  Many approximate Bayesian inference methods assume a particular parametric
form for approximating the posterior distribution. A multivariate Gaussian
distribution provides a convenient density for such approaches; examples
include the Laplace, penalized quasi-likelihood, Gaussian variational, and
expectation propagation methods. Unfortunately, these all ignore the potential
skewness of the posterior distribution. We propose a modification that accounts
for skewness, where key statistics of the posterior distribution are matched
instead to a multivariate skew-normal distribution. A combination of simulation
studies and benchmarking were conducted to compare the performance of this
skew-normal matching method (both as a standalone approximation and as a
post-hoc skewness adjustment) with existing Gaussian and skewed approximations.
We show empirically that for small and moderate dimensional cases, skew-normal
matching can be much more accurate than these other approaches. For post-hoc
skewness adjustments, this comes at very little cost in additional
computational time.
