Updating Variational Bayes: fast sequential posterior inference

Variational Bayesian (VB) methods usually 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 class 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 is repeatedly updated in the context of predicting the future behaviour of vehicles on a stretch of the US Highway 101.

[1]  Léon Bottou,et al.  Large-Scale Machine Learning with Stochastic Gradient Descent , 2010, COMPSTAT.

[2]  C. F. Beckmann,et al.  Estimating an Inverse Gamma distribution , 2016, 1605.01019.

[3]  D. Aldous.  Exchangeability and related topics , 1985 .

[4]  Michael I. Jordan,et al.  Graphical Models, Exponential Families, and Variational Inference , 2008, Found. Trends Mach. Learn..

[5]  David B. Dunson,et al.  Bayesian Data Analysis , 2010 .

[6]  Arto Klami,et al.  Importance Sampled Stochastic Optimization for Variational Inference , 2017, UAI.

[7]  Sean Gerrish,et al.  Black Box Variational Inference , 2013, AISTATS.

[8]  P. H. Garthwaite,et al.  Adaptive optimal scaling of Metropolis–Hastings algorithms using the Robbins–Monro process , 2010, 1006.3690.

[9]  Claudia Czado,et al.  Selecting and estimating regular vine copulae and application to financial returns , 2012, Comput. Stat. Data Anal..

[10]  Max Welling,et al.  Auto-Encoding Variational Bayes , 2013, ICLR.

[11]  H. Robbins.  A Stochastic Approximation Method , 1951 .

[12]  Hedvig Kjellström,et al.  Advances in Variational Inference , 2017, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[13]  Václav Smídl,et al.  The variational Bayes approach in signal processing , 2004 .

[14]  Masa-aki Sato,et al.  Online Model Selection Based on the Variational Bayes , 2001, Neural Computation.

[15]  Nando de Freitas,et al.  Sequential Monte Carlo Methods in Practice , 2001, Statistics for Engineering and Information Science.

[16]  J. Sethuraman.  A CONSTRUCTIVE DEFINITION OF DIRICHLET PRIORS , 1991 .

[17]  Gary Koop,et al.  Variational Bayesian Inference in Large Vector Autoregressions with Hierarchical Shrinkage , 2019, SSRN Electronic Journal.

[18]  Zoubin Ghahramani,et al.  Propagation Algorithms for Variational Bayesian Learning , 2000, NIPS.

[19]  Peter Green,et al.  Markov chain Monte Carlo in Practice , 1996 .

[20]  R. A. Leibler,et al.  On Information and Sufficiency , 1951 .

[21]  Chong Wang,et al.  Stochastic variational inference , 2012, J. Mach. Learn. Res..

[22]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[23]  Sumeetpal S. Singh,et al.  Filtering via approximate Bayesian computation , 2010, Statistics and Computing.

[24]  P. Moral,et al.  The Alive Particle Filter and Its Use in Particle Markov Chain Monte Carlo , 2015 .

[25]  David J. Nott,et al.  Gaussian variational approximation with sparse precision matrices , 2016, Statistics and Computing.

[26]  Hagai Attias,et al.  A Variational Bayesian Framework for Graphical Models , 1999 .

[27]  Francis R. Bach,et al.  Online Learning for Latent Dirichlet Allocation , 2010, NIPS.

[28]  Le Song,et al.  Meta Particle Flow for Sequential Bayesian Inference , 2019, ICML 2019.

[29]  S. Lauritzen.  Propagation of Probabilities, Means, and Variances in Mixed Graphical Association Models , 1992 .

[30]  Robert Kohn,et al.  Variational Bayes approximation of factor stochastic volatility models , 2021, International Journal of Forecasting.

[31]  D. Nott,et al.  Gaussian Variational Approximation With a Factor Covariance Structure , 2017, Journal of Computational and Graphical Statistics.

[32]  Nasser M. Nasrabadi,et al.  Pattern Recognition and Machine Learning , 2006, Technometrics.

[33]  Jeffrey W. Miller.  An elementary derivation of the Chinese restaurant process from Sethuraman’s stick-breaking process , 2018, Statistics & Probability Letters.

[34]  W. Gilks,et al.  Adaptive Rejection Metropolis Sampling Within Gibbs Sampling , 1995 .

[35]  David B. Dunson,et al.  Online Variational Bayes Inference for High-Dimensional Correlated Data , 2011, 1108.1079.

[36]  N. Chopin.  A sequential particle filter method for static models , 2002 .

[37]  Miguel Lázaro-Gredilla,et al.  Doubly Stochastic Variational Bayes for non-Conjugate Inference , 2014, ICML.

[38]  David M. Blei,et al.  Variational Inference: A Review for Statisticians , 2016, ArXiv.

[39]  Jimmy Ba,et al.  Adam: A Method for Stochastic Optimization , 2014, ICLR.

[40]  Andre Wibisono,et al.  Streaming Variational Bayes , 2013, NIPS.

[41]  Fernando A. Quintana,et al.  Bayesian Nonparametric Data Analysis , 2015 .

[42]  Michael I. Jordan,et al.  An Introduction to Variational Methods for Graphical Models , 1999, Machine Learning.

[43]  A. Gelman,et al.  Weak convergence and optimal scaling of random walk Metropolis algorithms , 1997 .

[44]  Simon P. Wilson,et al.  Sequential Bayesian inference for static parameters in dynamic state space models , 2014, 1408.4559.

[45]  Chong Wang,et al.  Online Variational Inference for the Hierarchical Dirichlet Process , 2011, AISTATS.