Contributions to the Theory of Dirichlet Processes

Korwar, Ramesh M.; Hollander, Myles

doi:10.1214/aop/1176996898

Cited by 163 publications

(74 citation statements)

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“…1 with probability 1 as n ! 1 (Korwar & Hollander, 1973), the number of components increases approximately logarithmically with the number of observations. This is illustrated in Fig.…”

Section: Learning the Dispersion Of Datamentioning

confidence: 94%

Modeling individual differences using Dirichlet processes

Navarro

Griffiths

Steyvers

et al. 2006

Journal of Mathematical Psychology

112

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We introduce a Bayesian framework for modeling individual differences, in which subjects are assumed to belong to one of a potentially infinite number of groups. In this model, the groups observed in any particular data set are not viewed as a fixed set that fully explains the variation between individuals, but rather as representatives of a latent, arbitrarily rich structure. As more people are seen, and more details about the individual differences are revealed, the number of inferred groups is allowed to grow. We use the Dirichlet process-a distribution widely used in nonparametric Bayesian statistics-to define a prior for the model, allowing us to learn flexible parameter distributions without overfitting the data, or requiring the complex computations typically required for determining the dimensionality of a model. As an initial demonstration of the approach, we present three applications that analyze the individual differences in category learning, choice of publication outlets, and web-browsing behavior. r 2005 Elsevier Inc. All rights reserved.

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“…1 with probability 1 as n ! 1 (Korwar & Hollander, 1973), the number of components increases approximately logarithmically with the number of observations. This is illustrated in Fig.…”

Section: Learning the Dispersion Of Datamentioning

confidence: 94%

Modeling individual differences using Dirichlet processes

Navarro

Griffiths

Steyvers

et al. 2006

Journal of Mathematical Psychology

112

View full text Add to dashboard Cite

show abstract

“…From the form of the predictive distribution, proceeding as in Korwar and Hollander (1973) and Antoniak (1974, Lemma 1 ), it can be shown that P, i.e. the joint probability law of (X1 ..... X,), gives positive probability to certain collections of hyperplanes in the sample space R ~.…”

Section: The Likelihood Of a Sample From A Diriehlet Processmentioning

confidence: 99%

“…the joint probability law of (X1 ..... X,), gives positive probability to certain collections of hyperplanes in the sample space R ~. Let us introduce the set Cg (D, nl ..... riD I~(D'nl,'",no) Korwar and Hollander (1973) show that, given the number D of distinct values, these are i.i.d. according to fo.…”

Section: The Likelihood Of a Sample From A Diriehlet Processmentioning

confidence: 99%

A note on the Dirichlet process prior in Bayesian nonparametric inference with partial exchangeability

Petrone

Raftery

1997

Statistics & Probability Letters

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We consider Bayesian nonparametric inference for continuous-valued partially exchangeable data, when the partition of the observations into groups is unknown. This includes change-point problems and mixture models. As the prior, we consider a mixture of products of Dirichlet processes. We show that the discreteness of the Dirichlet process can have a large effect on inference (posterior distributions and Bayes factors), leading to conclusions that can be different from those that result from a reasonable parametric model. When the observed data are all distinct, the effect of the prior on the posterior is to favor more evenly balanced partitions, and its effect on Bayes factors is to favor more groups. In a hierarchical model with a Dirichlet process as the second-stage prior, the prior can also have a large effect on inference, but in the opposite direction, towards more unbalanced partitions. (~) 1997 Elsevier Science B.V.

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“…Another distinctive feature, if compared with the Dirichlet process, is represented by the asymptotic behaviour of the number of groups K n generated by the first n observations, as n → ∞. For the Dirichlet process, as shown in Korwar and Hollander (1973), K n ∼ θ log(n) almost surely as n → ∞. Hence, the number of distinct observations increases at a logarithmic rate.…”

Section: Example 9 (The Two Parameter Poisson-dirichlet Process) Onmentioning

confidence: 99%