Skip to main content
SpringerLink
Log in

Dirichlet process and its developments: a survey

  • Survey Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

Abstract

The core of the nonparametric/semiparametric Bayesian analysis is to relax the particular parametric assumptions on the distributions of interest to be unknown and random, and assign them a prior. Selecting a suitable prior therefore is especially critical in the nonparametric Bayesian fitting. As the distribution of distribution, Dirichlet process (DP) is the most appreciated nonparametric prior due to its nice theoretical proprieties, modeling flexibility and computational feasibility. In this paper, we review and summarize some developments of DP during the past decades. Our focus is mainly concentrated upon its theoretical properties, various extensions, statistical modeling and applications to the latent variable models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aldous D J. Exchangeability and related topics, In: École d’Éte de Probabilités de Saint-Flour XIII-1983, Lecture Notes in Math., Vol. 1117, New York: Springer-Verlag, 1985, 23–34

    Chapter  Google Scholar 

  2. Antoniak C E. Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Statist., 1974, 2(6): 1152–1174

    Article  MathSciNet  MATH  Google Scholar 

  3. Basu S, Chib S. Marginal likelihood and Bayes factors for Dirichlet process mixture models, J. Amer. Statist. Assoc., 2003, 98(461): 224–235

    Article  MathSciNet  MATH  Google Scholar 

  4. Bentler P M, Wu E J C. EQS6: Structural Equations Program Manual. Encino, CA: Multivariate Software, 2006

    Google Scholar 

  5. Blackwell D. Discreteness of Ferguson selections. Ann. Statist., 1973, 1(2): 356–358

    MathSciNet  MATH  Google Scholar 

  6. Blackwell D, MacQueen J B. Ferguson distributions via polya urn schemes. Ann. Statist., 1973, 1(2): 353–355

    MathSciNet  MATH  Google Scholar 

  7. Bollen K A. Structural Equations with Latent Variables. New York: John Wiley & Sons, 1989

    Book  MATH  Google Scholar 

  8. Bush C A, MacEachern S N. A semiparametric Bayesian model for randomised block designs. Biometrika, 1996, 83(2): 275–285

    Article  MATH  Google Scholar 

  9. Carota C, Parmigiani G. Semiparametric regression for count data. Biometrika, 2002, 89(2): 265–281

    Article  MathSciNet  MATH  Google Scholar 

  10. Chow S M, Tang N S, Yuan Y, Song X Y, Zhu H T. Bayesian estimation of semiparametric nonlinear dynamic factor analysis models using the Dirichlet process prior. Br. J. Math. Stat. Psychol., 2011, 64(1): 69–106

    Article  MathSciNet  MATH  Google Scholar 

  11. Cifarelli D, Regazzini E. Problemi statistici non parametrici in condizioni di scambialbilita parziale: impiego di medie associative. Technical Report, Quad. Insitit. Mat. Finana. Univ. Torino III, 1978, 1–13 (in Italian)

  12. Connor R J, Mosimann J E. Concepts of independence for proportions with a generalization of the Dirichlet distribution. J. Amer. Statist. Assoc., 1969, 64(325): 194–206

    Article  MathSciNet  MATH  Google Scholar 

  13. Crandell L J, Dunson D B. Posterior simulation across nonparametric models for functional clustering. Sankhya B, 2011, 73(1): 42–61

    Article  MathSciNet  MATH  Google Scholar 

  14. Dalal S R. Dirichlet invariant processes and applications to nonparametric estimation of symmetric distribution functions. Stochastic Process. Appl., 1979, 9(1): 99–107

    Article  MathSciNet  MATH  Google Scholar 

  15. De Iorio M, Müller P, Rosner G L, MacEacher S N. An ANOVA model for dependent random measures. J. Amer. Statist. Assoc., 2004, 99(465): 205–215

    Article  MathSciNet  MATH  Google Scholar 

  16. Doss H. Bayesian nonparametric estimation of the median: Part I. Computation of the estimates. Ann. Statist., 1985, 13(4): 1432–1444

    MathSciNet  MATH  Google Scholar 

  17. Doss H. Bayesian nonparametric estimation of the median: Part II. Asymptotic properties of the estimates. Ann. Statist., 1985, 13(4): 1445–1464

    MathSciNet  MATH  Google Scholar 

  18. Doss H. Bayesian nonparametric estimation for incomplete data via successive substitution sampling. Ann. Statist., 1994, 22(4): 1763–1786

