  Bivariate partial information decompositions (PIDs) characterize how the
information in a "message" random variable is decomposed between two
"constituent" random variables in terms of unique, redundant and synergistic
information components. These components are a function of the joint
distribution of the three variables, and are typically defined using an
optimization over the space of all possible joint distributions. This makes it
computationally challenging to compute PIDs in practice and restricts their use
to low-dimensional random vectors. To ease this burden, we consider the case of
jointly Gaussian random vectors in this paper. This case was previously
examined by Barrett (2015), who showed that certain operationally
well-motivated PIDs reduce to a closed form expression for scalar messages.
Here, we show that Barrett's result does not extend to vector messages in
general, and characterize the set of multivariate Gaussian distributions that
reduce to closed-form. Then, for all other multivariate Gaussian distributions,
we propose a convex optimization framework for approximately computing a
specific PID definition based on the statistical concept of deficiency. Using
simplifying assumptions specific to the Gaussian case, we provide an efficient
algorithm to approximately compute the bivariate PID for multivariate Gaussian
variables with tens or even hundreds of dimensions. We also theoretically and
empirically justify the goodness of this approximation.
