The free energy principle for action and perception: A mathematical review

Abstract

The 'free energy principle' (FEP) has been suggested to provide a unified theory of the brain, integrating data and theory relating to action, perception, and learning. The theory and implementation of the FEP combines insights from Helmholtzian 'perception as inference', machine learning theory, and statistical thermodynamics. Here, we provide a detailed mathematical evaluation of a suggested biologically plausible implementation of the FEP that has been widely used to develop the theory. Our objectives are (i) to describe within a single article the mathematical structure of this implementation of the FEP; (ii) provide a simple but complete agent-based model utilising the FEP; (iii) disclose the assumption structure of this implementation of the FEP to help elucidate its significance for the brain sciences. be shown that minimising IFE makes the R-density a good approximation to posterior density of environmental variables given sensory data. Under this interpretation the surprisal term in the IFE becomes more akin to the negative of log model evidence defined in more standard implementations of variational Bayes [30].130 The Action-Perception CycleMinimising IFE by updating the R-density provides an upper-bound on surprisal but cannot minimise it directly. The FEP suggests that organisms also act on their environment to change sensory input, and thus minimise surprisal indirectly [1,2]. The mechanism underlying this process is formally symmet-135 ric to perceptual inference, i.e., rather than inferring the cause of sensory data an organism must infer actions that best make sensory data accord with an internal environmental model [9]. Thus, the mechanism is often referred to as

A predictive processing model of episodic memory and time perception

“…In that study, model-based estimates replicated key features of human duration estimates of the exact same videos, including biases related to scene type (busy city scenes estimated as longer that quiet office scenes, for example; see also 12,13 ). The model displayed conceptual similarities with the predictive processing account of perception wherein perception is proposed to operate as a function of both sensory predictions and current sensory stimulation, with perceptual content understood as the brain's "best guess" (Bayesian posterior) of the causes of current sensory input given the prior expectations or predictions [14][15][16][17][18] . However, this previous model 4 had only very simple "memory", in the form of content-less tokens marking the occurrence of salient events in perception.…”

Dopamine role in learning and action inference

“…Although there are multiple internal 39 dimensions which animals need to optimize, e.g. temperature (Buckley, Kim, McGregor, & Seth, 2017), 40 internal salt levels (Cone et al, 2016), etc., for simplicity, we will consider a single dimension of food 41 reserves. As there exists an optimal level of food reserves, an animal should only seek food resources, 42 if its reserves are below the optimum value.…”

A probabilistic interpretation of PID controllers using active inference

“…In this work we will not provide a complete derivation of the active inference scheme, referring to previous treatments [26,16,8] for more details. Here we will begin from the Laplace encoded variational free energy for a univariate case:…”

“…The constants within the free energy will not be discussed here since they play no role in the minimisation scheme we present. As previously shown [16,8], to minimise equation (3) we need to specify the agent's generative density P (ρ, µ x ) = P (ρ|µ x )P (µ x ) introducing a likelihood P (ρ|µ x ) and a prior P (µ x ) in terms of an agent's beliefs µ x . These probabilities can be specified by a generative model in the form of a state space model.…”

“…Probabilistic approaches to the study of living systems and cognition are becoming increasingly popular in the natural sciences. In particular for the brain sciences, theories inspired the Bayesian brain hypothesis such as predictive coding, the free energy principle and active inference have been used to explain mechanism of the brain and cognition including perception, action and higher order functions [14,32,30,21,11,27,7,8].…”