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 an abstract representation of the world are consolidated, remains controversial. Using two-photon calcium imaging of CA3 outputs combined with simultaneous local field potential recordings, we show that the statistics of replay deviate significantly from those of experience: highly salient environmental features may be either selected or suppressed in replay depending on their statistical generalizability. We propose a parsimonious 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--that this mechanism enables inference of the generalizable structure of the world given noisy observations. We analyze mathematically how sSTDP shapes sequence dynamics in a recurrent network, and show that replay is a consistent estimator of a latent sequence under an injection noise model. Finally, we combine a novel sparse optogenetic strategy with calcium imaging to experimentally test the model prediction that artificially induced nongeneralizable representations will be suppressed from SWRs. We find that these "artificial cue cells" are suppressed during SWRs compared to non-induced cells, consistent with our predictions. Our experimental and theoretical work here outlines a potential direct link between the synaptic and cognitive levels of memory consolidation.