Measuring Sample Quality with Stein's Method

By Jackson Gorham, Lester W. Mackey. 2015

252 citations recorded by the source corpus.

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 quantifies the maximum 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.

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

A-NICE-MC: Adversarial Training for MCMC (2017, methodology): “Even evaluating an existing MCMC technique is often challenging, with natural performance metrics intractable to compute [8–10].”
A Kernelized Stein Discrepancy for Goodness-of-fit Tests (2016, methodology): “As a result, S ( p, q ) is rarely used in practical machine learning; perhaps the only exception is the recent work of Gorham & Mackey (2015) who obtained a computationally tractable form by enforcing smoothness constraints only on a ﬁnite number of points, turning the optimization into a linear…”
A Kernel Test of Goodness of Fit (2016, methodology): “Recently, in the multivariate setting, Gorham & Mackey (2015) proposed an elegant measure of sample quality with respect to a target.”
Control functionals for Monte Carlo integration (2014, methodology): “Friel et al., 2015). (ii) An interesting direction would be to use the discrepancy D(D 0;D 1) as a tool for assessment of MCMC convergence, providing a reproducing kernel Hilbert space alternative to Gorham and Mackey (2015). Finally we note that Oates and Girolami (2016) provide a complementary stu”
Learning to Draw Samples: With Application to Amortized MLE for Generative Adversarial Learning (2016, methodology): “One exception is a very recent paper (Ranganath et al., 2016) that avoids calculating q ( x ) using an idea related to Stein discrepancy (Gorham & Mackey, 2015; Liu et al., 2016; Oates et al., 2014; Chwialkowski et al., 2016).”
Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm (2016, background): “Traditionally, F is taken to be sets of functions with bounded Lipschitz norms, which unfortunately casts a challenging functional optimization problem that is computationally intractable or requires special considerations (see Gorham and Mackey [11] and reference therein).”
How to Train Your Energy-Based Models (2021, methodology): “With some regularity conditions (Gorham and Mackey, 2015; Liu et al., 2016), we have D Stein ( p data ( x ) (cid:107) p θ ( x )) ≥ 0 , where the equality holds if and only if p data ( x ) ≡ p θ ( x ) .”
On Maximum Likelihood Training of Score-Based Generative Models (2021, methodology): “More generally, the generator method [Barbour, 1988] uses the inﬁnitesimal generator of a Markov process to construct a Stein operator and an associated Stein discrepancy [Gorham and Mackey, 2015, Section 3.1].”
An introduction to sampling via measure transport (2016, background): “It is a non-trivial task to assess the stationarity of an ergodic Markov chain, let alone to measure the distance between the empirical measure given by the MCMC samples and the target distribution [36].”
Operator Variational Inference (2016, background): “[22], Gorham and Mackey [7] for related derivations.”
Monte Carlo Gradient Estimation in Machine Learning (2019, methodology): “Stein’s identity (Gorham and Mackey, 2015; Liu et al., 2016) tells us that if a distribution p(x) is continuously differentiable then the gradient of its log-probability with respect to the random variable x can be related to the gradient of a set of test functions t(x) by the integral: Ep(x)…”
Action-depedent Control Variates for Policy Optimization via Stein's Identity (2017, unknown): “An incomplete list includes Gorham & Mackey (2015), Oates et al. (2017), Oates et al. (2016), Chwialkowski et al. (2016), Liu et al. (2016), Sedghi et al. (2016), Liu & Wang (2016), Feng et al. (2017), Liu & Lee (2017).”
Stein's method for comparison of univariate distributions (2014, background): “Recent notable contributions to a general multivariate theory are to be found in [45, 69], [11] and [10].”
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.”
Measuring Sample Quality with Diffusions (2016, unknown): “MEASURING SAMPLE QUALITY WITH DIFFUSIONS”
Gradient Estimators for Implicit Models (2017, unknown): “If the test function set H is sufﬁciently rich, then one can deﬁne a Stein discrepancy measure by S ( q, ˆ q ) := sup see [11] for an example derivation.”
Robust generalised Bayesian inference for intractable likelihoods (2021, unknown): “…stronger assumptions, Stein discrepancy provides convergence con-trol, meaning that a sequence ( Q n ) ∞ n = 1 ⊂  (  ) converges in a specified sense to Q whenever SD ( Q || Q n ) → 0; see Gorham and Mackey (2015 Theorem 2, Proposition 3) and Gorham and Mackey (2017) Theorem 8, Proposition 9).”
Understanding MCMC Dynamics as Flows on the Wasserstein Space (2019, background): “For instance the standard Langevin dynamics induces the Stein’s operator, and it in turn produces a metric called the Stein discrepancy (Gorham & Mackey, 2015), which inspires SVGD, and Liu & Zhu (2017) consider the Riemannian counterparts.”