Sampling from Dirichlet process mixture models with unknown concentration parameter: mixing issues in large data implementations

We consider the question of Markov chain Monte Carlo sampling from a general stick-breaking Dirichlet process mixture model, with concentration parameter α . This paper introduces a Gibbs sampling algorithm that combines the slice sampling approach of Walker (Communications in Statistics - Simulatio...

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Vydané v:	Statistics and computing Ročník 25; číslo 5; s. 1023 - 1037
Hlavní autori:	Hastie, David I., Liverani, Silvia, Richardson, Sylvia
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.09.2015
Predmet:	Algorithms Artificial Intelligence Computation Computer simulation Dirichlet problem Mathematical models Mathematics and Statistics Probability and Statistics in Computer Science Sampling Source code Statistical Theory and Methods Statistics Statistics and Computing/Statistics Programs Mixture model Profile regression Dirichlet process Bayesian clustering
ISSN:	0960-3174, 1573-1375
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Abstract	We consider the question of Markov chain Monte Carlo sampling from a general stick-breaking Dirichlet process mixture model, with concentration parameter α . This paper introduces a Gibbs sampling algorithm that combines the slice sampling approach of Walker (Communications in Statistics - Simulation and Computation 36:45–54, 2007 ) and the retrospective sampling approach of Papaspiliopoulos and Roberts (Biometrika 95(1):169–186, 2008 ). Our general algorithm is implemented as efficient open source C++ software, available as an R package, and is based on a blocking strategy similar to that suggested by Papaspiliopoulos (A note on posterior sampling from Dirichlet mixture models, 2008 ) and implemented by Yau et al. (Journal of the Royal Statistical Society, Series B (Statistical Methodology) 73:37–57, 2011 ). We discuss the difficulties of achieving good mixing in MCMC samplers of this nature in large data sets and investigate sensitivity to initialisation. We additionally consider the challenges when an additional layer of hierarchy is added such that joint inference is to be made on α . We introduce a new label-switching move and compute the marginal partition posterior to help to surmount these difficulties. Our work is illustrated using a profile regression (Molitor et al. Biostatistics 11(3):484–498, 2010 ) application, where we demonstrate good mixing behaviour for both synthetic and real examples.
AbstractList	We consider the question of Markov chain Monte Carlo sampling from a general stick-breaking Dirichlet process mixture model, with concentration parameter $\alpha $. This paper introduces a Gibbs sampling algorithm that combines the slice sampling approach of Walker (Communications in Statistics - Simulation and Computation 36:45-54, 2007) and the retrospective sampling approach of Papaspiliopoulos and Roberts (Biometrika 95(1):169-186, 2008). Our general algorithm is implemented as efficient open source C++ software, available as an R package, and is based on a blocking strategy similar to that suggested by Papaspiliopoulos (A note on posterior sampling from Dirichlet mixture models, 2008) and implemented by Yau et al. (Journal of the Royal Statistical Society, Series B (Statistical Methodology) 73:37-57, 2011). We discuss the difficulties of achieving good mixing in MCMC samplers of this nature in large data sets and investigate sensitivity to initialisation. We additionally consider the challenges when an additional layer of hierarchy is added such that joint inference is to be made on $\alpha $. We introduce a new label-switching move and compute the marginal partition posterior to help to surmount these difficulties. Our work is illustrated using a profile regression (Molitor et al. Biostatistics 11(3):484-498, 2010) application, where we demonstrate good mixing behaviour for both synthetic and real examples. We consider the question of Markov chain Monte Carlo sampling from a general stick-breaking Dirichlet process mixture model, with concentration parameter $$\alpha $$ α. This paper introduces a Gibbs sampling algorithm that combines the slice sampling approach of Walker (Communications in Statistics - Simulation and Computation 36:45–54, 2007) and the retrospective sampling approach of Papaspiliopoulos and Roberts (Biometrika 95(1):169–186, 