Merging Kalman Filtering with the information bottleneck technique for optimal state inference

Living systems must make estimates of a state of a stimulus in order to make decisions relevant for survival. One approach living systems can take to constructing these estimates involves leveraging past state estimates and present sensory responses in a Kalman filter. However, unlike in engineering, living systems are capable of changing their sensory response over evolutionary timescales, subject to constraints on metabolism, computing power, and other biologically relevant limitations. The optimal choice for a sensory model navigates the resource constraints, providing information about the current state, and providing information about future states for later inference. To determine how this sensory model should be constructed, we use the Information Bottleneck method. By connecting Kalman filter state inference with Information Bottleneck-based sensory models, we can demonstrate how biological systems can optimally take advantage of both memory and sensory information simultaneously. We show this in both one- and two-dimensional Gaussian stimuli. We present the analytic solution to one-dimensional stimuli, and observe several regimes for the sensory model as a function of correlation time and resource constraints.