Let $\kappa(x, x')=\exp(-\frac{1}{2\sigma^2}||x-x'||)$ be the Gaussian kernel with the multivariate random variable $x, x' \in \mathbb{R}^D$ and scalar parameter $\sigma \in \mathbb{R}_+$. I am interested in the expectation $\mathbb{E}_{x,x'} [\kappa(x,x')]$ with $p(x)=p(x')=\mathcal{N}(\mu, \Sigma)$. Is there any closed form solution to it?
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1Start by taking advantage of the fact that $X - X'$ is distributed as $\mathcal{N}(0, 2\Sigma)$. – A rural reader Feb 24 '21 at 15:13
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Thx, this solves the issue – Andreas Look Feb 24 '21 at 19:31