The general consensus seems to be that lifted inference is concerned with exploiting model symmetries and grouping indistinguishable objects at inference time. Since first-order probabilistic formalisms are essentially template languages providing a more compact representation of a corresponding ground model, lifted inference tends to work especially well in these models. We show that the notion of indistinguishability manifests itself on several different levels – the level of constants, the level of ground atoms (variables), the level of formulas (features), and the level of assignments (possible worlds). We discuss existing work in the MCMC literature on exploiting symmetries on the level of variable assignments and relate it to novel results in lifted MCMC.
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