    MathSciNet  MATH  Google Scholar 

  19. Duan J A, Guindani M, Gelfand A E. Generalized spatial Dirichlet process models. Biometrika, 2007, 94(4): 809–825

    Article  MathSciNet  MATH  Google Scholar 

  20. Dunson D B. Nonparametric Bayes local partition models for random effects. Biometrika, 2009, 96(2): 249–262

    Article  MathSciNet  MATH  Google Scholar 

  21. Dunson D B, Park J H. Kernel stick-breaking processes. Biometrika, 2008, 95(2): 307–323

    Article  MathSciNet  MATH  Google Scholar 

  22. Dunson D B, Pillai N, Park J H. Bayesian density regression. J. R. Stat. Soc. Ser. B. Stat. Methodol., 2007, 69(2): 163–183

    Article  MathSciNet  MATH  Google Scholar 

  23. Escobar M D. Estimating the means of several normal populations by estimating the distribution of the means, Ph.D. Thesis. New Haven: Yale Univ., 1988

    Google Scholar 

  24. Escobar M D. Estimating normal means with a Dirichlet process prior. J. Amer. Statist. Assoc., 1994, 89(425): 268–277

    Article  MathSciNet  MATH  Google Scholar 

  25. Escobar M D, West M. Bayesian density estimation and inference using mixtures. J. Amer. Statist. Assoc., 1995, 90(430): 577–588

    Article  MathSciNet  MATH  Google Scholar 

  26. Ewens W J. Population Genetics Theory — The Past and the Future. In: Lessard S. (eds) Mathematical and Statistical Developments of Evolutionary Theory. NATO ASI Series (Series C: Mathematical and Physical Sciences), vol 299. Dordrecht:Springer, 1990

    Google Scholar 

  27. Fabius J. Asymptotic behavior of Bayes’ estimates. Ann. Math. Statist., 1964, 35(2): 846–856

    Article  MathSciNet  MATH  Google Scholar 

  28. Ferguson T S. A Bayesian analysis of some nonparametric problems. Ann. Statist., 1973, 1(2): 209–230

    Article  MathSciNet  MATH  Google Scholar 

  29. Ferguson T S. Prior distributions on spaces of probability measures. Ann. Statist., 1974, 2(4): 615–629

    Article  MathSciNet  MATH  Google Scholar 

  30. Fong D K H, Pammer S E, Arnold S F, Bolton G E. Reanalyzing ultimatum bargaining: comparing nondecreasing curves without shape constraints. J. Busin. Econom. Statist., 2002, 20(3): 423–430

    Article  MathSciNet  Google Scholar 

  31. Freedman D A. On the asymptotic behavior of Bayes’ estimates in the discrete case II. Ann. Math. Statist., 1963, 34(4): 1386–1403

    Article  MathSciNet  MATH  Google Scholar 

  32. Gelfand A E, Kottas A. A computational approach for full nonparametric Bayesian inference under Dirichlet Process mixture models. J. Comput. Graph. Stat., 2002, 11(2): 289–305

    Article  MathSciNet  Google Scholar 

  33. Gelfand A E, Kottas A. Bayesian semiparametric for median residual life. Scandinavian Journal of Statistics, 2003, 30(4): 651–665

    Article  MathSciNet  MATH  Google Scholar 

  34. Gelfand A E, Kottas A, MacEachern S N. Bayesian nonparametric spatial modeling with Dirichlet process mixing. J. Amer. Statist. Assoc., 2005, 100(471): 1021–1035

    Article  MathSciNet  MATH  Google Scholar 

  35. Gelfand A E, Kuo L. Nonparametric Bayesian bioassay including ordered polytomous response. Biometrika, 1991, 78(3): 657–666

    Article  MathSciNet  MATH  Google Scholar 

  36. Gelfand A E, Smith A F M. Sampling-based approaches to calculating marginal densities. J. Amer. Statist. Assoc., 1990, 85(410): 398–409

    Article  MathSciNet  MATH  Google Scholar 

  37. Geman S, Geman D. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. Trans. Pattern Anal. Mach. Intell., 1984, PAMI-6(6): 721–741

