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 entice statisticians generally. The e-book is basically geared toward graduate scholars and will be used because the textual content for a graduate direction in Bayesian non-parametrics.
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Extra info for Bayesian Nonparametrics (Springer Series in Statistics)
With hazard rate fθ (x) = 1 − Fθ (x) a if 0 < x < θ b if x > θ Typically a is much bigger than b. This density has been used to model electronic components with initial high hazard rate and cancer relapse times. . ˆ 1 , X2 , . . , Xn ) be the MLE of θ. It can be shown that φn = n−1 is the right Let θ(X norming constant and that the IH conditions hold. But the necessary condition that ξ(u − W ) is nonrandom fails. 2. 3. Ghosal et al. 2 fails. 4. 3 of  imply consistency of the posterior under conditions of IH.
Hence by weak (ω ). consistency at θ0 , for both i = 1, 2 the right-hand side converges to Ω φ(ω ) dPθ∞ 0 This yields the conclusion. Further connections between merging and posterior consistency is explored in Diaconis and Freedman. Note a few technical remarks: According to the deﬁnition, posterior consistency is a property that is speciﬁc to the ﬁxed version Π(·|Xn ). Measure theoretically, the posterior is unique only up to λn null sets. So the posterior is uniquely deﬁned up to Pθ0 if Pθn0 is dominated by λn .
7) 38 1. PRELIMINARIES AND THE FINITE DIMENSIONAL CASE Next consider t2 hn t2 hn t π(θˆn + √ )e 2 +Rn − π(θˆn )e− 2 dt n A2 t2 hn t2 hn t ≤ π(θˆn + √ )e 2 +Rn dt + π(θˆn )e− 2 dt n A2 A2 The second integral is ≤2π(θˆn )e− hn c log 2 ≤ Kπ(θˆn ) √ n √ √ √ [δ n − c log n] n nchn /4 so that by choosing c large, the integral goes to 0√ in Pθ0 probability. because t ∈ A , and c log n < |t| < δ n, we have |t|/ n < 2 ... 2 δ. Thus |Rn | = ( √|t|n )3 16 Ln (θ ) ≤ δ t6 n1 Ln (θ ) ... 8) and then by working with this δ in steps 1 and 3.
Bayesian Nonparametrics (Springer Series in Statistics) by J.K. Ghosh