[Paper Review] Measuring Sample Quality with Stein's Method

Jackson Gorham, Lester Mackey|arXiv (Cornell University)|Jun 9, 2015

Markov Chains and Monte Carlo MethodsMathematics37 references68 citations

TL;DR

This paper introduces a new computable quality measure for Markov chain Monte Carlo (MCMC) samples based on Stein's method, which quantifies the maximum discrepancy between sample and target expectations over a broad class of test functions. The method enables comparison of exact, biased, and deterministic samplers by solving a linear program, offering a practical tool for assessing bias-variance tradeoffs and convergence in Bayesian inference.

ABSTRACT

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.

Motivation & Objective

• To address the lack of reliable sample quality metrics in biased MCMC methods that trade asymptotic exactness for faster sampling.
• To develop a computable, interpretable measure that detects both convergence and asymptotic bias in sample sequences.
• To enable fair comparison between exact, biased, and deterministic sampling sequences in Monte Carlo estimation.
• To support practical tasks such as hyperparameter selection, convergence rate assessment, and bias-variance tradeoff quantification in posterior inference.

Proposed method

• The method defines a quality measure as the supremum of the absolute difference between sample and target expectations over a class of test functions, derived from Stein's method.
• It formulates the quality measure as a linear program, making it computationally tractable for finite samples.
• The approach uses geometric spanners to efficiently approximate the solution, reducing computational cost in high dimensions.
• The test function class is constructed using gradient and Hessian constraints to ensure smoothness and control over the discrepancy.
• The method leverages the Whitney-Glaeser extension theorem to extend functions defined on sample points while preserving gradient and Hessian bounds.
• A dual formulation allows efficient computation by solving for the optimal function in a reproducing kernel Hilbert space framework.

Experimental results

Research questions

• RQ1 How can we quantify the quality of a sample sequence when the sampler is asymptotically biased?
• RQ2 Can we detect convergence to the target distribution even when the sampler is biased?
• RQ3 How does the proposed measure compare to standard diagnostics like effective sample size in the presence of bias?
• RQ4 To what extent can the measure be used to optimize hyperparameters in biased MCMC algorithms?
• RQ5 What is the tradeoff between bias and variance in posterior inference, and how can it be quantified using this measure?

Key findings

• The proposed Stein-based quality measure detects both convergence and asymptotic bias, providing a reliable alternative to traditional diagnostics in biased MCMC.
• The measure is computable via linear programming, enabling practical application to real-world sampling problems.
• The method successfully identifies when a sequence of samples fails to converge to the target distribution, even under bias.
• The use of geometric spanners reduces computational complexity, making the method scalable to high-dimensional problems.
• The quality measure enables effective hyperparameter selection in biased samplers by quantifying the bias-variance tradeoff.
• Theoretical analysis confirms that convergence of the measure to zero implies weak convergence of the sampling distribution to the target.