# Approximation Properties of Variational Bayes for Vector Autoregressions

## TL;DR

This paper analyzes the approximation accuracy of Variational Bayes in vector autoregressions, showing it accurately estimates the posterior mean and predictive densities, with insights into variance underestimation.

## Contribution

It derives the approximation error of VB for linear Gaussian regressions, revealing conditions for accurate posterior mean and predictive densities.

## Key findings

- VB approximates the posterior mean perfectly.
- VB underestimates posterior variance and mode.
- VB estimates predictive densities accurately.

## Abstract

Variational Bayes (VB) is a recent approximate method for Bayesian inference. It has the merit of being a fast and scalable alternative to Markov Chain Monte Carlo (MCMC) but its approximation error is often unknown. In this paper, we derive the approximation error of VB in terms of mean, mode, variance, predictive density and KL divergence for the linear Gaussian multi-equation regression. Our results indicate that VB approximates the posterior mean perfectly. Factors affecting the magnitude of underestimation in posterior variance and mode are revealed. Importantly, We demonstrate that VB estimates predictive densities accurately.