Consider the setup of the theorem on asymptotic normality of posterior distribution (Theorem 4.2 of the...

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Consider the setup of the theorem on asymptotic normality of posterior distribution (Theorem 4.2 of the book by Ghosh et al. (2006)), proved in the class for a proper prior. (a) Suppose that the prior is improper but there is an no such that the posterior distribution of 0 given x1, . . . , xn0 is proper for a.e. (x1, . . . , xn0 ). Show that the theorem holds also in this case. (b) In addition to the assumptions of Theorem 4.2, assume that the prior density (0) has a finite expectation. Proceeding as in the proof of Theorem 4.2 and using the assumption of finite expectation for π, show that VI(00)-1(00)| dt → 0 • Stela (X₁, ... , Xxn) - with Peo-probability one. √2π Consider the setup of the theorem on asymptotic normality of posterior distribution (Theorem 4.2 of the book by Ghosh et al. (2006)), proved in the class for a proper prior. (a) Suppose that the prior is improper but there is an no such that the posterior distribution of 0 given x1, . . . , xn0 is proper for a.e. (x1, . . . , xn0 ). Show that the theorem holds also in this case. (b) In addition to the assumptions of Theorem 4.2, assume that the prior density (0) has a finite expectation. Proceeding as in the proof of Theorem 4.2 and using the assumption of finite expectation for π, show that VI(00)-1(00)| dt → 0 • Stela (X₁, ... , Xxn) - with Peo-probability one. √2π

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