Griffin, Jim E. and Steel, Mark F. J. (2009) Time-dependent stick-breaking processes. Working Paper. University of Warwick. Centre for Research in Statistical Methodology, Coventry.

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Abstract

This paper considers the problem of defining a time-dependent nonparametric prior. 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 have interesting interpretations that are further investigated. 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 stick-breaking process form increasing sequences. We derive a P´olya urn scheme type representation of the Dirichlet process construction, which allows us to develop a marginal MCMC method for this case. The results section shows the relative performance of the two MCMC schemes for the Dirichlet process case and contains three real data examples.

Item Type:	Working or Discussion Paper (Working Paper)
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Distribution (Probability theory)
Series Name:	Working papers
Publisher:	University of Warwick. Centre for Research in Statistical Methodology
Place of Publication:	Coventry
Date:	2009
Volume:	Vol.2009
Number:	No.5
Number of Pages:	36
Status:	Not Peer Reviewed
Access rights to Published version:	Open Access
References:	Antoniak, C. E. (1974): “Mixtures of Dirichlet processes with applications to non-parametric problems,” Journal of the American Statistical Association, 2, 1152-1174. Carlin, B. P., A. E. Gelfand and A. F. M. Smith (1992): “Hierarchical Bayesian analysis of changepoint problems,” Applied Statistics, 41, 389-405. Caron, F., M. Davy and A. Doucet (2007): “Generalized P´olya Urn for Time-varying Dirichlet Process Mixtures”, 23rd Conference on Uncertainty in Artificial Intelligence (UAI 2007). De Iorio, M., P. M¨uller, G. L. Rosner and S. N. MacEachern (2004): “An ANOVA model for dependent random measures, ” Journal of the American Statistical Association, 99, 205-215. Doucet, A., N. de Freitas and N. J. Gordon (2001): “Sequential Monte Carlo Methods in Practice,” Springer-Verlag: New York. Dunson, D. B. (2006): “Bayesian dynamic modeling of latent trait distributions,” Biostatistics, 7, 551-568. Dunson, D. B., N. Pillai and J. H. Park (2007): “Bayesian density regression, ” Journal of the Royal Statistical Society B, 69, 163-183. Grazia Pittau, M. and R. Zelli (2006): “Empirical Evidence of Income Dynamics Across EU Regions,” Journal of Applied Econometrics, 21, 605-628. Griffin, J. E. (2007): “The Ornstein-Uhlenbeck Dirichlet Process and other time-varying processes for Bayesian nonparametric inference,” Working Paper 07-03, CRiSM, University of Warwick. Griffin, J. E. and M. F. J. Steel (2006): “Order-based Dependent Dirichlet Processes,” Journal of the American Statistical Association, 101, 179-194. Ishwaran, H. and L. F. James (2001): “Gibbs Sampling Methods for Stick-Breaking Priors,” Journal of the American Statistical Association, 96, 161-173. Ishwaran, H. and L. F. James (2003). “Some further developments for stick-breaking priors: finite and infinite clustering and classification,” Sankhya, A, 65, 577-592. Ishwaran, H. and M. Zarepour (2000): “Markov chain Monte Carlo in approximate Dirichlet and two-parameter process hierarchical models,” Biometrika, 87, 371-390. Jacquier, E., N. G. Polson and P. E. Rossi (2004): “Bayesian analysis of stochastic volatility models with fat tails and correlated errors,” Journal of Econometrics, 122, 185-212. James, L.F. (2008): “Large sample asymptotics for the two-parameter PoissonDirichlet process,” in Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, B. Clarke and S. Ghosal, eds., IMS: Beachwood, 187-199. James, L. F., A. Lijoi and I. Pr¨unster (2005): “Bayesian inference via classes of normalized random measures,” Technical Report. Doucet, A., N. de Freitas and N. J. Gordon (2001): “Sequential Monte Carlo Methods in Practice,” Springer-Verlag: New York. Dunson, D. B. (2006): “Bayesian dynamic modeling of latent trait distributions,” Biostatistics, 7, 551-568. Dunson, D. B., N. Pillai and J. H. Park (2007): “Bayesian density regression, ” Journal of the Royal Statistical Society B, 69, 163-183. Grazia Pittau, M. and R. Zelli (2006): “Empirical Evidence of Income Dynamics Across EU Regions,” Journal of Applied Econometrics, 21, 605-628. Griffin, J. E. (2007): “The Ornstein-Uhlenbeck Dirichlet Process and other time-varying processes for Bayesian nonparametric inference,” Working Paper 07-03, CRiSM, University of Warwick. Griffin, J. E. and M. F. J. Steel (2006): “Order-based Dependent Dirichlet Processes,” Journal of the American Statistical Association, 101, 179-194. Ishwaran, H. and L. F. James (2001): “Gibbs Sampling Methods for Stick-Breaking Priors,” Journal of the American Statistical Association, 96, 161-173. Ishwaran, H. and L. F. James (2003). “Some further developments for stick-breaking priors: finite and infinite clustering and classification,” Sankhya, A, 65, 577-592. Ishwaran, H. and M. Zarepour (2000): “Markov chain Monte Carlo in approximate Dirichlet and two-parameter process hierarchical models,” Biometrika, 87, 371-390. Jacquier, E., N. G. Polson and P. E. Rossi (2004): “Bayesian analysis of stochastic volatility models with fat tails and correlated errors,” Journal of Econometrics, 122, 185-212. James, L.F. (2008): “Large sample asymptotics for the two-parameter PoissonDirichlet process,” in Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, B. Clarke and S. Ghosal, eds., IMS: Beachwood, 187-199. James, L. F., A. Lijoi and I. Pr¨unster (2005): “Bayesian inference via classes of normalized random measures,” Technical Report.
URI:	http://wrap.warwick.ac.uk/id/eprint/35197

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