Implicit Copula Variational Inference
提出一种调整后的隐式联结函数模型，使变分近似对目标密度的位置和尺度不变，并利用椭圆联结函数的生成表示实现高效一阶优化，通过混合效应逻辑回归和正则化相关矩阵两个例子展示其有效性。
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, Loaiza-Maya, and Nott (2020 Smith, M. S., Loaiza-Maya, R., and Nott, D. J. (2020), “High-Dimensional Copula Variational Approximation through Transformation,” Journal of Computational and Graphical Statistics, 29, 729–743. DOI: 10.1080/10618600.2020.1740097.[Taylor & Francis Online] , [Google Scholar]) 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.