AbstractModel-based decision making relies on the construction of an accurate representation of the underlying state-space, and localization of one’s current state within it. One way to localize is to recognize the state with which incoming sensory observations have been previously associated. Another is to update a previous state estimate given a known transition. In practice, both strategies are subject to uncertainty and must be balanced with respect to their relative confidences; robust learning requires aligning the predictions of both models over historic observations. Here, we propose a dual-systems account of the hippocampal-entorhinal system, where sensory prediction errors between these models duringonlineexploration of state space initiateofflineprobabilistic inference.Offlineinference computes ametricembedding on grid cells of anassociativeplace graph encoded in the recurrent connections between place cells, achieved by message passing between cells representing non-local states. We provide testable explanations for coordinated place and grid cell ‘replay’ as efficient message passing, and for distortions, partial rescaling and direction-dependent offsets in grid patterns as the confidence weighted balancing of model priors, and distortions to grid patterns as reflecting inhomogeneous sensory inputs across states.Author SummaryMinimising prediction errors between transition and sensory input (observation) models predicts partial rescaling and direction-dependent offsets in grid cell firing patterns.Inhomogeneous sensory inputs predict distortions of grid firing patterns duringonlinelocalisation, and local changes of grid scale duringofflineinference.Principled information propagation duringofflineinference predicts coordinated place and grid cell ‘replay’, where sequences propagate between structurally related features.