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Time-dependent stick-breaking processes

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Griffin, Jim E. and Steel, Mark F. J. (2009) Time-dependent stick-breaking processes. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. Working papers, Vol.2009 (No.5).

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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

Official Date:

2009

Dates:

Date	Event
2009	Published

Volume:

Vol.2009

Number:

No.5

Number of Pages:

Status:

Not Peer Reviewed

Access rights to Published version:

Open Access

References:

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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.