Download Bayesian Nonparametrics (Springer Series in Statistics) by J.K. Ghosh PDF

By J.K. Ghosh

This ebook is the 1st systematic therapy of Bayesian nonparametric equipment and the speculation in the back of them. it's going to additionally attract statisticians ordinarily. The e-book is essentially geared toward graduate scholars and will be used because the textual content for a graduate path in Bayesian non-parametrics.

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Extra resources for Bayesian Nonparametrics (Springer Series in Statistics)

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This result for the compact case can now be used to establish consistency of θˆn . 6. Suppose Θ is a convex open subset of Rp and for θ ∈ Θ, p log fθ (xi ) = A(θ) + θj xi + ψ(xi ) 1 and ∂ log fθ ∂θ , ∂ 2 log fθ ∂θ2 I(θ) = Eθ exist. Then by Lehman[123] ∂ log fθ ∂θ 2 = −Eθ ∂ 2 log fθ ∂θ2 = d2 A(θ) >0 dθ2 Thus the likelihood is log concave. 4. Start with a bounded open rectangle around θ0 and let K be its closure. Because K is compact, the MLE θˆK , with K as the parameter space exists and given any open neighborhood V ⊂ K of θ0 , θˆK lies in V with probability tending to 1.

Xn that is a function of the empirical measure. 1. For each n, let Π(·|Xn ) be a posterior given X1 , X2 , . . , Xn . 1. When Θ is a metric space {θ : ρ(θ, θ0 ) < 1/n : n ≥ 1} forms a base for the neighborhoods of θ0 , and hence one can allow the set of measure 1 to depend on U . e. Pθ0 . Thus the posterior is consistent at θ0 , if with Pθ0 probability 1, as n gets large, the posterior concentrates around θ0 . Why should one require consistency at a particular θ0 ? A Bayesian may think of θ0 as a plausible value and question what would happen if θ0 were indeed the true value and the sample size n increases.

Let pi (θ2 |θ1 ) be a given conditional prior. Our first object is to maximize the entropy in θ1 and find the marginal p(θ1 ). 14) 50 1. 13) of S(X, p(θ1 , θ2 ), S(X, pi (θ1 )) = n d1 log + 2 2πe pi1 (θ1 ) log Ki1 ψi (θ1 ) pi (θ1 ) where det I(θ) det I22 (θ) pi (θ2 |θ1 ) log ψi (θ1 ) = exp Ki1 dθ1 + o(1) 1/2 dθ2 Maximizing S(X, pi (θ1 )) asymptotically, pi (θ1 ) = const ψi (θ1 ) on Ki1 where the constant is for normalization. Then constant ψi (θ1 )pi (θ2 |θ1 ) on Ki1 × Ki2 pi (θ1 , θ2 ) = 0 elsewhere Finally take p(θ2 |θ1 ) = ci (θ1 ) {det I22 (θ)}1/2 0 on Ki2 otherwise To choose a limit in some sense, fix θ0 = (θ10 , θ20 ) and assume lim pi (θ1 , θ2 )/pi (θ10 , θ20 ) = p(θ1 , θ2 ) exists for all θ ∈ Θ.

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