    Article  MATH  Google Scholar 

  38. Ghosh J K, Ramamoorthi R V. Bayesian Nonparametrics, New York: Springer-Verlag, 2003

    MATH  Google Scholar 

  39. Giudici P, Mezzetti M, Muliere P. Mixtures of products of Dirichlet processes for variable selection in survival analysis. J. Statist. Plann. Inference, 2003, 111(1/2): 101–115

    Article  MathSciNet  MATH  Google Scholar 

  40. Gou J W, Xia Y M, Jiang D P. Bayesian analysis of two-part nonlinear latent variable model: Semiparametric method. Statistical Modelling, 2021, https://doi.org/10.1177/1471082X211059233

  41. Griffin J E, Steel M F J. Order-based dependent Dirichlet processes. J. Amer. Statist. Assoc., 2006, 101(473): 179–194

    Article  MathSciNet  MATH  Google Scholar 

  42. Halmos P R. Random alms. Ann. Math. Statist., 1944, 15(2): 182–189

    Article  MathSciNet  MATH  Google Scholar 

  43. Hanson T E. Inference for mixtures of finite Polya tree models. J. Amer. Statist. Assoc., 2006, 101(476): 1548–1565

    Article  MathSciNet  MATH  Google Scholar 

  44. Hastings W K. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 1970, 57(1): 97–109

    Article  MathSciNet  MATH  Google Scholar 

  45. Ishwaran H, James L F. Gibbs sampling methods for stick-breaking priors. J. Amer. Statist. Assoc., 2001, 96(453): 161–173

    Article  MathSciNet  MATH  Google Scholar 

  46. Ishwaran H, James L F. Approximate Dirichlet process computing in finite normal mixtures: smoothing and prior information. J. Comput. Graph. Stat., 2002, 11(3): 508–532

    Article  MathSciNet  Google Scholar 

  47. Ishwaran H, James L F. Generalized weighted Chinese restaurant processes for species sampling mixture models. Statist. Sin., 2003, 13(4): 1211–1235

    MathSciNet  MATH  Google Scholar 

  48. Ishwaran H, James L F. Computational methods for multiplicative intensity models using weighted Gamma process: proportional hazards, marked point processes, and panel count data. J. Amer. Statist. Assoc., 2004, 99(465): 175–190

    Article  MathSciNet  MATH  Google Scholar 

  49. Ishwaran H, Takahara G. Independent and identically distributed Monte Carlo algorithms for semiparametric linear mixed models. J. Amer. Statist. Assoc., 2002, 97(460): 1154–1166

    Article  MathSciNet  MATH  Google Scholar 

  50. Ishwaran H, Zarepour M. Markov chain Monte Carlo in approximate Dirichlet and beta two-parameter process hierarchical models. Biometrika, 2000, 87(2): 371–390

    Article  MathSciNet  MATH  Google Scholar 

  51. Jöreskog K, Sörbom D. LISREL 8: Structural Equation Modeling with the SIMPLIS Command Language. Hove and London: Scientific Software International, 1996

    Google Scholar 

  52. Kelloway E K. Using Mplus for Structural Equation Modeling. Canadian Psychology, 1998, 40(4):381–383

    Google Scholar 

  53. Kingman J F C, Taylor S J, Hawkes A G, Walker A M, Cox D R, Smith A F M, Hill B M, Burville P J, Leonard T. Random discrete distributions. J. R. Stat. Soc. Ser. B., 1975, 37: 1–22

    MathSciNet  Google Scholar 

  54. Kleinman K P, Ibrahim J G. A semiparametric Bayesian approach to the random effects model. Biometrics, 1998, 54(3): 921–938

    Article  MATH  Google Scholar 

  55. Kleinman K P, Ibrahim J G. A semi-parametric Bayesian approach to generalized linear mixed models. Statist. Med., 1998, 17(22): 2579–2596

    Article  Google Scholar 

  56. Kolmogorov A N. Foundations of the Theory of Probability, 2nd ed., trans. Nathan Morrison (1956). Chelsea: New-York, 1933. J. Amer. Statist. Assoc., 1994, 89(425): 278–288

    Google Scholar 

  57. Kong A, Liu J S, Wong W H. Sequential imputations and Bayesian missing data problems. J. Amer. Statist. Assoc., 1994, 89(425): 278–288

