Beyond Bayesian Inference: The Correlation Integral Likelihood Framework and Gradient Flow Methods for Deterministic Sampling

TL;DR

This paper introduces the Correlation Integral Likelihood framework and gradient flow methods for deterministic sampling, offering a new approach to parameter inference in complex biological PDE models with noisy and heterogeneous data.

Contribution

It presents a unified mathematical framework for parameter estimation in biological systems, extending beyond classical Bayesian methods, and compares stochastic and deterministic sampling strategies within this framework.

Findings

CIL framework effectively handles chaotic and heterogeneous dynamics.

Deterministic gradient flow methods improve inference efficiency.

Framework enhances model calibration with noisy, incomplete data.

Abstract

Calibrating mathematical models of biological processes is essential for achieving predictive accuracy and gaining mechanistic insight. However, this task remains challenging due to limited and noisy data, significant biological variability, and the computational complexity of the models themselves. In this method's article, we explore a range of approaches for parameter inference in partial differential equation (PDE) models of biological systems. We introduce a unified mathematical framework, the Correlation Integral Likelihood (CIL) method, for parameter estimation in systems exhibiting heterogeneous or chaotic dynamics, encompassing both pattern formation models and individual-based models. Departing from classical Bayesian inverse problem methodologies, we motivate the development of the CIL method, demonstrate its versatility, and highlight illustrative applications within mathematical biology. Furthermore, we compare stochastic sampling strategies, such as Markov Chain Monte Carlo (MCMC), with deterministic gradient flow approaches, highlighting how these methods can be integrated within the proposed framework to enhance inference performance. Our work provides a practical and theoretically grounded toolbox for researchers seeking to calibrate complex biological models using incomplete, noisy, or heterogeneous data, thereby advancing both the predictive capability and mechanistic understanding of such systems.