Is Multiple Integration Over Random Variables NP HARD In a seminar, in a passing moment a presenter mentioned that multiple integrals over random variables is NP-Hard. This means there is no efficient algorithm to compute the integral. 
So 

Theorem: Integral of p(x1, x2, x3,...,xn) over x1, x2, ...., xn is np
  hard.

A preliminary search over the internet didn't give anything to see. 
My question is:


*

*Is it correct? It looks like correct because in many machine learning models they say that estimating max likelihood over latent variables is np-hard, so can't do it, but I don't really feel confident asserting this.

*How to go about proving (showing the operation to be of exponential complexity) or disproving the result?

*Are there tractable or intractable probability distributions too? What do they imply?
 A: 1.Yes, it is correct .
2.Here is a sketch. Think of a simple subset of problem instances - binary (0/1) random vars (so the integral become a sum) .What we want to show is that ,given access to some oracle P, with the property that it returns for any n binary values $x_{1}, x_{2},...,x_{n}$ , in time at most polynomial in n, a value from the interval $[0,1]$ , it is NP-hard to compute the (partial) sums over subsets of variables. For these instances we can construct a reduction from SAT. Checking the satisfiability for fixed values of the variables in a formula can be done in polynomial time . We will use such an algorithm as the oracle. Assuming we have an efficient procedure to compute the partial sums , given the oracle , we can use it to efficiently decide the SAT problem : if the total sum is greater then 1, the answer is "yes" , otherwise the answer is "no")
3.Yes, some instances are easy, but in most practical settings one needs to rest on approximations (like Monte Carlo or variational methods  )
