I need to estimate $$ \mathbb{E}_x[f_i(x)] = \int_{\mathbb{R}^n} f_i(x) p(x) dx $$ for many functions $f_i(x)$, where $p(x)$ is the density of a normal distribution. The evaluation of all the functions $f_i(x)$ is expensive. I was looking into Bayesian Quadrature but I cannot find an (approximately) optimal way of choosing sample points.
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$\begingroup$ How accurate does this estimate need to be? Highly accurate or somewhat accurate? Also: what properties about $f$ do you know? Is it $L$-Lipschitz or smooth in some other sense, or does $f$ vary widely? $\endgroup$– cdipaoloAug 18, 2019 at 15:41
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$\begingroup$ The approximation doesn't need to be higly accurate, $f$ is L-Lipschitz. $\endgroup$– RoshAug 18, 2019 at 16:23
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$\begingroup$ in that case, you can always get a Monte-Carlo estimate (depending on your internal definition of “highly accurate.”) Since $f$ is Lipschitz, $f(x)$ is sub-Gaussian and obeys nice concentration properties. $\endgroup$– cdipaoloAug 19, 2019 at 5:29
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$\begingroup$ The problem is that Monte-Carlo converges slowly. I was looking to find a way of defining the sampling points (with the corresponding weights) to converge faster than Monte-Carlo. $\endgroup$– RoshAug 19, 2019 at 8:27
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Please have a look at the answers to this question on CrossValidated. There you can find a worked out solution (including R-code) based on Gauss-Hermite quadrature, which is especially designed for this problem (note that it requires a variable transformation) and -in a different answer- a comparsion of MonteCarlo (slow) with the R builtin function integrate (fast).
