The hippocampus reinstates learned sequences in temporally compressed "replay" epochs inside sharp-wave ripples (SWRs), which occur while an animal is at rest. Replay is required for the long-term consolidation of memories. However, the content of these replay episodes, and in particular whether specific experiences or a task-optimized representation of the world are consolidated, remains controversial. Using two-photon calcium imaging of CA3 outputs combined with simultaneous local field potential (LFP) recordings, our group has shown that the statistics of replay deviate significantly from those of experience: environmental features which are highly salient in experience may be either selected or suppressed in replay depending on their behavioral importance. We propose a parsimonious novel mechanism for this phenomenon: the reuse of the symmetric spike-time dependent plasticity rule (sSTDP), previously reported at CA3 excitatory synapses, to remodel inhibitory synapses as well. We show using three levels of modeling—spiking network, detailed biophysical, and abstract normative—that this mechanism enables efficient inference of the latent statistical structure of the world given noisy observations. We develop a mathematical theory of how sSTDP shapes sequence dynamics in a recurrent network, and prove that replay, viewed as a statistical estimator of a latent sequence, converges asymptotically to the true sequence. Finally, we make a number of predictions illustrating the power of inhibitory plasticity as a conceptual advance in our understanding of hippocampal dynamics and memory consolidation, foremost that CA3 replay consolidates "world structure" rather than specific experience. Our experimental and theoretical work here outlines a potential direct link between the synaptic and cognitive levels of memory consolidation.