dictionary D in the following section. Our approach differs from the related work of (Chen et al, 2019a) in that we thin the entire dictionary at each step. In contrast, no prior samples are thinned in the Stein Point MCMC algorithm presented by (Chen et al, 2019a).…”