  Combining the Information Bottleneck model with deep learning by replacing
mutual information terms with deep neural nets has proved successful in areas
ranging from generative modelling to interpreting deep neural networks. In this
paper, we revisit the Deep Variational Information Bottleneck and the
assumptions needed for its derivation. The two assumed properties of the data
$X$, $Y$ and their latent representation $T$ take the form of two Markov chains
$T-X-Y$ and $X-T-Y$. Requiring both to hold during the optimisation process can
be limiting for the set of potential joint distributions $P(X,Y,T)$. We
therefore show how to circumvent this limitation by optimising a lower bound
for $I(T;Y)$ for which only the latter Markov chain has to be satisfied. The
actual mutual information consists of the lower bound which is optimised in
DVIB and cognate models in practice and of two terms measuring how much the
former requirement $T-X-Y$ is violated. Finally, we propose to interpret the
family of information bottleneck models as directed graphical models and show
that in this framework the original and deep information bottlenecks are
special cases of a fundamental IB model.
