Measuring Sample Quality with Stein's Method (1506.03039v6)

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.

• The paper introduces a novel computational measure based on Stein's method to evaluate the quality of samples from biased Monte Carlo procedures, accounting for asymptotic bias ignored by traditional metrics.
• This method transforms sample quality evaluation into a computationally tractable linear programming problem and demonstrates convergence properties with standard probability metrics like Wasserstein distance.
• The approach is efficiently implementable and practically applicable for selecting hyperparameters in biased MCMC algorithms where traditional diagnostics fail.

An Analytical Framework for Sample Quality Evaluation Using Stein's Method

The paper "Measuring Sample Quality with Stein's Method," authored by Jackson Gorham and Lester Mackey, addresses the challenge of evaluating the quality of samples drawn from biased Markov chain Monte Carlo (MCMC) procedures. As recent trends in Monte Carlo estimation have shifted towards biased sampling to increase computational speed, traditional metrics like effective sample size fall short in accounting for the introduced asymptotic bias. This research presents an innovative approach utilizing Stein's Method to address these inadequacies by providing a new computable sample quality measure.

Core Contributions

The authors introduce a computational method based on Stein's method, designed to quantify the discrepancy between sample and target expectations over a broad class of test functions. This can be particularly beneficial in cases involving exact, biased, and deterministic sample sequences. The proposed method addresses several significant aspects:

1. Establishment of a New Quality Measure: By leveraging Stein's method, this research develops a measurable quantity that reflects maximum discrepancy over an extensive scope of test functions while avoiding explicit calculation under the target distribution.
2. Optimization Framework: This approach transforms the problem of evaluating sample quality into a linear programming task, making it computationally tractable.
3. Strong Convergence Properties: The paper establishes convergence relationships between the new quality measure and standard probability metrics like the Wasserstein distance, particularly in the context of strongly log-concave distributions.
4. Efficient Implementation: By adopting geometric spanners, the method achieves computational efficiency, enabling its application to larger sample sets with reduced complexity.
5. Application to Hyperparameter Selection: The paper demonstrates practical applications in selecting hyperparameters for biased MCMC algorithms like Stochastic Gradient Langevin Dynamics, where standard methods might fail due to bias.

Implications and Future Directions

From a theoretical perspective, this research provides a robust framework for understanding and evaluating sample quality under asymptotic bias conditions. Practically, its applications extend to hyperparameter tuning in scalable inference methods, offering significant improvements over traditional diagnostics that ignore long-term bias.

The paper hints at several possible avenues for future research. An immediate next step could be extending the method to broader classes of distributions beyond those analyzed. Additionally, the application of this measure to more complex models and real-world datasets could further validate its utility. There's also potential in exploring the incorporation of this measure into automated systems for inference algorithm selection, which could streamline the deployment of scalable Bayesian inference solutions.

In conclusion, this paper contributes a mathematically rigorous and computationally efficient framework for evaluating sample quality in the context of biased MCMC procedures. It sets the stage for further advancements in the development of sampling algorithms that balance bias and computational efficiency effectively.