Let ₁,...,, iid from N(u, o2) where o² is known. (a) Show that the Jeffreys prior for the normal likelihood is n(u) = C₁√n/0², μER for some constant c₁ > 0. (b) Is this a proper prior or improrer prior? Explain. (c) Derive the posterior density for u under the normal likelihood N(μ, o) and Jeffreys prior for u. Plot the density. (d) Simulate 1,000 draws from the posterior derived in (c) and plot a histogram of the simulated values. (e) Let = exp(μ). Find the posterior density of analytically and plot the density. (f) Estimate by Monte Carlo integration. (g) Compute a 95% equal tail interval for analytically and by simulation.
Let ₁,...,, iid from N(u, o2) where o² is known. (a) Show that the Jeffreys prior for the normal likelihood is n(u) = C₁√n/0², μER for some constant c₁ > 0. (b) Is this a proper prior or improrer prior? Explain. (c) Derive the posterior density for u under the normal likelihood N(μ, o) and Jeffreys prior for u. Plot the density. (d) Simulate 1,000 draws from the posterior derived in (c) and plot a histogram of the simulated values. (e) Let = exp(μ). Find the posterior density of analytically and plot the density. (f) Estimate by Monte Carlo integration. (g) Compute a 95% equal tail interval for analytically and by simulation.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 7CR
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VIEWStep 2: Find the Jeffreys prior for the normal likelihood
VIEWStep 3: Check whether the given prior probability for μ is a proper prior or improper prior and Explain
VIEWStep 4: Derive the posterior density for μ under the normal likelihood N(μ,σ^2) and Jeffreys prior for μ
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