Abstract
In view of the limitation of the Dirichlet process that it selects a discrete probability distribution with probability one, efforts were made to discover some alternatives. Tailfree and Polya tree processes introduced in this chapter provide a partial solution. In the first part of the chapter, we formally define the tailfree process on the real line based on a sequence of nested partitions in which the probabilities assigned to individual partitions are assumed to be independent across different levels of partitions. We then illustrate the construction of a dyadic tailfree process on the interval (0,1] as an example, and discuss several properties of the process including the conjugacy property. In particular we show that the Dirichlet process is tailfree with respect to every sequence of nested measurable partitions. In the second part we introduce the Polya tree process as a special case of the tailfree process in which the probabilities assigned to all the partitions are assumed to be independent and its properties are discussed in greater details. It is pointed out that the Polya tree processes are flexible and are particularly useful when it is desired to assign greater weights to the regions where it is deemed appropriate, by selecting the partitions accordingly. Its advantage over the Dirichlet process highlighted is that with a proper choice of parameters it can select an absolutely continuous distribution with positive probability. Its main weakness identified is that the inferential results obtained are very much influenced by the partitions chosen, although it is noted that it can be addressed partially by introducing randomness in defining the partitions. Potentials of using finite and mixtures of Polya trees in modeling statistical data are indicated.
Extensions of prior processes to higher dimension always pose significant challenges. Several attempts made in this direction with limited success are briefly reviewed. Thereafter it is indicated that in the case of tailfree processes it is relatively straightforward. This is demonstrated by constructing a bivariate tailfree process on the unit square.
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References
Antoniak, C. (1974). Mixtures of Dirichlet processes with applications Bayesian nonparametric problems. Annals of Statistics, 2, 1152–1174.
Blackwell, D. (1973). Discreteness of Ferguson selections. Annals of Statistics, 1, 356–358.
Bulla, P., Muliere, P., & Walker, S. (2007). Bayesian nonparametric estimation of a bivariate survival function. Statistica Sinica, 17, 427–444.
Bulla, P., Muliere, P., & Walker, S. (2009). A Bayesian nonparametric estimator of a multivariate survival function. Journal of Statistical Planning and Inference, 139, 3639–3648.
Christensen, R., Hanson, T., & Jara, A. (2008). Parametric nonparametric statistics: An introduction to mixtures of finite Polya trees. Annals of Statistics, 62, 296–306.
Dabrowska, D. M. (1988). Kaplan-Meier estimate on the plane. Annals of Statistics, 15, 1475–1489.
Dalal, S. R., & Phadia, E. G. (1983). Nonparametric Bayes inference for concordance in Bivariate distributions. Communications in Statistics - Theory & Methods, 12(8), 947–963.
Doksum, K. A. (1974). Tailfree and neutral random probabilities and their posterior distributions.Annals of Probability, 2, 183–201.
Dråghici, L., & Ramamoorthi, R. V. (2000). A note on the absolute continuity and singularity of Polya tree priors and posteriors. Scandinavian Journal of Statistics, 27, 299–303.
Dubins, L. E., & Freedman, D. A. (1966). Random distribution functions. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (Vol. 2, pp. 183–214).
Fabius, J. (1964). Asymptotic behavior of Bayes estimates. Annals of Mathematical Statistics, 35, 846–856.
Fabius, J. (1973). Neutrality and Dirichlet distributions. In Transactions of the 6th Prague Conference on Information Theory, Statistical Decision Functions and Random Processes (pp. 175–181).
Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Annals of Statistics, 1, 209–230.
Ferguson, T. S. (1974). Prior distributions on spaces of probability measures. Annals of Statistics, 2, 615–629.
Freedman, D. A. (1963). On the asymptotic behavior of Bayes estimates in the discrete case. Annals of Mathematical Statistics, 34, 1386–1403.
Ghosh, J. K., Hjort, N. L., Messan, C., & Ramamoorthi, R. V. (2006). Bayesian bivariate survival estimation. Journal of Statistical Planning and Inference, 136, 2297–2308.