    Article  MATH  Google Scholar 

  58. Korwar R M, Hollander M. Contributions to the theory of Dirichlet processes. Ann. Probab., 1973, 1(4): 705–711

    Article  MathSciNet  MATH  Google Scholar 

  59. Kuo L. Computations of mixtures of Dirichlet processes. SIAM J. Sci. Stat. Comput., 1986, 7(1): 60–71

    Article  MathSciNet  MATH  Google Scholar 

  60. Lavine M. Some aspects of Polya tree distributions for statistical modelling. Ann. Statist., 1992, 20(3): 1222–1235

    Article  MathSciNet  MATH  Google Scholar 

  61. Lavine M. More aspects of Polya tree distributions for statistical modelling. Ann. Statist., 1994, 22(3): 1161–1176

    Article  MathSciNet  MATH  Google Scholar 

  62. Lee S Y. Structural Equation Modeling: A Bayesian Approach. Chichester: John Wiley & Sons., 2007

    Book  Google Scholar 

  63. Lee S Y, Lu B, Song X Y. Semiparametric Bayesian analysis of structural equation models with fixed covariates. Statist. Med., 2008, 27(13): 2341–2360

    Article  MathSciNet  Google Scholar 

  64. Lennox K P, Dahl D B, Vannucci M, Day R, Tsai J W. A Dirichlet process mixture of hidden Markov Models for protein structure prediction. Ann. Appl. Stat., 2010, 4(2): 916–942

    Article  MathSciNet  MATH  Google Scholar 

  65. Li Y S, Lin X H, Muöller P. Bayesian inference in semiparametric mixed models for longitudinal data. Biometrics, 2010, 66(1): 70–78

    Article  MathSciNet  Google Scholar 

  66. Liu J S. Nonparametric hierarchical Bayes via sequential imputations. Ann. Statist., 1996, 24(3): 911–930

    Article  MathSciNet  MATH  Google Scholar 

  67. Lo A Y. On a class of Bayesian nonparametric estimates: I. Density estimates. Ann. Statist., 1984, 12(1): 351–357

    Article  MathSciNet  MATH  Google Scholar 

  68. MacEachern S N. Estimating normal means with a conjugate style Dirichlet process prior. Comm. Stat. Simulat. Comput., 1994, 23(3): 727–741

    Article  MathSciNet  MATH  Google Scholar 

  69. MacEachern S N. Dependent Dirichlet processes, In: ASA Proceedings of the Section on Bayesian Statistical Science. Alexandria, VA: Amer. Statist. Assoc., 1999: 50–55

    Google Scholar 

  70. MacEachern S N. Decision theoretic aspects of dependent nonparametric processes. In: Bayesian Methods with Applications to Science, Policy and Official Statistics, Crete: International Society for Bayesian Analysis, 2000: 551–560

    Google Scholar 

  71. MacEachern S N, Clyde M, Liu J S. Sequential importance sampling for nonparametric Bayes models: The next generation. Canad. J. Statist., 1999, 27(2): 251–267

    Article  MathSciNet  MATH  Google Scholar 

  72. MacEachern S N, Müller P. Estimating mixture of Dirichlet process models. J. Comput. Graph. Stat., 1998, 7(2): 223–238

    Google Scholar 

  73. MacEachern S N, Müller P. Efficient MCMC schemes for robust model extensions using encompassing Dirichlet process mixture models. In: Robust Bayesian Analysis, Lecture Notes in Statist., Vol. 152. New York: Springer-Verlag, 2000: 295–315

    MATH  Google Scholar 

  74. McCloskey J W. A model for the distribution of individuals by species in an environment. Ph.D. Thesis, East Lansing, MI: Michigan State Univ., 1965

    Google Scholar 

  75. Metropolis N, Rosenbluth A W, Rosenbluth M N, Teller A H, Teller E. Equation of state calculations by fast computing machines. J. Chem. Phys., 1953, 21(6): 1087–1092

    Article  MATH  Google Scholar 

  76. Mira A, Petrone S. Bayesian hierarchical non-parametric inference for change-point problems. In: Bayesian Statistics 5, Oxford: Oxford Univ. Press, 1996: 693–703

    Google Scholar 

  77. Muliere P, Petrone S. A Bayesian predictive approach to sequential search for an optimal dose: parametric and nonparametric models. J. Ital. Statist. Soc., 1993, 2(3): 349–364

    Article  MATH  Google Scholar 

  78. Muliere P, Tardella L. Approximating distributions of random functionals of Ferguson-Dirichlet priors. Canadian J. Statist., 1998, 26(2): 283–297

