I wonder if the following integral has an analytical solution. $$ \int_{-\infty}^{\infty}\frac{w_1 N_1(x)}{\sum_{i=1}^{n} w_i N_i(x)} N_0(x) dx $$ where $w_1, \ldots, w_n$ are positive constants, $N_0, N_1, \ldots, N_n$ are Gaussian functions with different parameters.

If no analytical solution exists, what would be the best way to efficiently approximate it? I'm considering importance sampling by sampling from $N_0$. Any suggestion of a more efficient yet not too complicated method would be appreciated.

A previous question on this site seems quite related to mine, but no solution has been given there: How to solve an integral with a Gaussian Mixture denominator?

  • $\begingroup$ Just out of curiosity, what is the context where this problem arose? $\endgroup$ – mikkola Mar 13 '16 at 9:18
  • $\begingroup$ @mikkola I'm trying to do maximum likelihood estimation of a complicated model that contains conditional Gaussian mixture models, which requires the computation of this integration. $\endgroup$ – took Mar 13 '16 at 23:37

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