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Here we consider the problem of Bayesian inference for μ and σ based on these data. μ

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

Below are n = 10 values generated from a normal distribution.

4.90791, 4.83425, 5.19801, 6.85308, 4.07085, 4.66076, 4.16201, 5.49752, 4.15438, 3.72703

Here we consider the problem of Bayesian inference for μ and σ based on these data. μ

(a) There is some R code below that can be used to make a contour plot for the log- likelihood here, log(L, (u,σ)). Run it and obtain a contour plot for the log-likelihood.

(b) A number of (joint) prior densities for (u,σ) are specified below. Since л(μ,σ) L(μ,σ)л(μ,σ), it follows the contour plot for log posterior density can be made by contour plotting log(7, (μ,0)) ∞ log(L, (μ,0))+log(л(μ,σ)) For each of the priors pdf given below, make contour plots for both log(,,(,)) and log(L, (u,σ))+log(л(μ,σ))

You can modify the R code below as you see fit. You will have to change the "levels" in order to get informative plots. You may type ?contour() in R to learn more about the contour function.

How would one try to "see" from the contour plots for the log priors that priors i-iii are ones of "independence" and the last two are not?

the product of Jeffreys improper priors, (u,o)=1/σ2. a priori ~

N(0,100)

and

~Inv-Gamma(1/2, 50)

independent.

o-Inv-Gamma(3/2, 2)

independent

a priori ~N(0,4) and

i.

ii.

111.

iv.

V.

vi.

Same as (v) but with 4 =0, K6 = 10, v1 = 1, and σ =2

the jointly conjugate prior discussed in Section 3.3 of the text, with Ho=0, K1 =1, V = 1, and σ = 2

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