RE: SAEM and IMP

Hi Emmanuel, While I am a strong advocate of using quasi-random rather than pseudo- random sequences for importance sampling in EM methods like IMP, there is a theoretical (and very real) problem with their use in the context you suggested in your message, namely with a multivariate t distribution as the importance sampling distribution. The 3S2 option implies you are using a Sobol quasi-random sequence, while the DF=7 implies the use of a multivariate T-distribution with 7 degrees of freedom. The standard way of generating a p-dimensional multivariate t -random variable with DF degrees of freedom is to generate a p-dimensional multivariate normal and then divide by an additional independent random variable which is basically the square root of a 1-d chi square random variable with DF degrees of freedom. Thus to generate a p-dimensional importance sample, you actually need to use p+1 independent random variables. If you simply use a p+1 dimensional Sobol vector as the base quasi-random draw, the nonlinear mapping from p+1 dimensions to the final p dimensional result destroys the low discrepancy property of the final sequence in the p-dimensional space and in fact introduces a significant amount of bias in the final result. The problem arises directly from the p+1 vs p dimensional mismatch. There is no problem if the final p-dimensional result can be generated from a p-dimensional quasi-random sequence, which is the case for multivariate normal Importance samples. So quasi random sequences should really only be used for the DF=0 multivariate normal importance sampling distribution case, not the multivariate DF>0 multivariate t case. I ran across this effect in testing the Sobol-based importance sampling EM algorithm QRPEM in Phoenix NLME. It is very real and the net effect is to introduce a significant bias. There is a partial fix that works but gives up some of the benefit of using low-discrepancy sequences - namely use a p-dimensional quasi-random vector to generate the p-dimensional multivariate normal, but then use a 1-d pseudo-random sequence to generate the chi-square random variable.