  Variational methods are attractive for computing Bayesian inference for
highly parametrized models and large datasets where exact inference is
impractical. They approximate a target distribution - either the posterior or
an augmented posterior - using a simpler distribution that is selected to
balance accuracy with computational feasibility. Here we approximate an
element-wise parametric transformation of the target distribution as
multivariate Gaussian or skew-normal. Approximations of this kind are implicit
copula models for the original parameters, with a Gaussian or skew-normal
copula function and flexible parametric margins. A key observation is that
their adoption can improve the accuracy of variational inference in high
dimensions at limited or no additional computational cost. We consider the
Yeo-Johnson and G&H transformations, along with sparse factor structures for
the scale matrix of the Gaussian or skew-normal. We also show how to implement
efficient reparametrization gradient methods for these copula-based
approximations. The efficacy of the approach is illustrated by computing
posterior inference for three different models using six real datasets. In each
case, we show that our proposed copula model distributions are more accurate
variational approximations than Gaussian or skew-normal distributions, but at
only a minor or no increase in computational cost.
