I need to calculate the average outputs ($Y_j$) from a non-linear simulation model ($M_j$) which takes a large number of possible input strings ($B_i$).
$$Y_j = E(M_j(B_i))$$
Each $B_i=(b_{i,1},b_{i,2},...,b_{i,N})$ where $N \ge 12$, and the $b_{i,k}$ are the possible values of a Poisson random variable with known mean.
The calculation of $M_j$ is very expensive so I want to find a way to subsample the $B_i$ to get an accurate estimate of $Y_j$ without having to calculate it too exhaustively (something in the spirit of Latin Hypercube Sampling or Gaussian Quadrature or Sparse Grid method perhaps).
I think $M_j$ is monotonic on $b_{i,k}$ but the $B_i$ are not ordered.
Any ideas?