Implicit copula variational inference

TL;DR

This paper introduces an improved implicit copula variational inference method that is invariant to scale and location, enabling scalable and efficient estimation for complex models like mixed effects logistic regression and regularized correlation matrices.

Contribution

The paper proposes a scale- and location-invariant adjustment to implicit copula transformations and demonstrates their application with elliptical copulas for efficient variational inference.

Findings

Effective estimation of mixed effects logistic regression

Scalable inference for regularized correlation matrices

Application to income inequality data in U.S. states

Abstract

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.