# High-dimensional copula variational approximation through transformation

## TL;DR

This paper introduces a novel variational approximation method using copula models with transformations like Yeo-Johnson, improving high-dimensional Bayesian inference accuracy with minimal additional computational cost.

## Contribution

It proposes a new copula-based variational approximation framework that enhances accuracy in high-dimensional models without significant computational overhead.

## Key findings

- Copula models outperform Gaussian and skew-normal approximations in accuracy.
- The method is computationally efficient and scalable to high dimensions.
- Successful application to three models with six real datasets.

## Abstract

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.