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 the use of recursive algorithms to update a sequence of VB posterior approximations in an online, time series setting, with the computation of each posterior update requiring only the data observed since the previous update. We show how importance sampling can be incorporated into online variational inference allowing the user to trade accuracy for a substantial increase in computational speed. The proposed methods and their properties are detailed in two separate simulation studies. Additionally, two empirical illustrations 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.
Highlights
Time series data often arrives in high frequency streams in applications that may require a response within a very short period of time
This paper proposes a framework to extend the use of Stochastic Variational Bayes (SVB) inference to a sequential posterior updating setting
Updating Variational Bayes (UVB) is a variational analogue to exact Bayesian updating, where the previous posterior distribution, taken as an updated prior, is replaced with an approximation itself derived from an earlier SVB approximation
Summary
Time series data often arrives in high frequency streams in applications that may require a response within a very short period of time. Any data observed from the data generating process may be used within the MFVB coordinate descent algorithm, which is applied online with newly observed data substituted in as it becomes available Each of these approaches results in only a single posterior distribution conditioned on data up to some pre-specified time period T1, and do not provide a mechanism for the approximation to be updated at a later time period T2 following the availability of additional observations. Smidl (2004) and Broderick et al (2013) each consider VB approximations for Bayesian updating, resulting in a progressive sequence of approximate posterior distributions that each condition on data up to any given time period Tn Their approaches update to the time Tn+1 by substitution of the time Tn posterior with MFVB approximations, which are feasibly obtained due to assuming the model and approximation each adheres to a suitably defined exponential family form. UVB is applied to a vehicle DPM model in Section 7, and Section 8 concludes the paper
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