A probabilistic interpretation of PID controllers using active inference

Abstract In the past few decades, probabilistic interpretations of brain functions have become widespread in cognitive science and neuroscience. The Bayesian brain hypothesis, predictive coding, the free energy principle and active inference are increasingly popular theories of cognitive functions that claim to unify understandings of life and cognition within general mathematical frameworks derived from information theory, statistical physics and machine learning. Furthermore, it has been argued that one such proposal, active inference, combines both information and control theory and has its roots in cybernetics studies of the brain. The connections between information and control theory have been discussed since the 1950’s by scientists like Shannon and Kalman and have recently risen to prominence in modern stochastic optimal control theory. How-ever, the implications of the confluence of these two theoretical frame-works for the biological sciences have been slow to emerge. Here we argue that if the active inference proposal is to be taken as a general process theory for biological systems, we need to consider how existing control theoretical approaches to biological systems relate to it. In this work we will focus on PID (Proportional-Integral-Derivative) controllers, one of the most common types of regulators employed in engineering and more recently used to explain behaviour in biological systems, e.g. chemotaxis in bacteria and amoebae or robust adaptation in biochemical networks. Using active inference, we derive a probabilistic interpretation of PID controllers, showing how they can fit a more general theory of life and cognition under the principle of (variational) free energy minimisation once we use only simple linear generative models.