Paper List

Journal: ArXiv Preprint
Published: Unknown
Mathematical BiologyComputational Statistics

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

Institute of Mathematics, Polish Academy of Sciences | Interdisciplinary Centre for Mathematical and Computational Modelling, University of Warsaw | Institute for Mathematics, Heidelberg University

Piotr Gwiazda, Alexey Kazarnikov, Anna Marciniak-Czochra, Zuzanna Szymańska
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The 30-Second View

IN SHORT: This paper addresses the core challenge of calibrating complex biological models (e.g., PDEs, agent-based models) with incomplete, noisy, or heterogeneous data, where traditional pointwise comparison methods fail due to system sensitivity and intrinsic variability.

Innovation (TL;DR)

  • Methodology Introduces the Correlation Integral Likelihood (CIL) framework, a unified approach for parameter estimation in systems with heterogeneous or chaotic dynamics (e.g., pattern formation, individual-based models), moving beyond classical Bayesian methods.
  • Methodology Proposes integration of deterministic gradient flow methods within the CIL framework to enhance inference efficiency and accuracy, compared to traditional stochastic sampling (e.g., MCMC).
  • Theory Generalizes the concept of correlation dimension from chaos theory to construct a robust metric for comparing the global geometric structure of model outputs (e.g., attractors, spatial patterns) rather than relying on unstable pointwise comparisons.

Key conclusions

  • The CIL method provides a theoretically grounded framework for parameter estimation in systems where solution heterogeneity (e.g., in Turing patterns or chaotic attractors) makes conventional likelihoods ineffective.
  • Integrating deterministic gradient flow sampling with the CIL framework can potentially enhance computational efficiency and inference accuracy compared to purely stochastic methods like MCMC, especially for high-dimensional parameter spaces.
  • The approach enables reliable model calibration and validation even with incomplete, noisy, or single-snapshot data, advancing the predictive capability and mechanistic understanding of complex biological systems.
Background and Gap： Current parameter inference methods struggle with systems exhibiting high sensitivity to initial conditions, intrinsic stochasticity, or spatial heterogeneity (e.g., chaotic dynamics, pattern formation models), where limited or noisy data prevents reliable calibration using traditional likelihoods based on direct trajectory matching.

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.