Active inference is a normative principle underwriting perception, action, planning, decision-making and learning in biological or artificial agents. From its inception, its associated process theory has grown to incorporate complex generative models, enabling simulation of a wide range of complex behaviours. Due to successive developments in active inference, it is often difficult to see how its underlying principle relates to process theories and practical implementation. In this paper, we try to bridge this gap by providing a complete mathematical synthesis of active inference on discrete state-space models. This technical summary provides an overview of the theory, derives neuronal dynamics from first principles and relates this dynamics to biological processes. Furthermore, this paper provides a fundamental building block needed to understand active inference for mixed generative models; allowing continuous sensations to inform discrete representations. This paper may be used as follows: to guide research towards outstanding challenges, a practical guide on how to implement active inference to simulate experimental behaviour, or a pointer towards various in-silico neurophysiological responses that may be used to make empirical predictions.

Multisensory perception is regarded as one of the most prominent examples where human behaviour conforms to the computational principles of maximum likelihood estimation (MLE). In particular, observers are thought to integrate auditory and visual spatial cues weighted in proportion to their relative sensory reliabilities into the most reliable and unbiased percept consistent with MLE. Yet, evidence to date has been inconsistent. The current pre-registered, large-scale (N = 36) replication study investigated the extent to which human behaviour for audiovisual localization is in line with maximum likelihood estimation. The acquired psychophysics data show that while observers were able to reduce their multisensory variance relative to the unisensory variances in accordance with MLE, they weighed the visual signals significantly stronger than predicted by MLE. Simulations show that this dissociation can be explained by a greater sensitivity of standard estimation procedures to detect deviations from MLE predictions for sensory weights than for audiovisual variances. Our results therefore suggest that observers did not integrate audiovisual spatial signals weighted exactly in proportion to their relative reliabilities for localization. These small deviations from the predictions of maximum likelihood estimation may be explained by observers' uncertainty about the world's causal structure as accounted for by Bayesian causal inference.

We use our eyes to assess the value of objects around us and carefully fixate options that we are about to choose. Neurons in the prefrontal cortex reliably encode the value of fixated options, which is essential for decision making. Yet as a decision unfolds, it remains unclear how prefrontal regions determine which option should be fixated next. Here we show that anterior cingulate cortex (ACC) encodes the value of options in the periphery to guide subsequent fixations during economic choice. In an economic decision-making task involving four simultaneously presented cues, we found rhesus macaques evaluated cues using their peripheral vision. This served two distinct purposes: subjects were more likely to fixate valuable peripheral cues, and more likely to choose valuable options whose cues were never even fixated. ACC, orbitofrontal cortex, dorsolateral pre-frontal cortex, and ventromedial prefrontal cortex neurons all encoded cue value post-fixation. ACC was unique, however, in also encoding the value of cues before fixation and even cues that were never fixated. This pre-saccadic value encoding by ACC predicted which cue was next fixated during the decision process. ACC therefore conducts simultaneous processing of peripheral information to guide information sampling and choice during decision making.