. In, you were asked to derive Gibbs samplers under two model parameterizations. The goal of this…

. In,
you were asked to derive Gibbs samplers under two model parameterizations. The
goal of this problem is to compare the performance of the samplers. The website
for this book provides a dataset on the moisture content in the manufacture of
pigment paste [58]. Batches of the pigment were produced, and the moisture
content of each batch was tested analytically. Consider data from 15 randomly selected
batches of pigment. For each batch, two independent samples were randomly selected
and each of these samples was measured twice. For the analyses below, let Implement
the two Gibbs samplers described below. To facilitate comparisons between the
samplers, use the same number of iterations, random seed, starting values, and
burn-in period for both implementations.

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a. Analyze
these data by applying the Gibbs sampler from part (a). Implement the sampler
in blocks. For example,α = (α1, . . . , α15) is one block where all
parameters can be updated simultaneously because their conditional
distributions are independent. Update the blocks using a deterministic order
within each cycle. For example, generate μ(0), α(0),
β
(0) in sequence, followed by μ(1), α(1),
β
(1), and so on.

b. Analyze
these data by applying the Gibbs sampler from part (b). Implement the sampler
and update the blocks using a deterministic order within each cycle, updating μ(0),
γ
(0), η(0) in sequence, followed by μ(1),
γ
(1), η(1), and so on.

c. Compare
performance of the two algorithms by constructing the following diagnostics for
each of the above implementations.

i. After
deleting the burn-in iterations, compute the pairwise correlations between all
parameters.

ii. Select several of the parameters in each implementation, and
construct an autocorrelation plot for each parameter.

iii. Compare the effective sample size for several parameters for the two
implementations of the algorithm.

You may also wish to explore other
diagnostics to facilitate these comparisons. For this problem, do you recommend
the standard or the reparameterized model?