Random vectors $(X,Y)$ are distributed over $\mathbb{R}^n \times \mathbb{R}^m$ with zero mean and covariance $\Sigma \in \mathbb{R}^{(n+m)\times (n+m)}_+.$What distributions attain the maximum value for $h(Y|X)$?Really, what do pathological/nonstandard distributions look like?
Similar-but-not-quite facts:
- The max of $h(Y)$ is attained iff(!) $Y$ is jointly-Gaussian.
- Among those $(X,Y)$ with fixed conditional covariance $K_{Y|X} = E[(Y-E[Y|X])(Y-E[Y|X])']$, the max of $h(Y|X)$ is attained at a joint-Gaussian distribution (El Gamal and Kim, Sec 2.2, eq 2.7). This implies it is sufficient, but not necessary the distribution be multivariate Gaussian. (Indeed, fixing the covariance is properly more stringent constraint than fixing the conditional covariance matrix)
- If $m=1$ then the max is attained with a joint-Gaussian distribution. (Sufficient, not necessary)https://tselab.stanford.edu/mirror/ee376a_winter1617/Lecture_18.pdf