  Key to effective generic, or "black-box", variational inference is the
selection of an approximation to the target density that balances accuracy and
speed. Copula models are promising options, but calibration of the
approximation can be slow for some choices. Smith et al. (2020) suggest using
tractable and scalable "implicit copula" models that are formed by element-wise
transformation of the target parameters. We propose an adjustment to these
transformations that make the approximation invariant to the scale and location
of the target density. We also show how a sub-class of elliptical copulas have
a generative representation that allows easy application of the
re-parameterization trick and efficient first order optimization. We
demonstrate the estimation methodology using two statistical models as
examples. The first is a mixed effects logistic regression, and the second is a
regularized correlation matrix. For the latter, standard Markov chain Monte
Carlo estimation methods can be slow or difficult to implement, yet our
proposed variational approach provides an effective and scalable estimator. We
illustrate by estimating a regularized Gaussian copula model for income
inequality in U.S. states between 1917 and 2018. An Online Appendix and MATLAB
code to implement the method are available as Supplementary Materials.