Hanson, T. E. (2006). Inference for mixtures of finite Polya tree models. Journal of the American Statistical Association, 101, 1548–1565.
Hanson, T. E. (2007). Polya trees and their use in reliability and survival analysis. In Encyclopedia of statistcs in quality and reliability (pp. 1385–1390). New York: Wiley.
Hanson, T. E., Branscum, A., & Gardner, I. (2008). Multivariate mixtures of Polya trees for modelling ROC data. Statistical Modelling, 8, 81–96.
Hanson, T. E., & Johnson, W. O. (2002). Modeling regression error with a mixture of Polya trees. Journal of the American Statistical Association, 97 , 1020–1033.
Hjort, N. L. (1990). Nonparametric Bayes estimators based on Beta processes in models for life history data. Annals of Statistics, 18(3), 1259–1294.
Ishwaran, H., & James, L. F. (2001). Gibbs sampling methods for Stick-breaking priors. Journal of the American Statistical Association, 96 , 161–173.
Kaplan, E. L., & Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53, 457–481.
Kraft, C. H. (1964). A class of distribution function processes which have derivatives. Journal of Applied Probability, 1, 385–388.
Lavine, M. (1992). Some aspects of Polya tree distributions for statistical modelling. Annals of Statistics, 20, 1222–1235.
Lavine, M. (1994). More aspects of Polya trees for statistical modelling. Annals of Statistics, 22, 1161–1176.
Lo, A. Y. (1984). On a class of Bayesian nonparametric estimates; I. Density estimates. Annals of Statistics, 12, 351–357.
Lo, A. Y. (1986). Bayesian statistical inference for sampling a finite population. Annals of Statistics, 14, 1226–1233.
Mauldin, R. D., Sudderth, W. D., & Williams, S. C. (1992). Polya trees and random distributions. Annals of Statistics, 20, 1203–1221.
Muliere, P., & Walker, S. (1997). A Bayesian non-parametric approach to survival analysis using Polya trees. Scandinavian Journal of Statistics, 24, 331–340.
Paddock, S., Ruggeri, F., Lavine, M., & West, M. (2003). Randomised Polya tree models for nonparametric Bayesian inference. Statistica Sinica, 13, 443–460.
Phadia, E. G. (2007). On bivariate tailfree processes. In Proceedings of the 56th Session of the International Statistical Institute, Lisbon (2007) (electronic version)
Phadia, E. G., & Susarla, V. (1983). Nonparametric Bayesian estimation of a survival curve with dependent censoring mechanism. Annals of the Institute of Statistical mathematics, 35, 389–400.
Pruitt, R. C. (1992). An inconsistent Bayes estimate in bivariate survival curve analysis. Statistics and Probability Letters, 15(3), 177–180.
Salinas-Torres, V. H., Pereira, C. A. B., & Tiwari, R. C. (2002). Bayesian nonparametric estimation in a series system or a competing-risks model. Nonparametric Statistics, 14, 449–458.
Tsai, W. Y. (1986). Estimation of survival curves from dependent censorship models via a generalized self-consistent property with nonparametric Bayesian estimation application. Annals of Statistics, 14, 238–249.
Walker, S. G., & Mallick, B. K. (1997). Hierarchical generalized linear models and frailty models with Bayesian nonparametric mixing. Journal of the Royal Statistical Society B, 59, 845–860.
Walker, S. G., & Mallick, B. K. (1999). Semiparametric accelerated life time models. Biometrics, 55, 477–483.
Walker, S. G., & Muliere, P. (1997b). A characterization of Polya tree distributions. Statistics & Probability Letters, 31, 163–168.
Walker, S. G., & Muliere, P. (2003). A bivariate Dirichlet process. Statistics & Probability Letters, 64, 1–7.
Yang, M., Hanson, T., & Christensen, R. (2008). Nonparametric Bayesian estimation of a bivariate density with interval censored data. Computational Statistics & Data Analysis, 52(12), 5202–5214.
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Phadia, E.G. (2016). Tailfree Processes. In: Prior Processes and Their Applications. Springer Series in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-32789-1_5
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