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

Area of Science:

• Mathematical Biology
• Computational Biology
• Systems Biology

Background:

• Calibrating mathematical models of biological processes is crucial for predictive accuracy and mechanistic insight.
• Challenges include limited/noisy data, biological variability, and model computational complexity.
• Parameter inference in partial differential equation (PDE) models is particularly difficult.

Purpose of the Study:

• To introduce a unified mathematical framework for parameter inference in biological systems.
• To address challenges in calibrating complex biological models with heterogeneous or chaotic dynamics.
• To provide a practical and theoretically grounded toolbox for researchers.

Main Methods:

• Introduction of the Correlation Integral Likelihood (CIL) method.
• Development of a unified mathematical framework for parameter estimation.
• Comparison of stochastic sampling (e.g., Markov Chain Monte Carlo) with deterministic gradient flow approaches.

Main Results:

• The CIL method is versatile and applicable to pattern formation and individual-based models.
• Demonstration of the CIL method's utility in mathematical biology applications.
• Integration strategies for stochastic and deterministic methods to enhance inference performance.

Conclusions:

• The CIL method offers a robust approach for parameter inference in complex biological models.
• The framework facilitates model calibration using incomplete, noisy, or heterogeneous data.
• This work advances predictive capabilities and mechanistic understanding in mathematical biology.