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Existing theories of control under uncertainty typically treat inference as cost-free, despite the substantial computational and energetic burden it imposes in both artificial and biological systems. To remedy this problem, we introduce a novel variant of the POMDP framework in which the information acquired through inference is treated as a resource that must be optimized alongside utility. Solving a local linear-Gaussian approximation of the resulting problem reveals three general principles of resource-efficient control. First, when information is costly, inference shifts from a Bayes-optimal (lossless) compression of the past to a lossy regime that strategically leaves some uncertainty unresolved to optimize resource use. 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