    Article  MathSciNet  MATH  Google Scholar 

  79. Müller P, Erkanli A, West M. Bayesian curving fitting using multivariate normal mixtures. Biometrika, 1996, 83(1): 67–79

    Article  MathSciNet  MATH  Google Scholar 

  80. Müller P, Quintana F, Rosner G. A method for combining inference across related non-parametric Bayesian models. J. R. Stat. Soc. Ser. B. Stat. Methodol., 2004, 66(3): 735–749

    Article  MathSciNet  MATH  Google Scholar 

  81. Müller P, Quintana F, Rosner G. A product partition model with regression on covariates. Journal of Computational and Graphical Statistics, 2011, 20, 260–278.

    Article  MathSciNet  Google Scholar 

  82. Müller P, Quintana F A, Rosner G L, Maitland M L. Bayesian inference for longitudinal data with non-parametric treatment effects. Biostatistics, 2014, 15(2): 341–352

    Article  Google Scholar 

  83. Muthén L K, Muthén B O. Mplus user’s guild. Los Angels, CA: Muthén & Muthé, 1998. Biostatistics, 2014, 15(2): 341–352

    Google Scholar 

  84. Neal R M. Markov chain sampling methods for Dirichlet process mixture models. J. Comput. Graph. Statist., 2000, 9(2): 249–265

    MathSciNet  Google Scholar 

  85. Papaspiliopoulos O, Roberts G O. Retrospective Markov Chain Monte Carlo methods for Dirichlet process hierarchical models. Biometrika, 2008, 95(1): 169–186

    Article  MathSciNet  MATH  Google Scholar 

  86. Petrone S, Guindani M, Gelfand A E. Hybrid dirichlet mixture models for functional data. J. R. Stat. Soc. Ser. B. Stat. Methodol., 2009, 71(4): 755–782

    Article  MathSciNet  MATH  Google Scholar 

  87. Pitman J. Some developments of the Blackwell-MacQueen urn scheme. In: Statistics, Probability and Game Theory, Papers in honor of David Blackwell, Hayward, CA: IMS, 1996, 245–267

    Google Scholar 

  88. Pitman J. Random discrete distributions invariant under size-biased permutation. Adv. Appl. Probab., 1996, 28(2): 525–539

    Article  MathSciNet  MATH  Google Scholar 

  89. Reich B J, Fuentes M. A multivariate semiparametric Bayesian spatial modeling framework for hurricane surface wind fields. Ann. Appl. Stat., 2007, 1(1): 249–264

    Article  MathSciNet  MATH  Google Scholar 

  90. Ripley B D. Stochastic Simulation. Chichester: John Wiley & Sons, 1987

    Book  MATH  Google Scholar 

  91. Rodríguez, A, Dunson D B, Gelfand A E. The nested Dirichlet process. J. Amer. Statist. Assoc., 2008, 103(483): 1131–1154

    Article  MathSciNet  MATH  Google Scholar 

  92. Rodriguez A, Dunson D B, Gelfand A E. Bayesian nonparametric functional data analysis through density estimation. Biometrika, 2009, 96(1): 149–162

    Article  MathSciNet  MATH  Google Scholar 

  93. Scarpa B, Dunson D B. Enriched stick-breaking processes for functional data. J. Amer. Statist. Assoc., 2014, 109(506): 647–660

    Article  MathSciNet  MATH  Google Scholar 

  94. Sethuraman J. A constructive definition of Dirichlet priors. Statist. Sin., 1994, 4(2): 639–650

    MathSciNet  MATH  Google Scholar 

  95. Sethuraman J, Tiwari R C. Convergence of Dirichlet measures and the interpretation of their parameters. In: Statistical Decision Theory and Related Topics III, New York: Academic Press, 1982: 305–316

    MATH  Google Scholar 

  96. Skrondal A, Rabe-Hesketh S. Generalized Latent Variable Modeling: Multilevel, Longitudinal, and Structural Equation Models. New York: Chapman & Hall/CRC, 2004

    Book  MATH  Google Scholar 

  97. Song X Y, Lee S Y. Basic and Advanced Bayesian Structural Equation Modeling: With Applications in the Medical and Behavioral Sciences. New York: John Wiley & Sons, 2012

