Let x₁,..., n iid from N(μ, o2) where o2 is known. (a) Show that the Jeffreys prior for the normal likelihood is p(μ) =₁₁√√√n/o², µER for some constant c₁ > 0. (b) Is this a proper prior or improrer prior? Explain. (c) Derive the posterior density for µ under the normal likelihood N(µ, o²) 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 0= 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 x₁,..., n iid from N(μ, o2) where o2 is known. (a) Show that the Jeffreys prior for the normal likelihood is p(μ) =₁₁√√√n/o², µER for some constant c₁ > 0. (b) Is this a proper prior or improrer prior? Explain. (c) Derive the posterior density for µ under the normal likelihood N(µ, o²) 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 0= 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.
MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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please answer part d,e,f,g only using r code showing the code and method
Transcribed Image Text:Let x₁,..., n iid from N(u, o²) where o² is known.
(a) Show that the Jeffreys prior for the normal likelihood is
p(μ) = c₁ √n/o², 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, o²) and Jeffreys
μ
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 0 = 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.
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VIEWStep 3: Find the posterior density of θ analytically and plot the density given θ = exp(μ)
VIEWStep 4: Estimate by Monte Carlo integration
VIEWStep 5: Compute a 95% equal tail interval for θ analytically and by simulation
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