Laplace-aided variational inference for differential equation models

TL;DR

This paper introduces a Laplace-aided variational inference method to improve covariance estimation in Bayesian models for ODEs, enhancing accuracy over previous variational approaches.

Contribution

It proposes integrating Laplace approximation into variational Bayes for ODE models to address covariance underestimation issues in existing methods.

Findings

Laplace approximation significantly improves covariance estimates.

The method yields posterior approximations comparable to MCMC.

Numerical experiments confirm enhanced accuracy in parameter estimation.

Abstract

An ordinary differential equation (ODE) model, whose regression curves are a set of solution curves for some ODEs, poses a challenge in parameter estimation. The challenge, due to the frequent absence of analytic solutions and the complicated likelihood surface, tends to be more severe especially for larger models with many parameters and variables. Yang and Lee (2021) proposed a state-space model with variational Bayes (SSVB) for ODE, capable of fast and stable estimation in somewhat large ODE models. The method has shown excellent performance in parameter estimation but has a weakness of underestimation of the posterior covariance, which originates from the mean-field variational method. This paper proposes a way to overcome the weakness by using the Laplace approximation. In numerical experiments, the covariance modified by the Laplace approximation showed a high degree of improvement when checked against the covariances obtained by a standard Markov chain Monte Carlo method. With the improved covariance estimation, the SSVB renders fairly accurate posterior approximations.