By Dasgupta A. (ed.)
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Thus Θ = Rp and the model for X = (X1 , . . , Xn ) is n P (dx|θ) = f (xi − θ)dxi i=1 on the sample space X = Rpn . With dx as Lebesgue measure on X , the density of P (dx|θ) with respect to dx is n p(x|θ) = f (xi − θ). i=1 Next take ν(dθ) = dθ on Θ = Rp and assume, for simplicity, that m(x) = p(x|θ)dθ Rp is in (0, ∞) for all x. Then a version of “Q(dθ|x)” is Q(dθ|x) = p(x|θ) dθ. m(x) Thus the transition function R is given by R(dθ|η) = X p(x|θ)p(x|η) dx dθ. m(x) Therefore, R(dθ|η) = r(θ|η)dθ where the density r(·|η) is r(θ|η) = X p(x|θ) p(x|η) dx.
2) −∞ then the Markov chain is recurrent and so the posterior distribution in this case is strongly admissible. 1 has a finite mean (see Eaton (1992) for details). When p = 2, a Chung–Fuchs-like argument also applies (see Revuz (1984)). 3) R2 then the Markov chain on R2 is recurrent so strong admissibility obtains. 3) holds. These results for p = 1, 2 are suggested by the work of Stein (1959) and James and Stein (1961). 1) can never be recurrent (see Guivarc’h, Keane, and Roynette (1977)) suggesting that the posterior distribution obtained from the improper prior dθ on Θ = Rp is suspect.
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