Download e-book for iPad: Bayesian Nonparametrics by J.K. Ghosh

By J.K. Ghosh

ISBN-10: 0387955372

ISBN-13: 9780387955377

Bayesian nonparametrics has grown drastically within the final 3 a long time, specifically within the previous few years. This ebook is the 1st systematic remedy of Bayesian nonparametric tools and the idea in the back of them. whereas the booklet is of targeted curiosity to Bayesians, it is going to additionally entice statisticians regularly simply because Bayesian nonparametrics bargains a complete non-stop spectrum of strong choices to basically parametric and simply nonparametric tools of classical records. The booklet is basically geared toward graduate scholars and will be used because the textual content for a graduate path in Bayesian nonparametrics. notwithstanding the emphasis of the publication is on nonparametrics, there's a titanic bankruptcy on asymptotics of classical Bayesian parametric versions.

Jayanta Ghosh has been Director and Jawaharlal Nehru Professor on the Indian Statistical Institute and President of the foreign Statistical Institute. he's at the moment professor of information at Purdue college. He has been editor of Sankhya and served at the editorial forums of a number of journals together with the Annals of facts. except Bayesian research, his pursuits comprise asymptotics, stochastic modeling, excessive dimensional version choice, reliability and survival research and bioinformatics.

R.V. Ramamoorthi is professor on the division of data and likelihood at Michigan nation college. He has released papers within the parts of sufficiency invariance, comparability of experiments, nonparametric survival research and Bayesian research. as well as Bayesian nonparametrics, he's at the moment attracted to Bayesian networks and graphical versions. he's at the editorial board of Sankhya.

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Because 0 π(θ) dθ < 1, there exists such a prior density π, which is also positive on (2, 3). 32 1. PRELIMINARIES AND THE FINITE DIMENSIONAL CASE We will argue that the posterior density fails to be consistent at θ0 by showing that the posterior probability of (2, 3) goes to one in Pθ0 probability. The proof rests on the following facts both of which are easy to verify: Let X(n) denote the maximum of X1 , X2 , . . , Xn . , under U (0, 1), n(X(n) − θ0 ) = OP (1). In fact, n(X(n) − θ0 ) converges to an exponential distribution.

PRELIMINARIES AND THE FINITE DIMENSIONAL CASE A different measure, also due to Shannon, was used by Lindley [128] and Bernardo [18]. Consider two random vectors V, W with joint density p. Then S(p) ≡ S(V, W ) = S(V ) + SV (W ) where SV (W ) = E(I(W |V )) I(W |V ) = −E{log p(W |V )|V } Here SV (W ) is the part of the entropy of W that can be explained by its dependence on V . The residual entropy is S(W ) − SV (W ) = E E log p(W |V ) |V p(W ) ≥0 Because p(w|v) log (p(w|v)/p(w)) µ(dw) ≥ 0 this quantity is taken as a measure of entropy in the construction of reference priors.

Thus, for a reader who is prepared to accept the existence theorems mentioned later, a cursory reading of this chapter would be adequate. On the other hand, for those who are interested in measure-theoretic aspects, a careful reading of this chapter will provide a working familiarity with the measure-theoretic subtleties involved. The last section where formal definitions of consistency are discussed, can be read independently. While we generally consider the case X =R, most of the arguments would go through when X is a complete separable metric space.

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Bayesian Nonparametrics by J.K. Ghosh

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