# Copula-like Variational Inference

## TL;DR

This paper introduces a new family of variational distributions inspired by copulas, enabling efficient sampling and better approximation of complex posteriors in Bayesian neural networks.

## Contribution

It proposes copula-like variational densities with efficient sampling and normalizing flows, improving approximation of non-Gaussian posteriors over traditional methods.

## Key findings

- Performs comparably to state-of-the-art variational methods on benchmarks.
- Can approximate non-Gaussian posteriors effectively.
- Sampling complexity is linear in the dimension.

## Abstract

This paper considers a new family of variational distributions motivated by Sklar's theorem. This family is based on new copula-like densities on the hypercube with non-uniform marginals which can be sampled efficiently, i.e. with a complexity linear in the dimension of state space. Then, the proposed variational densities that we suggest can be seen as arising from these copula-like densities used as base distributions on the hypercube with Gaussian quantile functions and sparse rotation matrices as normalizing flows. The latter correspond to a rotation of the marginals with complexity $\mathcal{O}(d \log d)$. We provide some empirical evidence that such a variational family can also approximate non-Gaussian posteriors and can be beneficial compared to Gaussian approximations. Our method performs largely comparably to state-of-the-art variational approximations on standard regression and classification benchmarks for Bayesian Neural Networks.