compare the performance of the samplers

In
Problem 7.7, 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.

a.
Analyze these data by applying the Gibbs sampler from part (a) of
Problem 7.7. 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) of
Problem 7.7. 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?

Problem
7.7

Consider
a hierarchical nested model

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