# Updating Variational Bayes: Fast sequential posterior inference

## TL;DR

This paper introduces Updating Variational Bayes (UVB), a recursive online algorithm for fast sequential posterior inference that updates VB approximations efficiently, with an extension (UVB-IS) trading accuracy for speed using importance sampling.

## Contribution

The paper proposes UVB, a novel recursive algorithm for online VB posterior updates, and UVB-IS, an importance sampling extension for faster computation with adjustable accuracy.

## Key findings

- UVB effectively updates VB posteriors in an online setting.
- UVB-IS significantly increases computational speed with controllable accuracy.
- Empirical results demonstrate UVB's applicability to complex models like Dirichlet Process Mixtures.

## Abstract

Variational Bayesian (VB) methods produce posterior inference in a time frame considerably smaller than traditional Markov Chain Monte Carlo approaches. Although the VB posterior is an approximation, it has been shown to produce good parameter estimates and predicted values when a rich classes of approximating distributions are considered. In this paper we propose Updating VB (UVB), a recursive algorithm used to update a sequence of VB posterior approximations in an online setting, with the computation of each posterior update requiring only the data observed since the previous update. An extension to the proposed algorithm, named UVB-IS, allows the user to trade accuracy for a substantial increase in computational speed through the use of importance sampling. The two methods and their properties are detailed in two separate simulation studies. Two empirical illustrations of the proposed UVB methods are provided, including one where a Dirichlet Process Mixture model with a novel posterior dependence structure is repeatedly updated in the context of predicting the future behaviour of vehicles on a stretch of the US Highway 101.