Gibbs sampling methods for stick-breaking priors

Journal of the American Statistical Association

Volumn 96, Issue 453, 2001, Pages 161-173

  Ishwaran, Hemant ^a   James, Lancelot F ^b  

^a LERNER RESEARCH INSTITUTE   (United States)

^b JOHNS HOPKINS UNIVERSITY   (United States)

Author keywords

Blocked Gibbs sampler; Dirichlet process; Generalized Dirichlet distribution; Pitman Yor process; Prediction rule; P lya urn Gibbs sampler; Random probability measure; Random weights; Stable law

Indexed keywords

EID: 1842816362     PISSN: 01621459     EISSN: 1537274X     Source Type: Journal    
DOI: 10.1198/016214501750332758     Document Type: Article

References (51)

1. Antoniak, C. E. (1974). “Mixtures of Dirichlet processes With Applications to Bayesian Nonparametric Problems” The Annals of Statistics. 2. 1152-1174.
2. Blackwell, D. and MacQueen, J. B. (1973). “Ferguson Distributions via Polya Um Schemes.” The Annals of Statistics. I. 353-355.
3. Bush, C. A., and MacEachcm, S. N. (1996). “A Semiparamctric Bayesian Model for randomised Block Designs.” Biometrika. 83. 275-285.
4. Carlton, M. A. (1999). “Applications of the Two-Parameter Poisson-Dirichlet Distribution.” unpublished Ph.D. thesis. University of California. Los Angeles. Dept, of Statistics.
5. Connor, R. J., and Mosimann, J. E. (1969). “Concepts of Independence for Proportions With a Generalization of the Dirichlet Distribution.” Journal of the American Statistical Association. 64. 194-206.
6. Damien, P., Laud, P. W., and Smith, A. F. M. (1996). “Implementation of Bayesian Non-Parametric Inference Based on Bela Processes” Scandinavian Journal of Statistics, 23. 27-36.
7. Donnelly, P. and Joyce, P. (1989). “Continuity and Weak Convergence of Ranked and Size-Biased Permutations on the Infinite Simplex.” Stochastic Processes and Applications. 31. 89-103.
8. Doss, H. (1994). “Bayesian Nonparametric Estimation for Incomplete Data via Successive Substitution Sampling.” The Annals of Statistics. 22, 1763-1786.
9. Escobar, M. D. (1988). “Estimating the Means of Several Normal Populations by Nonparametric Estimation of the Distribution of the Means.” unpublished Ph.D. thesis. Yale University. Dept, of Statistics.
10. Escobar, M. D. (1994). “Estimating Normal Means With a Dirichlet Process Prior.” Journal of the American Statistical Association. 89. 268-277.
11. Escobar, M. D., and West, M. (1995). “Bayesian Density Estimation and Inference Using Mixtures.” Journal of the American Statistical Association. 90. 577-588.
12. Escobar, M. D., and West, M. (1998). “Computing Nonparametric Hierarchical Models.” in Practical Nonparametric and Semiparametric Bayesian Statistics, eds, D. Dcy, P. Mueller, and D. Sinha. New York: Springer-verlag. 1-22.
13. Fabius, J. (1964). “Asymptotic Behavior of Bayes Estimates.” Annals of Mathematical Statistics. 35. 846-856.
14. Ferguson, T. S. (1973). “A Bayesian Analysis of Some Nonparametric Problems.” The Annals of Statistics. 1. 209-230.
15. Ferguson, T. S. (1974). “Prior Distributions on Spaces of Probability Measures." The Annals of Statistics. 2, 615-629.
16. Ferguson, T. S., and Klass, M. J. (1972), “A Representation of Independent Increment Processes Without Gaussian Components.” The Annals of Math-ema tica I Statistics. 43.1634-1643.
17. Freedman, D. (1963). “On the Asymptotic Behavior of Bayes Estimates in the Discrete Case.” Annals of Mathematical Statistics. 34. 1386-1403.
18. Green, P., and Richardson, S. (1999). “Modelling Heterogeneity With and Without the Dirichlet Process.” unpublished manuscript.
19. Haimos, P. (1944). “Random Alms.” The Annals of Mathematical Statistics. 15. 182-189.
20. Hjort, N. L. (1990). “Nonparametric Bayes Estimators Based on Beta Pro cesses in Models for Life History Data.” The Annals of Statistics. 18. 1259-1294.
21. Ishwaran, H., and James, L. F. (2000). “Approximate Dirichlet Process Computing for Finite Normal Mixtures: Smoothing and Prior Information.” unpublished manuscript.
22. Ishwaran, H., and Zarepour, M. (2000a). “Markov Chain Monte Carlo in Approximate Dirichlet and Beta TWo-Paramctcr Process Hierarchical Models.” Biometrika. 87. 371-390.
23. Ishwaran, H., and Zarepour, M. (2000b). “Exact and Approximate Sum-Representations for the Dirichlet Process.” unpublished manuscript.
24. Ishwaran, H., and Zarepour, M. (2000c). “Dirichlet Prior Sieves in Finite Normal Mixtures.” unpublished manuscript.
25. Kerov, S. (1995). “Coherent Random Allocations and the Ewcns-Pitman Formula.” PDMI Preprint. Steklov Institute of Mathematics. St. Petersburg.
