Stick-breaking autoregressive processes

This paper considers the problem of defining a time-dependent nonparametric prior for use in Bayesian nonparametric modelling of time series. A recursive construction allows the definition of priors whose marginals have a general stick-breaking form. The processes with Poisson-Dirichlet and Dirichlet process marginals are investigated in some detail. We develop a general conditional Markov Chain Monte Carlo (MCMC) method for inference in the wide subclass of these models where the parameters of the marginal stick-breaking process are nondecreasing sequences. We derive a generalised Polya urn scheme type representation of the Dirichlet process construction, which allows us to develop a marginal MCMC method for this case. We apply the proposed methods to financial data to develop a semi-parametric stochastic volatility model with a time-varying nonparametric returns distribution. Finally, we present two examples concerning the analysis of regional GDP and its growth.

[1]  Mark J. Jensen,et al.  Bayesian Semiparametric Stochastic Volatility Modeling , 2008 .

[2]  Fernando A. Quintana,et al.  Nonparametric Bayesian data analysis , 2004 .

[3]  Enrique ter Horst,et al.  Bayesian dynamic density estimation , 2008 .

[4]  Siddhartha Chib,et al.  Semiparametric Bayes analysis of longitudinal data treatment models , 2002 .

[5]  Michael,et al.  On a Class of Bayesian Nonparametric Estimates : I . Density Estimates , 2008 .

[6]  Adrian F. M. Smith,et al.  Hierarchical Bayesian Analysis of Changepoint Problems , 1992 .

[7]  N. Pillai,et al.  Bayesian density regression , 2007 .

[8]  Keisuke Hirano,et al.  Semiparametric Bayesian Inference in Autoregressive Panel Data Models , 2002 .

[9]  Purushottam W. Laud,et al.  Bayesian Nonparametric Inference for Random Distributions and Related Functions , 1999 .

[10]  J. Geweke,et al.  Smoothly mixing regressions , 2007 .

[11]  Lancelot F. James,et al.  Gibbs Sampling Methods for Stick-Breaking Priors , 2001 .

[12]  Jun S. Liu,et al.  Monte Carlo strategies in scientific computing , 2001 .

[13]  Michael I. Jordan,et al.  An HDP-HMM for systems with state persistence , 2008, ICML '08.

[14]  G. Roberts,et al.  Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical models , 2007, 0710.4228.

[15]  Matt Taddy,et al.  Markov switching Dirichlet process mixture regression , 2009 .

[16]  H. Ishwaran,et al.  Markov chain Monte Carlo in approximate Dirichlet and beta two-parameter process hierarchical models , 2000 .

[17]  J. E. Griffin,et al.  Order-Based Dependent Dirichlet Processes , 2006 .

[18]  S. Chib,et al.  Additive cubic spline regression with Dirichlet process mixture errors , 2010 .

[19]  Albert Y. Lo,et al.  On a Class of Bayesian Nonparametric Estimates: I. Density Estimates , 1984 .

[20]  Yuan Ji,et al.  A Time‐Series DDP for Functional Proteomics Profiles , 2012, Biometrics.

[21]  Peter E. Rossi,et al.  Bayesian analysis of stochastic volatility models with fat-tails and correlated errors , 2004 .

[22]  J. Lafferty,et al.  Time-Sensitive Dirichlet Process Mixture Models , 2005 .

[23]  P. Müller,et al.  A method for combining inference across related nonparametric Bayesian models , 2004 .

[24]  Lancelot F. James,et al.  Posterior Analysis for Normalized Random Measures with Independent Increments , 2009 .

[25]  M. G. Pittau,et al.  Empirical evidence of income dynamics across EU regions , 2006 .

[26]  J. Pitman,et al.  The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator , 1997 .

[27]  Lancelot F. James,et al.  Some further developments for stick-breaking priors: Finite and infinite clustering and classification , 2003 .

[28]  David S. Leslie,et al.  A general approach to heteroscedastic linear regression , 2007, Stat. Comput..

[29]  S. MacEachern,et al.  An ANOVA Model for Dependent Random Measures , 2004 .

[30]  D. Dunson.  Bayesian dynamic modeling of latent trait distributions. , 2006, Biostatistics.

[31]  M. Steel,et al.  Semiparametric Bayesian Inference for Stochastic Frontier Models , 2004 .

[32]  J. Pitman.  Poisson-Kingman partitions , 2002, math/0210396.

[33]  Arnaud Doucet,et al.  Generalized Polya Urn for Time-varying Dirichlet Process Mixtures , 2007, UAI.

[34]  D. Goldstein.  Statistics and science : a Festschrift for Terry Speed , 2003 .