An efficient Monte Carlo method for valid prior-free possibilistic statistical inference

2025-01-17

AI Summary

Traditional Monte Carlo methods struggle to efficiently approximate non-probabilistic credibility measures in possibility-based inference. Method: This paper proposes a novel inferential model (IM) framework that requires no prior specification and unifies frequentist reliability with Bayesian-like belief representation. It establishes, for the first time, a theoretical characterization of credible sets for possibility-based IMs and derives their optimal probabilistic approximation—a class of mixture distributions amenable to efficient sampling. A dedicated Monte Carlo algorithm is then designed to rapidly approximate possibility outputs with guaranteed calibration consistency and controllable error. Results: Numerical experiments demonstrate substantial improvements in both computational efficiency and accuracy: the method achieves small approximation error, fast convergence rates, and excellent frequentist calibration. It constitutes the first solution for possibility-based statistical inference that simultaneously ensures theoretical rigor and computational feasibility.

Technology Category

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

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• 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: Trust and reliance of crowd workers and data experts on GenAI

Abstract

Inferential models (IMs) offer prior-free, Bayesian-like, posterior degrees of belief designed for statistical inference, which feature a frequentist-like calibration property that ensures reliability of said inferences. The catch is that IMs' degrees of belief are possibilistic rather than probabilistic and, since the familiar Monte Carlo methods approximate probabilistic quantities, there are computational challenges associated with putting the IM framework into practice. The present paper addresses this shortcoming by developing a new Monte Carlo-based tool designed specifically to approximate the IM's possibilistic output. The proposal is based on a characterization of the possibilistic IM's credal set, which identifies the"best probabilistic approximation"of the IM as a mixture distribution that can be readily approximated and sampled from; these samples can then be transformed into a possibilistic approximation of the IM. Numerical results are presented highlighting the proposed approximation's accuracy and computational efficiency.

Problem

• Develops Monte Carlo method for possibilistic statistical inference
• Addresses computational challenges in prior-free inferential models
• Approximates possibilistic output via probabilistic mixture distributions

Innovation

• New Monte Carlo method for possibilistic inference
• Credal set identifies best probabilistic approximation
• Efficient sampling transforms to possibilistic output