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Let $W_1, W_2 \in \mathbb{R}^{n \times n}$, $x, b_1, b_2 \in \mathbb{R}^{n}$ and $\sigma \in \mathbb{R}^{+}$, I am trying to integrate a Gaussian over the intersection of two half-space: $$ f(x) = \frac{1}{\sqrt{2 \pi} \sigma} \int_{\Omega_1 \cap \Omega_2} \exp\left(-\frac{\left\Vert x - t \right\Vert_2^2}{2 \sigma^2}\right) d t $$ where $$ \Omega_1 = \left\{ z \in \mathbb{R}^{n}: W_1 z + b_1 \geq 0 \right\} $$ and $$ \Omega_2 = \left\{ z \in \mathbb{R}^{n}: W_2 z + b_2 \geq 0 \right\} $$

Related questions have been answered here and here but I still haven't figure out a proper way to compute this integral. I also wondering if the result from this paper can be of help.

Thank you for your help.

Alex
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  • I don't think that you have any hope of an analytic expression in terms of elementary functions. There might be something "nice" in terms of the $\operatorname{erf}$ function, but the best that I would expect would be a numerical result. – Xander Henderson Apr 25 '23 at 19:34
  • An expression with the erf function would be great, if that's possible – Alex Apr 25 '23 at 19:37
  • One point to simplify matters is to start with lower-dimensional examples. The case of $n=1$ is probably too simple to be interesting, but $n=2$ already looks quite difficult with the range of possible parameters. – Semiclassical Apr 25 '23 at 19:39
  • Not sure that "half-spaces" in your title is literally correct. Have you sketched examples in 2D and perhaps 3D? – MathFont Apr 25 '23 at 19:43
  • I am not sure why that would be incorrect. In 1 dimension, the set ${z \in \mathbf{R} : w x + b \geq 0 }$ defines a half-space right? Do you mean that $W_1$ and $W_2$ would need to be vectors? – Alex Apr 25 '23 at 19:58
  • In 3D I think $\Omega_1$ is like a coordinate octant that has been distorted by a linear transformation so that its walls do not meet at right angles, and the intersection of two such regions is complicated. In 2D the intersection is a quadrilateral formed by two V-shaped wedges. – MathFont Apr 25 '23 at 21:54
  • The vector b has the effect of shifting the vertex of the Vee-shaped region. – MathFont Apr 26 '23 at 00:28

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