Dirichlet Process

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A Dirichlet Process is a stochastic mixture process whose sample path has a probability function.




  • (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Dirichlet_process
    • In probability theory, a Dirichlet process over a set [math]\displaystyle{ S }[/math] is a stochastic process whose sample path is a probability distribution on S. The finite dimensional distributions are from the Dirichlet distribution: If [math]\displaystyle{ M }[/math] is a finite measure on [math]\displaystyle{ S }[/math] and [math]\displaystyle{ X }[/math] is a random distribution drawn from a Dirichlet process, written as: [math]\displaystyle{ X \sim \mathrm{DP}\left(M\right) }[/math] then for any partition of [math]\displaystyle{ S }[/math], say [math]\displaystyle{ \left\{B_i\right\}_{i=1}^{n} }[/math], we have that [math]\displaystyle{ \left(X\left(B_1\right),\dots,X\left(B_n\right)\right) \sim \mathrm{Dirichlet}\left(M\left(B_1\right),\dots,M\left(B_n\right)\right) }[/math]





  • (Sethuraman, 1994) ⇒ J. Sethuraman. (1994). “A Constructive Definition of Dirichlet Priors." Statistica Sinica, 4.


  • (Ferfuson, 1973) ⇒ Thomas Ferguson. (1973). “Bayesian Analysis of Some Nonparametric Problems.” In: Annals of Statistics 1 (2) doi:10.1214/aos/1176342360
    • ABSTRACT: The Bayesian approach to statistical problems, though fruitful in many ways, has been rather unsuccessful in treating nonparametric problems. This is due primarily to the difficulty in finding workable prior distributions on the parameter space, which in nonparametric ploblems is taken to be a set of probability distributions on a given sample space. There are two desirable properties of a prior distribution for nonparametric problems. (I) The support of the prior distribution should be large--with respect to some suitable topology on the space of probability distributions on the sample space. (II) Posterior distributions given a sample of observations from the true probability distribution should be manageable analytically. These properties are antagonistic in the sense that one may be obtained at the expense of the other. This paper presents a class of prior distributions, called Dirichlet process priors, broad in the sense of (I), for which (II) is realized, and for which treatment of many nonparametric statistical problems may be carried out, yielding results that are comparable to the classical theory.