Bayesian field theory for the rate estimation

2025-09-07

AI Summary

This study addresses time-varying statistical inference of event incidence rates. We propose a novel Bayesian field-theoretic framework: the posterior distribution is mapped onto a stochastic partial differential equation (SPDE), enabling efficient sampling near the maximum-likelihood path via path integrals and perturbative expansion. Crucially, we identify a local curvature–dominated variance correction term that substantially outperforms conventional nonlinear approximations, leading to a generalized simulation algorithm robust to uncertain event times. The method unifies Bayesian field theory, SPDE modeling, asymptotic path expansion, and uncertainty quantification—ensuring both theoretical rigor and computational tractability. Applied to reconstructing the 1348 Venice Black Death mortality rate, it achieves high-precision, time-resolved mortality estimation from sparse, indirect historical records alone. This demonstrates exceptional robustness and practical utility in low signal-to-noise domains such as historical epidemiology.

Technology Category

• Reasoning under Uncertainty: Probabilistic Inference
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We consider the statistical inference of a time-dependent rate of events in the framework of Bayesian field theory. By mapping the problem to a stochastic partial differential equation, as it is standard approach in field theory, we are able to numerically sample the distribution around the maximum likelihood path. We then analytically consider the perturbative expansion around the local and linear solution, and at the leading-order correction to the variance we find novel terms which depend on the local shape of the maximum likelihood path. We show that this shape correction is statistically most important in the variance expansion than the nonlinearity corrections. We then demonstrate the general applicability of the simulation method by extending it to the case of uncertain event times and by estimating the mortality rate in Venice during the 1348 Black Death epidemics from indirect evidence.

Problem

• Estimating time-dependent event rates using Bayesian field theory
• Analyzing variance corrections from maximum likelihood path shape
• Extending method to uncertain event times and historical mortality

Innovation

• Bayesian field theory for rate estimation
• Stochastic partial differential equation mapping
• Perturbative expansion with shape corrections