Calibrating Bayesian Inference

2025-10-30

AI Summary

Bayesian inference in small-sample disciplines (e.g., psychology) often suffers from long-term inferential failure due to misspecified priors, while the true data-generating mechanism remains unknown. Method: We propose a novel framework for calibrating Bayesian credible regions by embedding frequentist validity—specifically, nominal coverage preservation—into Bayesian inference. Our approach jointly leverages Bayesian modeling and a frequency-based calibration objective, solved via stochastic approximation to determine calibrated thresholds; systematic Monte Carlo experiments validate its performance. Contribution/Results: Uncalibrated Bayesian methods frequently yield overly permissive intervals with subnominal coverage. In contrast, our calibrated approach robustly maintains nominal coverage (e.g., 95%) across diverse data-generating mechanisms—including misspecified and nonstandard settings—without requiring knowledge of the true parameter-generating process. This substantially enhances the reliability and reproducibility of Bayesian inference in small-sample contexts.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Inference
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

While Bayesian statistics is popular in psychological research for its intuitive uncertainty quantification and flexible decision-making, its performance in finite samples can be unreliable. In this paper, we demonstrate a key vulnerability: When analysts' chosen prior distribution mismatches the true parameter-generating process, Bayesian inference can be misleading in the long run. Given that this true process is rarely known in practice, we propose a safer alternative: calibrating Bayesian credible regions to achieve frequentist validity. This latter criterion is stronger and guarantees validity of Bayesian inference regardless of the underlying parameter-generating mechanism. To solve the calibration problem in practice, we propose a novel stochastic approximation algorithm. A Monte Carlo experiment is conducted and reported, in which we observe that uncalibrated Bayesian inference can be liberal under certain parameter-generating scenarios, whereas our calibrated solution is always able to maintain validity.

Problem

• Bayesian inference becomes unreliable when priors mismatch true data-generating processes
• Calibrating credible regions ensures frequentist validity regardless of parameter-generating mechanisms
• A stochastic approximation algorithm solves calibration for practical Bayesian applications

Innovation

• Calibrating Bayesian credible regions for frequentist validity
• Proposing a novel stochastic approximation algorithm
• Ensuring validity regardless of parameter-generating mechanism