By J.K. Ghosh
This ebook is the 1st systematic therapy of Bayesian nonparametric equipment and the speculation at the back of them. it's going to additionally attract statisticians usually. The booklet 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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Additional info for Bayesian Nonparametrics (Springer Series in Statistics)
Xn goes to 0. For reﬁnements of such results see . 4. Multiparameter extensions follow in a similar way. 5. 5) that n log R 1 1 fθ (Xi )π(θ)dθ = Ln (θˆn ) + log Cn − log n 2 1 1 1 = Ln (θˆn ) − log n + log 2π − log I(θ0 ) + log π(θ0 ) + oP (1) 2 2 2 In the multiparameter case with a p dimensional parameter, this would become n log R 1 p 1 p fθ (Xi )π(θ)dθ = Ln (θˆn )− log n+ log 2π − log ||I(θ0 )||+log π(θ0 )+oP (1) 2 2 2 where ||I(θ0 )|| stands for the determinant of the Fisher information matrix.
Proof. Let G be the class of all functions on Ω that are ﬁnite linear combinations of functions of the form k φ(ω) = fi (ωi ) 1 where f1 , f2 , . . , fk are continuous functions on K. It is easy to see that if φ ∈ G then θ → φ(ω ) dPθ∞ (ω ) is continuous. Further, by the Stone-Weirstrass theorem G is dense in the space of all continuous functions on K ∞ . From the deﬁnition of λΠ1 (·|Xn ) and λΠ2 (·|Xn ), if Ω0 is the set where the posterior converges to δθ0 , then for ω ∈ Ω0 , for φ ∈ G, φ(ω )λΠi (dω |Xn (ω)) = Θ Ω φ(ω ) dPθ∞ (ω ) Πi (dθ|(Xn (ω)) 22 1.
In the last expression the posterior given X1 , X2 , . . , Xn is the same as that given a permutation Xπ(1) , Xπ(2) , . . , Xπ(n) . Said diﬀerently, the posterior depends only on the empirical measure (1/n) n1 δXi , where for any x, δx denotes the measure degenerate at x. This property holds also in the undominated case. A simple suﬃciency argument shows that there is a version of the posterior given X1 , X2 , . . , Xn that is a function of the empirical measure. 1. For each n, let Π(·|Xn ) be a posterior given X1 , X2 , .
Bayesian Nonparametrics (Springer Series in Statistics) by J.K. Ghosh