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How can one compute the mutual information of Gaussian mixture distribution \begin{equation} \mathrm{I}(X;C|Y)=H[C|Y]-H(C|X,Y)=-\int\int\sum_{C}P(X,C,Y)\log\frac{P(X,C|Y)}{P(C|Y)P(X|Y)}\mathrm{d}X\mathrm{d}Y \end{equation} where the prior distribution of $C$ and $X$ are given as $C\sim\mathrm{Cat}(\pi)$ and $P(X|c=k)=\mathcal{N}(\mu_k,\sigma^2_k\mathbb{I})$, Finally the distribution of $X$ is given as $P(X)=\mathcal{N}(\boldsymbol{0}|\mathbb{I})$?

Dalek
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  • Take the summation outside the integral. To evaluate the integrals, apply the [formula](https://stats.stackexchange.com/questions/7440). – whuber Oct 10 '21 at 17:46

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