6.4 Suppose that the posterior density Ct1y is N(0. I), as in Problem 6.1, and that z ly, 8 - N((3, 0.5).

The goal is to sample the m e t density zly.

(a) %rive the target density h(z 1,~) = Ih(z Iy, fl)h(8l y) db, by direct calculation.
(Hint: the marginalized density is normal).

(b) Use your result in

(a) to obtain M = 5000 drdws from h(z I,v) with the help of a standard normat generator. Estimate the mean and standard deviation of z Iv from this sample of draws.

(c) Use the method of composition to obtain M = 5000 draws fiom h(z(y), drawing 8 from 8ty using the algorithm you developed in problem 6.2 followed by the drawing of L from the distribution z Jy. 6. Find the mean and standard deviation of the sampled s and compare these estimates with those you found in part (b).
6 9 Consider the standard Gaussian model for univariate observations yj:

(a) Derive the full conditional distribution PI y, a’.

(b) Derive the full conditional distribution d I y, p.

(c) Explain how these full conditional distributions could be used to set up a