摘要：

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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