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In this Review, we discuss advances in computational neuroscience that relate to psychiatry. We review computational psychiatry in terms of the ambitions of investigators, emerging domains of application, and future work. Our focus is on theoretical formulations of brain function that put subjective beliefs and behaviour within formal (computational) frameworks—frameworks that can be grounded in neurophysiology down to the level of synaptic mechanisms. Understanding the principles that underlie the brain's functional architecture might be essential for an informed phenotyping of psychopathology in terms of its pathophysiological underpinnings. We focus on active (Bayesian) inference and predictive coding. Specifically, we show how basic principles of neuronal computation can be used to explain psychopathology, ranging from impoverished theory of mind in autism to abnormalities of smooth pursuit eye …