.
The goal of this problem is to investigate the role of the proposal distribution in a Metropolis–Hastings algorithm designed to simulate from the posterior distribution of a parameter
δ. In part (a), you are asked to simulate data from a distribution with
δ
known. For parts (b)–(d), assume
δ
is unknown with a Unif(0,1) prior distribution for
δ. For parts (b)–(d), provide an appropriate plot and a table summarizing the output of the algorithm. To facilitate comparisons, use the same number of iterations, random seed, starting values, and burn-in period for all implementations of the algorithm.
a.
Simulate 200 realizations from the mixture distribution in Equation (7.6) with
δ
= 0.7. Draw a histogram of these data.
b.
Implement an independence chain MCM procedure to simulate from the posterior distribution of
δ, using your data from part (a).
c.
Implement a random walk chain with
δ
∗ =
δ(t) +
_
with
_
∼Unif(−1,1).
d.
Reparameterize the problem letting
U
= log{δ/(1 −
δ)} and
U
∗ =
u(t) +
_. Implement
a random walk chain in
U-space as in Equation (7.8).
e.
Compare the estimates and convergence behavior of the three algorithms.