Control Theory and Experimental Design in Diffusion Processes

TL;DR

This paper develops control strategies for nonlinear diffusion processes to maximize information gain about parameters, using optimal control and filtering techniques, demonstrated on physics, neuroscience, and ecology models.

Contribution

It introduces a two-step approach combining stochastic control and filtering for optimal experimental design in diffusion processes.

Findings

Effective control policies increase Fisher information in tested models.

Method adapts to incomplete, noisy observations.

Demonstrated applicability across diverse scientific models.

Abstract

This paper considers the problem of designing time-dependent, real-time control policies for controllable nonlinear diffusion processes, with the goal of obtaining maximally-informative observations about parameters of interest. More precisely, we maximize the expected Fisher information for the parameter obtained over the duration of the experiment, conditional on observations made up to that time. We propose to accomplish this with a two-step strategy: when the full state vector of the diffusion process is observable continuously, we formulate this as an optimal control problem and apply numerical techniques from stochastic optimal control to solve it. When observations are incomplete, infrequent, or noisy, we propose using standard filtering techniques to first estimate the state of the system, then apply the optimal control policy using the posterior expectation of the state. We assess the effectiveness of these methods in 3 situations: a paradigmatic bistable model from statistical physics, a model of action potential generation in neurons, and a model of a simple ecological system.