.
shows some data on the number of coal-mining disasters per year between 1851 and 1962, available from the website for this book. These data originally appeared in [434] and were corrected in [349]. The form of the data we consider is given in [91]. Other analyses of these data include [445, 525]. The rate of accidents per year appears to decrease around 1900, so we consider a change-point model for these data. Let
j
= 1 in 1851, and index each year thereafter, so
j
= 112 in 1962. Let
Xj
be the number of accidents in year
j, with
X1, . . . , Xθ
∼ i.i.d. Poisson(λ1) and
Xθ+1, . . . , X112 ∼ i.i.d. Poisson(λ2). Thus the change-point occurs after the
θth year in the series, where
θ
∈ {1, . . . ,
111}. This model has parameters
θ,
λ1, and
λ2. Below are three sets of priors for a Bayesian analysis of this model. In each case, consider sampling from the priors as the first step of applying the SIR algorithm for simulating from the posterior for the model parameters. Of primary interest is inference about
θ.
a.
Assume a discrete uniform prior for
θ
on {1,
2, . . . ,
111}, and priors
λi|ai
∼ Gamma(3, ai) and
ai
∼ Gamma(10,
10) independently for
i
= 1,
2. Using the SIR approach, estimate the posterior mean for
θ, and provide a histogram and a credible
interval for
θ. Provide similar information for estimating
λ1 and
λ2. Make a scatterplot of
λ1 against
λ2 for the initial SIR sample, highlighting the points resampled at the second stage of SIR. Also report your initial and resampling sample sizes, the number of unique points and highest observed frequency in your resample, and a measure of the effective sample size for importance sampling in this case. Discuss your results.
b.
Assume that
λ2 =
αλ1. Use the same discrete uniform prior for
θ
and
λ1|a
∼ Gamma(3, a),
a
∼ Gamma(10,
10), and log
α
∼ Unif(log 1/8,
log 2). Provide the same results listed in part (a), and discuss your results.
c.
Markov chain Monte Carlo approaches (see Chapter 7) are often applied in the analysis of these data.Aset of priors that resembles the improper diffuse priors used in some such analyses is:
θ
having the discrete uniform prior,
λi|ai
∼ Gamma(3, ai), and
ai
∼ Unif(0,
100) independently for
i
= 1,
2. Provide the same result listed in part (a), and discuss your results, including reasons why this analysis is more difficult than the previous two.
6.5.
Prove the following results.
a.
If
h1 and
h2 are functions of
m
random variables
U1, . . . , Um, and if each function is monotone in each argument, then cov{h1(U1, . . . , Um), h2(1 −
U1, . . . ,
1 −
Um)} ≤ 0.
b.
Let
μˆ 1(X) estimate a quantity of interest,
μ, and let
μˆ 2(Y) be constructed from realizations
Y1, . . . ,
Y
n
chosen to be antithetic to
X1, . . . ,
X
n. Assume that both estimators are unbiased for
μ
and are negatively correlated. Find a control variate for
μˆ 1, say
Z, with mean zero, for which the control variate estimator
μˆCV =
μˆ 1(X) +
λZ
corresponds to the antithetic estimator based on
μˆ 1 and
μˆ 2 when the optimal
λ
is used. Include your derivation of the optimal
λ.