The ‘Bayesian Coding Hypothesis’ formalises the classic Helmholtzian picture of perception as inverse inference, stating that the brain uses Bayes’ rule to compute posterior belief distributions over states of the world. There is much behavioural evidence that human observers can behave Bayes-optimally, and there is theoretical work that shows how populations of neurons might perform the underlying computations. There are, however, many remaining questions, three of which are addressed in this thesis. First, we investigate the limits of optimality, demonstrating that observers can correctly integrate an external loss function with their uncertainty about a very simple stimulus, but behave suboptimally with respect to highly complex stimuli. Second, we use the same paradigm in a collaborative fMRI study, asking where along the path from sensory to motor areas a loss function is integrated with sensory uncertainty. Our results suggest that value a¿ects a fronto-striatal action selection network rather than directly impacting on sensory processing. Finally, we consider a major theoretical problem – the demonstrations of optimality that dominate the ¿eld have been obtained in tasks with a small number of objects in the focus of attention. When faced instead with a complex scene, the brain can’t be Bayes-optimal everywhere. We suggest that a general limitation on the representation of complex posteriors causes the brain to make approximations, which are then locally re¿ned by attention. This framework extends ideas of attention as Bayesian prior, and uni¿es apparently disparate attentional ‘bottlenecks’. We present simulations of three key paradigms, and discuss how such modelling could be extended to more detailed, neurally inspired settings. Broadening the Bayesian picture of perception and strengthening its connection to neuroscienti¿c and psychological literatures is critical to its future as a comprehensive theory of neural inference, and the thesis concludes with a brief discussion of future challenges in this direction.