  We develop a version of stochastic control that accounts for computational
costs of inference. Past studies identified efficient coding without control,
or efficient control that neglects the cost of synthesizing information. Here
we combine these concepts into a framework where agents rationally approximate
inference for efficient control. Specifically, we study Linear Quadratic
Gaussian (LQG) control with an added internal cost on the relative precision of
the posterior probability over the world state. This creates a trade-off: an
agent can obtain more utility overall by sacrificing some task performance, if
doing so saves enough bits during inference. We discover that the rational
strategy that solves the joint inference and control problem goes through phase
transitions depending on the task demands, switching from a costly but optimal
inference to a family of suboptimal inferences related by rotation
transformations, each misestimate the stability of the world. In all cases, the
agent moves more to think less. This work provides a foundation for a new type
of rational computations that could be used by both brains and machines for
efficient but computationally constrained control.
