Perception is often characterized as an inference process in which the brain unconsciously reasons about causes in the world based on their sensory effects. According to this "Bayesian Brain" hypothesis, perception is an expression of beliefs about latent variables in a statistical model of the world, and sensory neurons implement some form of approximate inference algorithm. The initial chapters of this dissertation summarize and link existing neural models of inference and suggest new ways to categorize and interpret them. Subsequent chapters derive predictions for both behavioral and neural consequences of approximate inference in the special case of visual categorization tasks. In particular, when approximate prior expectations impinge on sensory representations (a feature of optimal inference), behavioral consequence is the emergence of temporal biases, and a neural consequence is task-dependent noise covariability. The predictions for behavioral biases are verified in human subjects using a novel pair of visual psychophysics tasks. The predictions for neurons are explored using multi-electrode recordings from primary visual cortex of macaque monkeys, but with inconclusive results. We conclude with discussion of the nature of inference in the brain more broadly and sketch alternative perspectives on the results presented here.