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The Compound Information Bottleneck Outlook

We formulate and analyze the compound information bottleneck programming. In this problem, a Markov chain $ \mathsf{X} \rightarrow \mathsf{Y} \rightarrow \mathsf{Z} $ is assumed with fixed marginal distributions $\mathsf{P}_{\mathsf{X}}$ and $\mathsf{P}_{\mathsf{Y}}$, and the mutual information between $ \mathsf{X} $ and $ \mathsf{Z} $ is sought to be maximized over the choice of conditional probability of $\mathsf{Z}$ given $\mathsf{Y}$ from a given class, under the \textit{worst choice} of the joint probability of the pair $(\mathsf{X},\mathsf{Y})$ from a different class. We consider several classes based on extremes of: mutual information; minimal correlation; total variation; and the relative entropy class. We provide values, bounds, and various characterizations for specific instances of this problem: the binary symmetric case, the scalar Gaussian case, the vector Gaussian case and the symmetric modulo-additive case. Finally, for the general case, we propose a Blahut-Arimoto type of alternating iterations algorithm to find a consistent solution to this problem.