Optional Pólya tree and Bayesian inference. (English) Zbl 1189.62048

Summary: We introduce an extension of the Pólya tree approach for constructing distributions on the space of probability measures. By using optional stopping and optional choice of splitting variables, the construction gives rise to random measures that are absolutely continuous with piecewise smooth densities on partitions that can adapt to fit the data. The resulting “optional Pólya tree” distribution has large support in total variation topology and yields posterior distributions that are also optional Pólya trees with computable parameter values.


62F15 Bayesian inference
62G99 Nonparametric inference
60G57 Random measures
62G07 Density estimation
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[1] Blackwell, D. (1973). Discreteness of Ferguson selections. Ann. Statist. 1 356-358. · Zbl 0276.62009
[2] Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 353-355. · Zbl 0276.62010
[3] Breiman, L., Friedman, J. H., Olshen, R. A. and Stone, C. J. (1984). Classification and Regression Trees . Wadsworth Advanced Books and Software, Belmont, CA. · Zbl 0541.62042
[4] Denison, D. G. T., Mallick, B. K. and Smith, A. F. M. (1998). A Bayesian CART algorithm. Biometrika 85 363-377. JSTOR: · Zbl 1048.62502
[5] Diaconis, P. and Freedman, D. (1986). On inconsistent Bayes estimates of location. Ann. Statist. 14 68-87. · Zbl 0595.62023
[6] Fabius, J. (1964). Asymptotic behavior of Bayes’ estimates. Ann. Math. Statist. 35 846-856. · Zbl 0137.12604
[7] Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209-230. · Zbl 0255.62037
[8] Ferguson, T. S. (1974). Prior distributions on spaces of probability measures. Ann. Statist. 2 615-629. · Zbl 0286.62008
[9] Freedman, D. A. (1963). On the asymptotic behavior of Bayes’ estimates in the discrete case. Ann. Math. Statist. 34 1386-1403. · Zbl 0137.12603
[10] Ghosh, J. K. and Ramamoorthi, R. V. (2003). Bayesian Nonparametrics . Springer, New York. · Zbl 1029.62004
[11] Hanson, T. E. (2006). Inference for mixtures of finite Pólya tree models. J. Amer. Statist. Assoc. 101 1548-1565. · Zbl 1171.62323
[12] Hutter, M. (2009). Exact nonparametric Bayesian inference on infinite trees. Technical Report 0903.5342. Available at http://arxiv.org/abs/0903.5342.
[13] Kraft, C. H. (1964). A class of distribution function processes which have derivatives. J. Appl. Probab. 1 385-388. JSTOR: · Zbl 0203.19702
[14] Lavine, M. (1992). Some aspects of Pólya tree distributions for statistical modelling. Ann. Statist. 20 1222-1235. · Zbl 0765.62005
[15] Lavine, M. (1994). More aspects of Pólya tree distributions for statistical modelling. Ann. Statist. 22 1161-1176. · Zbl 0820.62016
[16] Lo, A. Y. (1984). On a class of Bayesian nonparametric estimates. I. Density estimates. Ann. Statist. 12 351-357. · Zbl 0557.62036
[17] Mauldin, R. D., Sudderth, W. D. and Williams, S. C. (1992). Pólya trees and random distributions. Ann. Statist. 20 1203-1221. · Zbl 0765.62006
[18] Nieto-Barajas, L. E. and Müller, P. (2009). Unpublished manuscript.
[19] Paddock, S. M., Ruggeri, F., Lavine, M. and West, M. (2003). Randomized Polya tree models for nonparametric Bayesian inference. Statist. Sinica 13 443-460. · Zbl 1015.62051
[20] Schwartz, L. (1965). On Bayes procedures. Z. Wahrsch. Verw. Gebiete 4 10-26. · Zbl 0158.17606
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