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Active inference is a probabilistic framework for modelling the behaviour ofbiological and artificial agents, which derives from the principle ofminimising free energy. In recent years, this framework has successfully beenapplied to a variety of situations where the goal was to maximise reward,offering comparable and sometimes superior performance to alternativeapproaches. In this paper, we clarify the connection between rewardmaximisation and active inference by demonstrating how and when activeinference agents perform actions that are optimal for maximising reward.Precisely, we show the conditions under which active inference produces theoptimal solution to the Bellman equation--a formulation that underlies severalapproaches to model-based reinforcement learning and control. On partiallyobserved Markov decision processes, the standard active inference scheme canproduce Bellman optimal actions for planning horizons of 1, but not beyond. Incontrast, a recently developed recursive active inference scheme (sophisticatedinference) can produce Bellman optimal actions on any finite temporal horizon.We append the analysis with a discussion of the broader relationship betweenactive inference and reinforcement learning.

Reward Maximisation through Discrete Active Inference