Reframing the expected free energy: four formulations and a unification

Active inference is a process theory of perception, learning, and decision making that is applied to a range of research fields, including neuroscience, robotics, psychology, and machine learning. Active inference rests on an objective function called the expected free energy, which can be justified by the intuitive plausibility of its formulations—for example, the risk plus ambiguity and information gain/pragmatic value formulations. This letter seeks to formalize the problem of deriving these formulations from a single root expected free energy definition—the unification problem. Then we analyze two approaches to defining expected free energy. More precisely, the expected free energy is either defined as (1) the risk over observations plus ambiguity or (2) the risk over states plus ambiguity. In the first setting, no rigorous mathematical justification for the expected free energy has been proposed to date, but all the formulations can be recovered from it by assuming that the likelihood of target distribution

(o|s) is the likelihood of the generative model

(o|s). Importantly, under this likelihood constraint, if the likelihood is lossless,1 then prior preferences over observations can be defined arbitrarily. However, in the more general case of partially observable Markov decision processes (POMDPs), we demonstrate that the likelihood constraint effectively restricts the set of valid prior preferences over observations. Indeed, only a limited class of prior preferences over observations is compatible with the likelihood mapping of the generative model. In the second setting, a justification of the root expected free energy definition exists, but this setting only accounts for two formulations: the risk over states plus ambiguity and entropy plus expected energy formulations. We conclude with a discussion of the conditions under which a unification of expected free energy formulations has been proposed in the literature by appeal to the free energy principle in the specific context of systems without random fluctuations.