Integrate mixture of multivariate normal distributions

[Neddie]

I have a mixture of multivariate normal distributions, and I want to compute the integral with the first element of the input vector varying between specified limits, and the other elements varying from -infinity to +infinity. See attached pdf for equations. I've done it numerically but would like an analytic solution. Any advice? Thanks!

 

[chiro]

Hey Neddie and welcome to the forums.

I'm not an expert by any means so take this with a large grain of salt but since your mixture function is linear, can you not just use the linear property of the integral?

That is given Integral over space X f(X) + g(X) = Integral over space X f(X) + Integral over space X g(X), if you partition your integral using each component of the linear sum could you not just separate the integral accordingly and use the normal rules to evaluate each piece by itself which then can be combined?

I don't know if this is possible but it seems that at a glance you could partition the integral based on the linear nature of your mixture model but I am probably missing something here.

 

[Neddie]

Thanks chiro. You're quite right -- the integral over the mixture is trivially the sum of the integrals over each mixture component. It's actually those integrals I'm having trouble with.

 

[chiro]

To my knowledge, I don't think you can actually get an analytic expression for the normal in both univariate or multivariate unless you use some special function of some sort that is a series expansion.

For the standard normal we use tables and my guess (although may be wrong) is that the generalized expression just gets more ugly and not any better.