I'm curious if there is a problem (e.g. something like perfect matchings on a given graph, number of solutions to a boolean equation, etc. for precise frameowork see JVV86) such that:
1) It is hard to sample approximately uniformly at random from the set of all solutions
2) It is easy to approximately integrate any (polynomially computable) function over the solutions.
I would also be happy with replacing 2) by the weaker:
2') It is easy to approximate the expectation of any polynomially computable function over the uniform distribution on the solutions.
If 2) holds, then by integrating the constant function $1$, one can approximately count the number of solutions to the problem.
If 2) holds but 1) does not, then the problem cannot be self-reducible, as otherwise by reductions in JVV86 there would be an efficient algorithm for approximate uniform sampling by using the observation of the previous bullet.
A reasonable candidate technique for accomplishing 2) without 1) is importance sampling.
I know of situations where sampling is easy and counting is hard (so integration is hard); the case of uniformly sampling solutions to a DNF equation is explained in JVV86 section 4.
A specific candidate problem I would be interested in is the problem of sampling directed simple cycles from a digraph. (It is shown in section 5. of JVV86 that this problem is hard.) However, I know that counting the number of directed simple cycles is NP-hard, by a similar argument to that presented in JVV, and therefore so is integrating over them. However, perhaps 2') can be solved in this case?
Any references that seem relevant would be very appreciated, even if they are not a direct answer to my question.