Measuring Sample Quality with Stein's Method
To improve the efficiency of Monte Carlo estimation, practitioners are turning to biased Markov chain Monte Carlo procedures that trade off asymptotic exactness for computational speed. The reasoning is sound: a reduction in variance due to more rapid sampling can outweigh the bias introduced. However, the inexactness creates new challenges for sampler and parameter selection, since standard measures of sample quality like effective sample size do not account for asymptotic bias. To address these challenges, we introduce a new computable quality measure based on Stein's method that bounds the discrepancy between sample and target expectations over a large class of test functions. We use our tool to compare exact, biased, and deterministic sample sequences and illustrate applications to hyperparameter selection, convergence rate assessment, and quantifying bias-variance tradeoffs in posterior inference.
The main contribution of the paper is a proposal for IPM with certain properties and geared towards a specific application. The IPM proposed is based on Stein operator, which is one of the standard methods, by now, for studying convergence in distribution. The motivating application is measuring the quality of biased MCMC procedures. The assumption are: (i) the expectation under the target distribution is zero (which is made because of the Stein's method) (ii) the operator used is the infinitesimal generator of Markov process and (iii) the set of functions in defining the IPM is what is called as 'classic Stein set' in page 3 of the paper. The main contributions of the paper are 1) Theorem 7 (and theorem 2) relating Stein and (Wasser)stein IPMs for log-concave densities under certain assumptions in the multivariate setting. 2) Providing computable optimization programs for Stein discrepancy - I especially liked the use ofWhitney's extension theorem for computing a graph based discrepancy. It would be interesting to explore rates of convergence results for Stein discrepancy rigorously. Furthermore, it would be interesting to explore relationship to MMD (as one of the main focus in on computability). Using MMD in this context would correspond to goodness-of-fit style testing for target distribution (say P) using MMD. The authors mention MMD would work when 'expected kernel evaluations are easily computed under the target'. But since the space of rkhs function is dense in L_2(P) essentially it could approximate a large class of functions (not in the sample sense as mentioned in the paper but in a population setting). I went over the proofs and they seem to be correct at least at a high level. The contributions of this paper are interesting and I would recommend accepting the paper. I think extending and exploring ideas along the lines of this paper should lead to interesting results.