WHY Bayesian filter? To estimate a “time-varying” or “time-dependent” statistic of time-series, traditional method is to use “moving/rolling data window” and/or “exponentially decayed time weights”. A straightforward and natural alternative is to use Bayesian framework: at each moment in time, using the last estimate as prior; conditional distribution of estimate (given statistical model for the estimate) as likelihood; and newly arrived/available time-series data point as observation. Then, the resulted posterior is a new estimate. In time-domain, such a Bayesian formulation is a filter.

WHY variational Bayes? The Bayesian filtering framework for estimating time-dependent statistics of time-series is straightforward and simple. The need of joint and conditional probability distribution functions, however, makes actual estimation complicated, difficult or even intractable. Discretized approximation, e.g. numerical particle filters, is computation-intensive and prone to cumulative errors in numerical distributions. A variational Bayes (VB) is an analytical approximation. After tedious derivation, a VB is fast, as long as the assumptions on distributions are appropriate.

WHAT next? In addition to a stand-alone filtering package for time-varying factor analysis on multiple time-series, the VBfFA algorithm will be employed as the underlying factor analysis engine of other machine learning packages here introduced earlier by i4cast LLC: LMDFM (long memory dynamic factor model); YWpcAR (Yule-Walker-PCA autoregressive model); LMVAR (long memory vector autoregressive model); and CTVARF (continuously trained vector autoregressive forecast model). We will introduce a published general-purpose multivariate variational Bayesian filter (VBF) algorithm as well.