2008). Our general algorithm is implemented as efficient open source C++ software, available as an R package, and is based on a blocking strategy similar to that suggested by Papaspiliopoulos (A note on posterior sampling from Dirichlet mixture models, 2008) and implemented by Yau et al. (Journal of the Royal Statistical Society, Series B (Statistical Methodology) 73:37–57, 2011). We discuss the difficulties of achieving good mixing in MCMC samplers of this nature in large data sets and investigate sensitivity to initialisation. We additionally consider the challenges when an additional layer of hierarchy is added such that joint inference is to be made on $$\alpha $$ α. We introduce a new label-switching move and compute the marginal partition posterior to help to surmount these difficulties. Our work is illustrated using a profile regression (Molitor et al. Biostatistics 11(3):484–498, 2010) application, where we demonstrate good mixing behaviour for both synthetic and real examples. We consider the question of Markov chain Monte Carlo sampling from a general stick-breaking Dirichlet process mixture model, with concentration parameter α . This paper introduces a Gibbs sampling algorithm that combines the slice sampling approach of Walker (Communications in Statistics - Simulation and Computation 36:45–54, 2007 ) and the retrospective sampling approach of Papaspiliopoulos and Roberts (Biometrika 95(1):169–186, 2008 ). Our general algorithm is implemented as efficient open source C++ software, available as an R package, and is based on a blocking strategy similar to that suggested by Papaspiliopoulos (A note on posterior sampling from Dirichlet mixture models, 2008 ) and implemented by Yau et al. (Journal of the Royal Statistical Society, Series B (Statistical Methodology) 73:37–57, 2011 ). We discuss the difficulties of achieving good mixing in MCMC samplers of this nature in large data sets and investigate sensitivity to initialisation. We additionally consider the challenges when an additional layer of hierarchy is added such that joint inference is to be made on α . We introduce a new label-switching move and compute the marginal partition posterior to help to surmount these difficulties. Our work is illustrated using a profile regression (Molitor et al. Biostatistics 11(3):484–498, 2010 ) application, where we demonstrate good mixing behaviour for both synthetic and real examples. We consider the question of Markov chain Monte Carlo sampling from a general stick-breaking Dirichlet process mixture model, with concentration parameter [Formula: see text]. This paper introduces a Gibbs sampling algorithm that combines the slice sampling approach of Walker (Communications in Statistics - Simulation and Computation 36:45-54, 2007) and the retrospective sampling approach of Papaspiliopoulos and Roberts (Biometrika 95(1):169-186, 2008). Our general algorithm is implemented as efficient open source C++ software, available as an R package, and is based on a blocking strategy similar to that suggested by Papaspiliopoulos (A note on posterior sampling from Dirichlet mixture models, 2008) and implemented by Yau et al. (Journal of the Royal Statistical Society, Series B (Statistical Methodology) 73:37-57, 2011). We discuss the difficulties of achieving good mixing in MCMC samplers of this nature in large data sets and investigate sensitivity to initialisation. We additionally consider the challenges when an additional layer of hierarchy is added such that joint inference is to be made on [Formula: see text]. We introduce a new label-switching move and compute the marginal partition posterior to help to surmount these difficulties. Our work is illustrated using a profile regression (Molitor et al. Biostatistics 11(3):484-498, 2010) application, where we demonstrate good mixing behaviour for both synthetic and real examples.We consider the question of Markov chain Monte Carlo sampling from a general stick-breaking Dirichlet process mixture model, with concentration parameter [Formula: see text]. This paper introduces a Gibbs sampling algorithm that combines the slice sampling approach of Walker (Communications in Statistics - Simulation and Computation 36:45-54, 2007) and the retrospective sampling approach of Papaspiliopoulos and Roberts (Biometrika 95(1):169-186, 2008). Our general algorithm is implemented as efficient open source C++ software, available as an R package, and is based on a blocking