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There is a growing body of evidence that the human brain may be organized according to principles of predictive processing. An important conjecture in neuroscience is that a brain organized in this way can effectively and efficiently approximate Bayesian inferences. Given that many forms of cognition seem to be well characterized as a form of Bayesian inference, this conjecture has great import for cognitive science. It suggests that predictive processing may provide a neurally plausible account of how forms of cognition that are modeled as Bayesian inference may be physically implemented in the brain. Yet, as we show in this paper, the jury is still out on whether or not the conjecture is really true. Specifically, we demonstrate that each key subcomputation invoked in predictive processing potentially hides a computationally intractable problem. We discuss the implications of these sobering results for the predictive processing account and propose a way to move forward.