26. Kingman, J. F. C. (1974), “Random Discrete Distributions.” Journal of the Royal Statistical Society. Series B. 37, 1-22.
27. Laud, P. W., Damien, P., and Smith, A. F. M. (1998). “Bayesian Nonparametric and Covariate Analysis of Failure Time Data.” in Practical Nonparametric and Semiparametric Bayesian Statistics, eds, D. Dey, P. Mueller, and D. Sinha. New York: Springer-\ferlag. pp. 213-225.
28. Liu, J. S. (1996). “Nonparametric Hierarchical Bayes via Sequential Imputations.” The Annals of Statistics. 24. 911-930.
29. Lo, A. Y. (1984). “On a Class of Bayesian Nonparametric Estimates: I. Density Estimates.” The Annals of Statistics. 12. 351-357.
30. MacEachcm, S. N. (1994). “Estimating Normal Means With a Conjugate Style Dirichlet Process Prior.” Communications in Statistics—Simulations. 23.727-741.
31. MacEachcm, S. N. (1998). “Computational Methods for Mixture of Dirichlet Process Models.” in Practical Nonparametric and Semiparametric Bayesian Statistics. eds, D. Dey, P. Mueller, and D. Sinha. New York: Springer-Verlag.
32. MacEachcm, S. N., and Muller, P. (1998). “Estimating Mixture of Dirichlet Process Models.” Journal of Computational and Graphical Statistics. 7. 223-238.
33. McCloskey, J. W. (1965). “A Model for the Distribution of Individuals by Species in an Environment.” unpublished Ph.D. thesis. Michigan State University.
34. Mekjian A. Z., and Chase, K. C. (1997). “Disordered Systems. Power Laws and Random Processes.” Physics Letters A. 229 340-346.
35. Muliere, R. and Secchi, P. (1995). “A Note on a Proper Bayesian Bootstrap.” Technical Report 18. Università degli Sudi di Pavia. Dipartimento di Economia Politica e Metodi Quantitativi.
36. Muliere, R. and Tardella, L. (1998). “Approximating Distributions of Random Functionals of Ferguson-Dirichlet Priors.” Canadian Journal of Statistics. 26. 283-297.
37. Neal, R. M. (2000). “Markov Chain Sampling Methods for Dirichlet Process Mixture Models.” Journal of Computational and Graphical Statistics, 9, 249-265.
38. Perman, M., Pitman, J., and Vor, M. (1992). “Size-Biased Sampling of Poisson Point Processes and Excursions.” Probability Theory and Related Fields. 92.21-39.
39. Pitman, J. (1995). “Exchangeable and Partially Exchangeable Random Partitions.” Probability Theory amd Related Fields. 102, 145-158.
40. Pitman, J. (1996a). “Some Developments of the Blackwell-.MacQuccn Um Scheme.” in Statistics. Probability and Game Theory, eds, T. S. Ferguson, L. S. Shapley. and J. B. MacQueen. IMS Lecture Notes—Monograph Series (Vol. 30). Hayward. CA: Institute of Mathematical Statistics, pp. 245-267.
41. Pitman, J. (1996b). “Random Discrete Distributions Invariant UndcrSize-Biased Permutation.” Advances in Applied Probability 28, 525-539.
42. Pitman, J. (1997). “Partition Structures Derived From Brownian Motion and Stable Subordinatone” Bernoulli. 3. 79-86.
43. Pitman, J. (1999). “Coalescents With Multiple Collisions.” The Annals of Probability. 27. 1870-1902.
44. Pitman, J. and Vor, M. (1997). “The Two-Parameter Poisson-Dirichlct Distribution Derived From a Stable Subordinator.” The Annals of Probability. 25. 855-900.
45. Scthuraman, J. (1994). “A Constructive Definition of Dirichlet Priors.” Statistica Sinica. 4. 639-650.
46. Scthuraman, J., and Tiwari, R. C. (1982). “Convergence of Dirichlet Measures and the Interpretation of Their Parameters.” Statistical Decision Theory and Related Topics III. 2, 305-315.
47. Tsilevich, N. V. (1997). “Distribution of the Mean Value for Certain Random Measures.” POMI Preprint 240. Steklov Institute of Mathematics. St. Petersburg.
48. Walker, S., and Damien, R (1998). “Sampling Methods for Bayesian Non-parametric Inference Involving Stochastic Processes.” in Practical Non-parametric and Semi parame trie Bayesian Statistics, ed, D. Dey. R Muller, and D. Sinha New York: Springer-verlag. pp. 243-254.
49. Walker, S., and Wakefield, J. (1998). “Population models With a Nonpara-metric Random Coefficient Distribution." Sankhyd. Ser, B. 60. 196-214.
50. West, M., Muller, R. and Escobar, M. D. (1994). “Hierarchical Priors and Mixture Models. With Applications in Regression and Density Estimation.” in A tribute to D. V Lindley. eds, A. F. M Smith and P. R. Freeman. New York: Wiley.
51. Zabell, S. L. (1997). ‘The Continuum of Inductive Methods Revisited.” in The Cosmos of Science, eds, J. Earman and J. D. Norton. Pittsburgh-Konstanz Series in the Philosophy and History of Science. Pittsburgh. PA: University of Pittsburgh Press, pp. 351-385.