strategy similar to that suggested by Papaspiliopoulos (A note on posterior sampling from Dirichlet mixture models, 2008) and implemented by Yau et al. (Journal of the Royal Statistical Society, Series B (Statistical Methodology) 73:37-57, 2011). We discuss the difficulties of achieving good mixing in MCMC samplers of this nature in large data sets and investigate sensitivity to initialisation. We additionally consider the challenges when an additional layer of hierarchy is added such that joint inference is to be made on [Formula: see text]. We introduce a new label-switching move and compute the marginal partition posterior to help to surmount these difficulties. Our work is illustrated using a profile regression (Molitor et al. Biostatistics 11(3):484-498, 2010) application, where we demonstrate good mixing behaviour for both synthetic and real examples. We consider the question of Markov chain Monte Carlo sampling from a general stick-breaking Dirichlet process mixture model, with concentration parameter [Formula: see text]. This paper introduces a Gibbs sampling algorithm that combines the slice sampling approach of Walker (Communications in Statistics - Simulation and Computation 36:45-54, 2007) and the retrospective sampling approach of Papaspiliopoulos and Roberts (Biometrika 95(1):169-186, 2008). Our general algorithm is implemented as efficient open source C++ software, available as an R package, and is based on a blocking strategy similar to that suggested by Papaspiliopoulos (A note on posterior sampling from Dirichlet mixture models, 2008) and implemented by Yau et al. (Journal of the Royal Statistical Society, Series B (Statistical Methodology) 73:37-57, 2011). We discuss the difficulties of achieving good mixing in MCMC samplers of this nature in large data sets and investigate sensitivity to initialisation. We additionally consider the challenges when an additional layer of hierarchy is added such that joint inference is to be made on [Formula: see text]. We introduce a new label-switching move and compute the marginal partition posterior to help to surmount these difficulties. Our work is illustrated using a profile regression (Molitor et al. Biostatistics 11(3):484-498, 2010) application, where we demonstrate good mixing behaviour for both synthetic and real examples.
Author	Richardson, Sylvia Liverani, Silvia Hastie, David I.
Author_xml	– sequence: 1 givenname: David I. surname: Hastie fullname: Hastie, David I. organization: Imperial College London – sequence: 2 givenname: Silvia surname: Liverani fullname: Liverani, Silvia organization: Imperial College London, MRC Biostatistics Unit – sequence: 3 givenname: Sylvia surname: Richardson fullname: Richardson, Sylvia email: sylvia.richardson@mrc-bsu.cam.ac.uk organization: MRC Biostatistics Unit
BackLink	https://www.ncbi.nlm.nih.gov/pubmed/26321800$$D View this record in MEDLINE/PubMed
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Cites_doi	10.1080/01621459.1995.10476550 10.1198/1061860043001 10.1214/07-BA219 10.1198/jasa.2009.0001 10.1214/aos/1176342360 10.1198/016214501750332758 10.1111/j.1467-9868.2010.00756.x 10.1289/ehp.1002118 10.1002/gepi.21661 10.1214/088342305000000016 10.1080/03610910601096262 10.1198/016214508000001039 10.1214/09-BA414 10.1111/1467-9868.00095 10.1186/1471-2288-13-129 10.1214/aop/1024404422 10.1214/aos/1176342372 10.1007/s11222-009-9150-y 10.1109/LSP.2010.2072919 10.1093/biomet/asp021 10.1093/biostatistics/kxq013 10.1214/aos/1176342871 10.2307/1390653 10.1021/es104017x 10.1093/biomet/asm086
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Snippet	We consider the question of Markov chain Monte Carlo sampling from a general stick-breaking Dirichlet process mixture model, with concentration parameter α .... We consider the question of Markov chain Monte Carlo sampling from a general stick-breaking Dirichlet process mixture model, with concentration parameter...
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SubjectTerms	Algorithms Artificial Intelligence Computation Computer simulation Dirichlet problem Mathematical models Mathematics and Statistics Probability and Statistics in Computer Science Sampling Source code Statistical Theory and Methods Statistics Statistics and Computing/Statistics Programs
Title	Sampling from Dirichlet process mixture models with unknown concentration parameter: mixing issues in large data implementations
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