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I have hard time on estimating the following integral on convex domain ($\mathcal D$) using Monte-Carlo approximation.

$$a = \int_{\mathcal D} dx f(x;\mu,\Sigma) $$

where $x \in \mathbb R^d$ and $f$ is Gaussian pdf. Note that this integral might not have a closed-form since $\mathcal D$ can be, for example, intersection of three disks. My naive approach is to sample $x_i \sim \mathcal N(\mu, \Sigma)$ and then verify whether it belongs to $\mathcal D$. If not, I discard it. After sampling for $N$ times, I have $n$ undiscarded samples $\{x_j\}_n$ that can be used for estimation,

$$\hat a \approx \frac 1 n\sum_{x_j} f(x_j;\mu,\Sigma)$$

However I am not sure if such approach can still yield consistent and unbiased estimator of $a$. If not, how can we do good estimation of $a$ in practice? Pointers to existing Python or R library can be helpful, thanks.

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    $\begingroup$ As a general problem, this can be difficult even for $d=1,$ because the expected rejection rate can be arbitrarily close to 100%. As an example of what specific cases might look like and of just one (out of myriad) ways of addressing them, see stats.stackexchange.com/questions/157963. $\endgroup$
    – whuber ♦
    Commented Mar 4 at 16:18
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    $\begingroup$ You need to find a manageable region $\mathcal R$containing $\mathcal D \subset\mathcal R$ and to simulate from an importance distribution restricted to this region. $\endgroup$
    – Xi'an
    Commented Mar 5 at 10:01

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