An importance sampling approach for reliable and efficient inference in Bayesian ordinary differential equation models

TL;DR

This paper introduces a new importance sampling method to verify the reliability of Bayesian inference in models involving solutions to nonlinear ODEs, addressing biases from numerical approximations.

Contribution

It presents a computationally efficient workflow for assessing the reliability of posterior inferences in Bayesian ODE models, including robust alternatives to adaptive ODE solvers.

Findings

The workflow effectively verifies inference reliability in simulated and real data.

It identifies issues with common adaptive ODE solvers and offers robust alternatives.

The method improves confidence in Bayesian inference results involving ODE solutions.

Abstract

Statistical models can involve implicitly defined quantities, such as solutions to nonlinear ordinary differential equations (ODEs), that unavoidably need to be numerically approximated in order to evaluate the model. The approximation error inherently biases statistical inference results, but the amount of this bias is generally unknown and often ignored in Bayesian parameter inference. We propose a computationally efficient method for verifying the reliability of posterior inference for such models, when the inference is performed using Markov chain Monte Carlo methods. We validate the efficiency and reliability of our workflow in experiments using simulated and real data, and different ODE solvers. We highlight problems that arise with commonly used adaptive ODE solvers, and propose robust and effective alternatives which, accompanied by our workflow, can now be taken into use without losing reliability of the inferences.