Korwar, R. M. and Hollander, M. (1973). Contributions to the theory of Dirichlet processes. Ann. Probability 1 705-711.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to the Blackwell MacQueen rule and the eppf for sampling from a Dirichlet( prior, which is the special case of the two parameter model with ff = 0 and 0. In this model, K n is a sum of independent indicator variables, which implies K n log n almost surely and K n is asymptotically normal [48]. In the model with 0 ff 1 and Gammaff the sequence (K n ) is an inhomogeneous Markov chain such that K n Sn ff almost surely, for a random variable S with a continuous density on (0; 1) depending on (ff; See [56, 60, 54] for this and other asymptotic results for the ....

....Xm does not equal X j for any 1 j m Gamma 1, and K n is the number of distinct values observed among X 1 ; Delta Delta Delta ; X n . The simple structure of the representation (56) in this setting can be read immediately from the Blackwell MacQueen urn scheme. As noted by Korwar Hollander [48], by application of standard limit theorems for sums of independent random variables, this leads to a law of large numbers and a central limit theorem governing the P asymptotic behaviour of K n for large n. In particular, for = 1, we recover the result of Goncharov [31] regarding the ....

R. M. Korwar and M. Hollander. Contributions to the theory of Dirichlet processes. Ann. Prob., 1:705--711, 1973.

....of the values of the random probability vectors could be repeated. Let N(i) be the number of unique values among the components of i, which we will call clusters. The weight c is closely related to the clustering structure. This relationship has been studied by Antoniak (1974) and particularly by Korwar and Hollander (1973), who showed that for a fixed value of c we have N(i) log(k) a.s. Gamma c as k 1. This suggests that larger values of c will tend to produce more clusters, and that N(i) asymptotically behaves like c log(k) In a model that treats the weight parameter as fixed, the value of c is typically ....

....for c on R , with density functions 1 and 2 . If for all x 0 we have Psi 1 (x) Psi 2 (x) then for all j = 1; 2; k Gamma 1 P 1 (N(i) j) P 2 (N(i) j) where P 1 and P 2 represent the marginal probabilities induced by the priors Psi 1 and Psi 2 for c. Proof: Following Korwar and Hollander (1973), let D 1 = 1, and for i = 2; 3; k, let D i = 0 if p i = p j for some j = 1; 2; i Gamma 1, and 1 otherwise. It follows (Korwar and Hollander 1973) that D 1 ; D k are conditionally independent given c, with P (D i = 1jc) c= c i Gamma 1) With this notation, we ....

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Korwar, R. M. and Hollander, M. (1973). Contributions to the theory of Dirichlet processes, The Annals of Probability 1: 705--711.

....locations is a random sample from ff. Clearly, a random partition is induced by considering in the same element of the partition, i.e. the same cluster, those individuals seated at the same table. The parameter c affects this distribution; for example N(z) log(n) a.s. Gamma c as n 1 (Korwar and Hollander 1973). Thus small values of c favor large clusters. The connection to nonparametric hierarchical Bayesian analysis is rather interesting. One models data y 1 ; y 2 ; y n from n experimental units as conditionally independent given unobserved parameters i 1 ; i 2 ; i n . These ....

Korwar, R. M. and Hollander, M. (1973). Contributions to the theory of Dirichlet processes, The Annals of Probability 1: 705--711.

....of the values of the random probability vectors could be repeated. Let N(i) be the number of unique values among the components of i, which we will call clusters. The weight c is closely related to the clustering structure. This relationship has been studied by Antoniak (1974) and particularly by Korwar and Hollander (1973), who showed that for a fixed value of c we have N(i) log(k) a.s. Gamma c as k 1. This suggests that larger values of c will tend to produce more clusters, and that N(i) asymptotically behaves like c log(k) For our model, we have the following statement. Lemma 1 Let F 1 and F 2 be two ....

....density functions f 1 and f 2 . If for all x 0 we have F 1 (x) F 2 (x) then for all j = 1; 2; k Gamma 1 P 1 (N(i) j) P 2 (N(i) j) where P 1 and P 2 represent the prior predictive probabilities induced by the prior distributions F 1 and F 2 for c, respectively. Proof: Following Korwar and Hollander (1973), let D 1 = 1, and for i = 2; 3; k let D i = 0 if p i = p j for some j = 1; 2; i Gamma 1, and 1 otherwise. It follows (Korwar and Hollander 1973) that D 1 ; D k are conditionally independent given c, with P (D i = 1jc) c= c i Gamma 1) With this notation, we have ....

[Article contains additional citation context not shown here]

Korwar, R. M. and Hollander, M. (1973). Contributions to the theory of Dirichlet processes, The Annals of Probability 1: 705--711.

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Korwar, R. M. and Hollander, M. (1973). Contributions to the theory of Dirichlet processes. Ann. Probability 1 705-711.

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Korwar, R. M. and Hollander, M. (1973) Contribution to the theory of Dirichlet processes, Ann. Probab., 1, 705-711.

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