Prediction-Centric Uncertainty Quantification via MMD

TL;DR

This paper introduces a novel uncertainty quantification method for deterministic models that uses a mixture distribution optimized via MMD gradient flow, improving predictive uncertainty estimates.

Contribution

It proposes a prediction-centric approach that addresses the limitations of Bayesian methods in misspecified deterministic models by using MMD-based mixture distributions.

Findings

Improved uncertainty quantification in toy and real biological models.

Consistent numerical approximation via MMD gradient flow.

Addresses deterministic posteriors becoming overly confident.

Abstract

Deterministic mathematical models, such as those specified via differential equations, are a powerful tool to communicate scientific insight. However, such models are necessarily simplified descriptions of the real world. Generalised Bayesian methodologies have been proposed for inference with misspecified models, but these are typically associated with vanishing parameter uncertainty as more data are observed. In the context of a misspecified deterministic mathematical model, this has the undesirable consequence that posterior predictions become deterministic and certain, while being incorrect. Taking this observation as a starting point, we propose Prediction-Centric Uncertainty Quantification, where a mixture distribution based on the deterministic model confers improved uncertainty quantification in the predictive context. Computation of the mixing distribution is cast as a (regularised) gradient flow of the maximum mean discrepancy (MMD), enabling consistent numerical approximations to be obtained. Results are reported on both a toy model from population ecology and a real model of protein signalling in cell biology.