  Underlying many Bayesian inference techniques that seek to approximate the
posterior as a Gaussian distribution is a fundamental linear algebra problem
that must be solved for both the mean and key entries of the covariance. Even
when the true posterior is not Gaussian (e.g., in the case of nonlinear
measurement functions) we can use variational schemes that repeatedly solve
this linear algebra problem at each iteration. In most cases, the question is
not whether a solution to this problem exists, but rather how we can exploit
problem-specific structure to find it efficiently. Our contribution is to
clearly state the Fundamental Linear Algebra Problem of Gaussian Inference
(FLAPOGI) and to provide a novel presentation (using Kronecker algebra) of the
not-so-well-known result of Takahashi et al. (1973) that makes it possible to
solve for key entries of the covariance matrix. We first provide a global
solution and then a local version that can be implemented using local message
passing amongst a collection of agents calculating in parallel. Contrary to
belief propagation, our local scheme is guaranteed to converge in both the mean
and desired covariance quantities to the global solution even when the
underlying factor graph is loopy; in the case of synchronous updates, we
provide a bound on the number of iterations required for convergence. Compared
to belief propagation, this guaranteed convergence comes at the cost of
additional storage, calculations, and communication links in the case of loops;
however, we show how these can be automatically constructed on the fly using
only local information.
