Name: -------------------------------------------
Bayesian Statistics, 22S:138
Midterm 2, 2005
PRACTICE PROBLEMS for Midterm 2
We have studied a hierarchical model for counts of the number of times that pupmps in
10 nuclear power plants failed. The attached WinBUGS code and output are for the same
data. The mo del is exactly the same as the one we studied except for the prior on alpha.
Three samplers were run using different sets of initial values. WinBUGS history plots, auto-
correlation plots, and BGR diagnostic plots, and density plots are given for the parameters
α,β,θ1,θ10, and thetamean. (The plots for the remaining θ’s are similar.) is shown. Use
the attached code, plots, and table of node statistics to answer the following questions.
1. Which line(s) of the WinBUGS code specify the first stage of the hierarchical mod el?
(Copy it or them here.)
2. Which line(s) of the WinBUGS code specify the second stage of the hierarchical
model? (Copy it or them here.)
3. Which (if any) of the parameters had almost no autocorrelation in their sampler
output?
4. What are the numeric estimates of the mean and standard deviation of the posterior
marginal distribution of theta[10]?
5. What is the 95% equal-tail credible set for the mean of the distribution from which
the failure rates of individual pu mps are drawn? (Numeric answer)
6. Interpret what this credible set means (one or two sentences).
1
7. A question of interest might be the variability between failure rates in different pumps,
or in other words, the variance of the distribution from which the failure rates of
individual pumps are drawn. Let’s refer to this variance as σ2
θ. Is it possible to
estimate the posterior distribution of σ2
θby monitoring any quantity in the WinBUGS
model as given? (yes/no)
8. If your answer to the previous question was “yes,” explain how you would do it.
If your answer to the previous question was “no,” write the line or lines of WinBUGS
code that you would need to add to the model in order to get samples from the
posterior distribution of σ2
θ.
9. Circle all of the true statements in the following list:
(a) The numbers in the “MC error” column help us assess the accuracy of the esti-
mated posterior means.
(b) In order to use the Gelman-Rubin convergence diagnostic, one must run more
than one chain.
(c) In choosing initial values for an MCMC sampler, one must not look at the current
dataset being analyzed.
(d) In this hierarchical model, the data from pumps numbered 2 to 10 play a role in
estimating theta[1].
(e) High autocorrelation in MCMC sampler output causes the Markov chain to con-
verge slowly to its stationary distribution.
(f) The Gelman Rubin d iagnostic plot for alpha shows failure to converge because
not all of the lines are on top of each other.
10. Derive the full conditional distribution of the parameter alpha based on the model
specification in the attached WinBugs code.
2