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• Dirichlet_process abstract "In probability theory, a Dirichlet process is a way of assigning a probability distribution over probability distributions. That is, a Dirichlet process is a probability distribution whose domain is itself a set of probability distributions. The probability distributions in the domain are almost surely discrete and may be infinite dimensional. Assigning an arbitrary probability distribution over a domain of infinite dimensional probability distributions would require an infinite amount of computational resources. The main function of the Dirichlet process is that it allows the specification of distribution over infinite dimensional distributions in a way that uses only finite resources.Given a Dirichlet process , where (the base distribution or base measure) is an arbitrary distribution and (the concentration parameter) is a positive real number, a draw from will return a random distribution over some of the values that can be drawn from . That is, the support of each draw of the output distribution is always a subset of the support of base distribution. The output distribution will be discrete, meaning that individual values drawn from the output distribution will sometimes repeat themselves even if the base distribution is continuous. The extent to which values will repeat is determined by , with higher values causing less repetition.A Dirichlet process is most easily defined implicitly by the distribution it induces on certain finite dimensional statistics. A common model for data is that observations are assumed to be identically and independently distributed according to some unknown distribution . We suppose that the unknown distribution is itself drawn randomly according to . If we wish to simulate observations a straightforward algorithm for doing so is: Draw a distribution from Draw observations independently from .The issue is that in many problems of interest the distribution may require an infinite number of parameters to specify. To avoid this we can simulate observations according the algorithm: Draw from measure . For With probability draw from . For each distinct in let be the number of such that . Let with probability .It can be shown that observations drawn according to this process are exchangeable and thus by de Finetti’s representation theorem the are identically and independently distributed according to some unknown distribution . This second algorithm is thus equivalent to first drawing a random distribution and then drawing observations independently from . The Dirichlet process with parameters and , , is defined to be the distribution over distributions from which is drawn; i.e. is the distribution over distributions which makes the two algorithms for simulating equivalent. Notice that by defining the Dirichlet process implicitly in this fashion we have sidestepped the impossible problem of explicitly defining a distribution over the infinite dimensional distributions .Dirichlet processes frequently appear in the context of Bayesian Non-parametric statistics where a typical task is to learn distributions on function spaces, which involve effectively infinitely many parameters. The key insight is that in many applications the infinite dimensional distributions appear only as an intermediary computational device and are in particular not required for either the initial specification of prior beliefs or for the statement of the final inference. The Dirichlet process can be used to circumvent infinite computational requirements as described above. A particularly important application of the Dirichlet process is as a prior probability in infinite mixture models; this is discussed in detail below.There are several equivalent views of the Dirichlet process. It can be formally defined as a stochastic process over the sigma algebra generated by , where the induced marginal distributions over any finite partition of the domain of are Dirichlet distributions. Alternatively the Dirichlet process may be defined implicitly through de Finetti’s theorem as described above; this is often called the Chinese restaurant process. The stick-breaking process defines the Dirichlet process constructively by writing a distribution sampled from the process as , where are samples from the base distribution and the are defined by a recursive scheme that repeatedly samples from a distribution.The Dirichlet process was formally introduced by Thomas Ferguson in 1973.".
• Dirichlet_process wikiPageExternalLink ClusterAnalysis.org.
• Dirichlet_process wikiPageExternalLink uai05tutorial-b.pdf.
• Dirichlet_process wikiPageExternalLink nips-tutorial05.ps.
• Dirichlet_process wikiPageExternalLink npbayes.
• Dirichlet_process wikiPageExternalLink dpcluster.html.
• Dirichlet_process wikiPageExternalLink Teh2010a.pdf.
• Dirichlet_process wikiPageExternalLink cribsheet.pdf.
• Dirichlet_process wikiPageExternalLink GreenCDP.pdf.
• Dirichlet_process wikiPageExternalLink UWEETR-2010-0006.pdf.
• Dirichlet_process wikiPageID "8330403".
• Dirichlet_process wikiPageRevisionID "604916647".
• Dirichlet_process hasPhotoCollection Dirichlet_process.
• Dirichlet_process subject Category:Non-parametric_Bayesian_methods.
• Dirichlet_process subject Category:Stochastic_processes.
• Dirichlet_process type Ability105616246.
• Dirichlet_process type Abstraction100002137.
• Dirichlet_process type Cognition100023271.
• Dirichlet_process type Concept105835747.
• Dirichlet_process type Content105809192.
• Dirichlet_process type Hypothesis105888929.
• Dirichlet_process type Idea105833840.
• Dirichlet_process type Know-how105616786.
• Dirichlet_process type Method105660268.
• Dirichlet_process type Model105890249.
• Dirichlet_process type Non-parametricBayesianMethods.
• Dirichlet_process type PsychologicalFeature100023100.
• Dirichlet_process type StochasticProcess113561896.
• Dirichlet_process type StochasticProcesses.
• Dirichlet_process comment "In probability theory, a Dirichlet process is a way of assigning a probability distribution over probability distributions. That is, a Dirichlet process is a probability distribution whose domain is itself a set of probability distributions. The probability distributions in the domain are almost surely discrete and may be infinite dimensional.".
• Dirichlet_process label "Dirichlet process".
• Dirichlet_process sameAs m.026_nb5.
• Dirichlet_process sameAs Q5280766.
• Dirichlet_process sameAs Q5280766.
• Dirichlet_process sameAs Dirichlet_process.
• Dirichlet_process wasDerivedFrom Dirichlet_process?oldid=604916647.
• Dirichlet_process isPrimaryTopicOf Dirichlet_process.