Information Bottleneck on General Alphabets

We prove rigorously a source coding theorem that can probably be considered folklore, a generalization to arbitrary alphabets of a problem motivated by the Information Bottleneck method. For general random variables (Y, X), we show essentially that for some n ∈ \mathbbN, a function f with rate limit log|f| ≤ nR and I(Yⁿ;f(Xⁿ)) ≥ nS exists if and only if there is a random variable U such that the Markov chain Y°-X°-U holds, I(U;X) ≤ R and I(U;Y) ≥ S. The proof relies on the well established discrete case and showcases a technique for lifting discrete coding theorems to arbitrary alphabets.