    Book  MATH  Google Scholar 

  98. Song X Y, Xia Y M, Lee S Y. Bayesian semiparametric analysis of structural equation models with mixed continuous and unordered categorical variables. Statist. Med., 2009, 28(17): 2253–2276

    Article  MathSciNet  Google Scholar 

  99. Song X Y, Xia Y M, Pan J H, Lee S Y. Model comparison of Bayesian semiparametric and parametric structural equation models. Struct. Equat. Model., 2011, 18(1): 55–72

    Article  MathSciNet  Google Scholar 

  100. Tang A M, Tang N S. Semiparametric Bayesian inference on skew-normal joint modeling of multivariate longitudinal and survival data. Statist. Med., 2015, 34(5): 824–843

    Article  MathSciNet  Google Scholar 

  101. Tanner M A, Wong W H. The calculation of posterior distributions by data augmentation. J. Amer. Statist. Assoc., 1987, 82(398): 528–540

    Article  MathSciNet  MATH  Google Scholar 

  102. Teh Y W, Jordan M I, Beal M J, Blei D M. Hierarchical Dirichlet processes, J. Amer. Statist. Assoc., 2006, 101(476): 1566–1581

    Article  MathSciNet  MATH  Google Scholar 

  103. Tomlinson G, Escobar M. Analysis of densities. Technical Report, Toronto: University of Toronto, 1999

    Google Scholar 

  104. Walker S G. Sampling the Dirichlet mixture model with slices, Comm. Statist. Simulation Comput., 2007, 36(1): 45–54

    Article  MathSciNet  MATH  Google Scholar 

  105. West M, Muöller P, Escobar M D. Hierarchical priors and mixtures models, with applications in regression and density estimates. In: Aspects of Uncertainty, A Tribute to D. V. Lindley. London: John Wiley & Sons, 1994: 363–386

    Google Scholar 

  106. Xia Y M, Gou J W. Assessing heterogeneity in multilevel factor analysis model: A semiparametric Bayesian approach. Acta Math. Sin., 2015, 38(4): 751–768 (in Chinese)

    MathSciNet  MATH  Google Scholar 

  107. Xia Y M, Gou J W. Bayesian semiparametric analysis for latent variable models with mixed continuous and ordinal outcomes. J. Korean Statist. Soc., 2016, 45(3): 451–465

    Article  MathSciNet  MATH  Google Scholar 

  108. Xia Y M, Gou J W, Liu Y A. Semi-parametric Bayesian analysis for factor analysis model mixed with hidden Markov model. Appl. Math. J. Chinese Univ. Ser. A, 2015, 30(1): 17–30 (in Chinese)

    Article  MathSciNet  Google Scholar 

  109. Xia Y M, Liu Y A. Bayesian semiparametric analysis and model comparison for confirmatory factor model. Chinese J. Appl. Probab. Statist., 2016, 32(2): 157–183

    MathSciNet  MATH  Google Scholar 

  110. Xia Y M, Pan M L. Bayesian analysis for confirmatory factor model with finite-dimensional Dirichlet prior mixing. Comm. Statist. Theory Methods, 2017, 46(9): 4599–4619

    Article  MathSciNet  MATH  Google Scholar 

  111. Xia Y M, Tang N S. Bayesian analysis for mixture of latent variable hidden Markov models with multivariate longitudinal data. Computational Statistics & Data Analysis, 2019, 132: 190–211

    Article  MathSciNet  MATH  Google Scholar 

  112. Yang M G, Dunson D B. Bayesian semiparametric structural equation models with latent variables. Psychometrika, 2010, 75(4): 675–693

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant No. 11471161) and the Technological Innovation Item in Jiangsu Province (No. BK2008156).

Author information

Authors and Affiliations

  1. School of Sciences, Nanjing Forestry University, Nanjing, 210037, China

    Yemao Xia & Jianwei Gou

  2. College of Information Science and Technology, Nanjing Forestry University, Nanjing, 210037, China

    Yingan Liu

Authors
  1. Yemao Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yingan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianwei Gou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianwei Gou.

Additional information

Translated from Advances in Mathematics (China), 2017, 46(5): 641–666

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xia, Y., Liu, Y. & Gou, J. Dirichlet process and its developments: a survey. Front. Math. China 17, 79–115 (2022). https://doi.org/10.1007/s11464-022-1004-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-022-1004-3

Keywords

MSC2020

Search

Navigation