  We take a closer look at the structure of bivariate dependency induced by a
pair of predictor random variables $(X_1, X_2)$ trying to synergistically,
redundantly or uniquely encode a target random variable $Y$. We evaluate a
recently proposed measure of redundancy based on the G\'acs-K\"{o}rner common
information (Griffith et al., Entropy 2014) and show that the measure, in spite
of its elegance is degenerate for most non-trivial distributions. We show that
Wyner's common information also fails to capture the notion of redundancy as it
violates an intuitive monotonically non-increasing property. We identify a set
of conditions when a conditional version of G\'acs and K\"{o}rner's common
information is an ideal measure of unique information. Finally, we show how the
notions of approximately sufficient statistics and conditional information
bottleneck can be used to quantify unique information.
