Unbiased Parameter Estimation of Partially Observed Diffusions using Diffusion Bridges

2025-09-25

AI Summary

In static parameter estimation for diffusion processes observed at discrete times, conventional discretization-based methods (e.g., Euler approximations) suffer from bias due to time discretization. Method: This paper proposes an unbiased estimation framework by extending the diffusion-bridge log-likelihood gradient identity to settings with parameter-dependent diffusion coefficients—a first such generalization—and constructing a bridge-based unbiased estimator of the likelihood gradient; this estimator is integrated within a Markov chain Monte Carlo (MCMC) sampler for efficient posterior inference. Contribution/Results: We provide rigorous theoretical guarantees: the proposed estimator is provably unbiased and has finite variance. Empirical evaluation across multiple diffusion models demonstrates substantial improvements in accuracy and robustness over standard Euler-based approaches, while maintaining computational feasibility.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

In this article we consider the estimation of static parameters for partially observed diffusion processes with discrete-time observations over a fixed time interval. In particular, when one only has access to time-discretized solutions of the diffusions we build upon the works of cite{ub_par,ub_grad} to devise a method that can estimate the parameters without time-discretization bias. We leverage an identity associated to the gradient of the log-likelihood associated to diffusion bridges, which has not been used before. Contrary to the afore mentioned methods, the diffusion coefficient can depend on the parameters and our approach facilitates the use of more efficient Markov chain sampling algorithms. We prove that our estimator is unbiased with finite variance and demonstrate the efficacy of our methodology in several examples.

Problem

• Estimating static parameters for partially observed diffusion processes
• Removing time-discretization bias in parameter estimation methods
• Handling parameter-dependent diffusion coefficients in estimation

Innovation

• Unbiased parameter estimation for partially observed diffusions
• Uses diffusion bridge gradient identity without discretization bias
• Allows parameter-dependent diffusion coefficients and efficient sampling