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A popular theory of perceptual processing holds that the brain learns both a generative model of the world and a paired recognition model using variational Bayesian inference. Most hypotheses of how the brain might learn these models assume that neurons in a population are conditionally independent given their common inputs. This simplification is likely not compatible with the type of local recurrence observed in the brain. Seeking an alternative that is compatible with complex inter-dependencies yet consistent with known biology, we argue here that the cortex may learn with an adversarial algorithm. Many observable symptoms of this approach would resemble known neural phenomena, including wake/sleep cycles and oscillations that vary in magnitude with surprise, and we describe how further predictions could be tested. We illustrate the idea on recurrent neural networks trained to model image and video datasets. This framework for learning brings variational inference closer to neuroscience and yields multiple testable hypotheses.