Measuring Sample Quality with Diffusions

By Jackson Gorham, Andrew B. Duncan, Sandra Jeanne Vollmer, Lester W. Mackey. 2016

123 citations recorded by the source corpus.

Standard Markov chain Monte Carlo diagnostics, like effective sample size, are ineffective for biased sampling procedures that sacrifice asymptotic correctness for computational speed. Recent work addresses this issue for a class of strongly log-concave target distributions by constructing a computable discrepancy measure based on Stein's method that provably determines convergence to the target. We generalize this approach to cover any target with a fast-coupling Ito diffusion by bounding the derivatives of Stein equation solutions in terms of Markov process coupling times. As example applications, we develop computable and convergence-determining diffusion Stein discrepancies for log-concave, heavy-tailed, and multimodal targets and use these quality measures to select the hyperparameters of biased samplers, compare random and deterministic quadrature rules, and quantify bias-variance tradeoffs in approximate Markov chain Monte Carlo. Our explicit multivariate Stein factor bounds may be of independent interest.

Subfields: Applied Mathematics, Statistics and Probability

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Citations in the corpus

Measuring Sample Quality with Kernels (2017, methodology): “However, the KSD assessment does not require access to an external trustworthy ground truth sample.”
Convergence rates for a class of estimators based on Stein’s method (2016, background): “The choice of Stein operator is not unique and some alternatives are listed in [17].”
Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula (2018, result): “stribution and the target distribution in terms of the step size in the algorithm. Our result complements those in the literature by relaxing the conditions on the increment distribution. We refer to [15] and the references therein for some related problems. We would like to mention two other distan”
Stochastic Gradient Markov Chain Monte Carlo (2019, methodology): “The design of appropriate diagnostic tests for SGMCMC is a relatively new area of research, and currently methods based on Stein’s discrepancy (Gorham and Mackey 2015, 2017; Gorham et al. 2019) are the most popular approach.”
Understanding MCMC Dynamics as Flows on the Wasserstein Space (2019, unknown): “This knowledge gives the expression Af = 1 p ∂j [ p ( Dij +Qij ) (∂if) ] , (8) which meets existing results (e.g. Gorham et al. (2016), Thm. 2).”
Underdamped Langevin MCMC: A non-asymptotic analysis (2017, background): “The continuous-time process has been studied extensively [12, 13, 14, 15, 16, 17, 18, 19, 20].”
Robust generalised Bayesian inference for intractable likelihoods (2021, unknown): “A more general formulation is the diffusion Stein operator (Gorham et al., 2019; Barp et al., 2019) SQ[h](x) = (M(x)h(x)) · ∇ log q(x) +∇ · (M(x)h(x)) , (6) parametrised by a matrix-valued function M ∈ C(1) b (Rd;Rd×d).”