For a propositional model (where each variable has a distinct name), you should have a look at probabilistic graphical models (in particular Markov networks). They are very closely related to SAT and CSP, since they are basically a generalization, but still fall into the same complexity class #P
.
If you are interested in concise, first order representation of these models, you should look into statistical relational learning or first order probabilistic models (synonyms). Here, the model is expressed in a "lifted" form. E.g. possibly probabilistic constraints of the following form, using variables ranging over some object domain:
on(?x,?y) => largerThan(?y,?x)
Inferences with these models that do not rely on generating the ground model are done in the field of lifted probabilistic inference.