Bayesian inference for ordinary differential equations models with heteroscedastic measurement error

2026-03-23

AI Summary

This work addresses the limitations of conventional ordinary differential equation (ODE) models in parameter inference, which typically assume homoscedastic Gaussian observation errors despite real-world data often exhibiting time-varying or state-dependent heteroscedasticity—leading to biased posterior estimates and miscalibrated uncertainty quantification. To overcome this, the authors propose a two-stage semiparametric Bayesian framework: first, a heteroscedastic Gaussian process is employed to nonparametrically model the time-varying error structure; this estimated error model is then incorporated into the likelihood function for subsequent Bayesian inference of ODE parameters. By avoiding prespecified parametric forms for the noise and circumventing the need to sample additional hyperparameters in MCMC, the method substantially improves both parameter estimation accuracy and predictive uncertainty calibration. Empirical evaluations on simulated and two real-world datasets demonstrate its clear superiority over standard homoscedastic approaches.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

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 to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Ordinary differential equation (ODE) models are widely used to describe systems in many areas of science. To ensure these models provide accurate and interpretable representations of real-world dynamics, it is often necessary to infer parameters from data, which involves specifying the form of the ODE system as well as a statistical model describing the observational process. A popular and convenient choice for the error model is a Gaussian distribution with constant variance. However, the choice may not be realistic in many systems, since the variance of the observational error may vary over time or have some dependence on the system state (heteroscedastic), reflecting changes in measurement conditions, environmental fluctuations, or intrinsic system variability. Misspecification of the error model can lead to substantial inaccuracies of the posterior estimates of the ODE model parameters and predictions. More elaborate parametric error models could be specified, but this would increase computational cost because additional parameters would need to be estimated within the MCMC procedure and may still be misspecified. In this work we propose a two-step semi-parametric framework for Bayesian parameter estimation of ODE model parameters when there exists heteroscedasticity in the error process. The first step applies a heteroscedastic Gaussian process to estimate the time-dependent error, and the second step performs Bayesian inference for the ODE model parameters using the estimated time-dependent error estimated from step one in the likelihood function. Through a simulation study and two real-world applications, we demonstrate that the proposed approach yields more reliable posterior inference and predictive uncertainty compared to the standard homoscedastic models. Although our focus is on heteroscedasticity, the framework could be applied to handle more complex error processes.

Problem

• ordinary differential equations
• Bayesian inference
• heteroscedasticity
• measurement error
• parameter estimation

Innovation

• heteroscedasticity
• Bayesian inference
• ordinary differential equations
• Gaussian process
• semi-parametric framework