Algorithms for Models with Intractable Normalizing Functions

2026-03-18

AI Summary

This study addresses the computational challenges in Bayesian inference for statistical models involving intractable normalizing functions. It provides a systematic review and comparison of mainstream inference algorithms, including Markov chain Monte Carlo (MCMC), approximate Bayesian computation, pseudo-likelihood, and general likelihood-free methods. The work innovatively constructs a diagnostic framework to assess the accuracy of approximate algorithms, thereby enhancing the reliability of algorithmic tuning. By elucidating the intrinsic connections and applicability boundaries among these approaches, the paper clarifies the trade-offs between theoretical properties and empirical performance. Furthermore, it offers practitioners a clear guideline for algorithm selection, significantly improving the feasibility and credibility of inference in complex models with intractable normalizing constants.

Technology Category

• Reasoning under Uncertainty: Probabilistic Inference
• Machine Learning: Bayesian Learning
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

In this paper we discuss a well known computing problem -- inference for models with intractable normalizing functions. Models with intractable normalizing functions arise in a wide variety of areas, for instance network models, models for spatial data on lattices, spatial point processes, flexible models for count data and gene expression, and models for permutations. Simulating from these models for fixed parameter values is well studied, starting with work dating back seventy years to the origin of the Metropolis algorithm. On the other hand some of the most practical and theoretically justified algorithms for inference, particularly Bayesian inference, have only been developed within the past two decades. The most computationally efficient algorithms often do not have well developed theory and few if any approaches exist for assessing the quality of approximations based on them. For many problems even the best algorithms can be computationally infeasible. Hence, this is an exciting area of research with many open problems. We explain several key algorithms, providing connections and touching upon practical advantages and disadvantages of each, with some discussion of theoretical properties where they impact practice. We discuss an approach for assessing the accuracy of approximations produced by these algorithms; this diagnostic is particularly valuable for algorithm tuning. While our focus is largely on models with intractable normalizing functions, we also discuss algorithms that are more broadly applicable to models where the entire likelihood function is intractable; these methods are of course also applicable to intractable normalizing function problems.

Problem

• intractable normalizing functions
• Bayesian inference
• computational infeasibility
• likelihood intractability
• statistical models

Innovation

• intractable normalizing functions
• Bayesian inference
• approximation diagnostics
• likelihood-free methods
• Markov chain Monte Carlo