  We define the relevant information in a signal $x\in X$ as being the
information that this signal provides about another signal $y\in \Y$. Examples
include the information that face images provide about the names of the people
portrayed, or the information that speech sounds provide about the words
spoken. Understanding the signal $x$ requires more than just predicting $y$, it
also requires specifying which features of $\X$ play a role in the prediction.
We formalize this problem as that of finding a short code for $\X$ that
preserves the maximum information about $\Y$. That is, we squeeze the
information that $\X$ provides about $\Y$ through a `bottleneck' formed by a
limited set of codewords $\tX$. This constrained optimization problem can be
seen as a generalization of rate distortion theory in which the distortion
measure $d(x,\x)$ emerges from the joint statistics of $\X$ and $\Y$. This
approach yields an exact set of self consistent equations for the coding rules
$X \to \tX$ and $\tX \to \Y$. Solutions to these equations can be found by a
convergent re-estimation method that generalizes the Blahut-Arimoto algorithm.
Our variational principle provides a surprisingly rich framework for discussing
a variety of problems in signal processing and learning, as will be described
in detail elsewhere.
