Description
HW3.1. Marietta Traffic Authority is concerned about the repeated accidents at
the intersection of Canton and Piedmont Roads. Bayes-inclined city-engineer would like to
estimate the accident rate, even better, find a credible set.
A well known model for modeling the number of road accidents in a particular location/time window is the Poisson distribution. Assume that X represents the number of
accidents in a 3 month period at the intersection od Canton and Piedmont Roads.
Assume that [X|θ] ∼ Poi(θ). Nothing is known a priori about θ, so it is reasonable to
assume the Jeffreys’ prior
π(θ) = 1
√
θ
1(0 < θ < ∞).
In the four most recent three-month periods the following realizations for X are observed:
1, 2, 0, and 2.
(a) Compare the Bayes estimator for θ with the MLE (For Poisson, recall, ˆθMLE = X¯).
(b) Compute (numerically) a 95% equitailed credible set.
(c) Compute (numerically) a 95% HPD credible set.
(d) Numerically find the mode of the posterior, that is, MAP estimator of θ.
(e) If you test the hypotheses
H0 : θ ≥ 1 vs H1 : θ < 1,
based on the posterior, which hypothesis will be favored?
HW3.2. Find the Jeffreys’ prior for the parameter α of the Maxwell distribution
p(x|α) = s
2
π
α
3/2x
2
exp(−
1
2
αx2
)
and find a transformation of this parameter in which the corresponding prior is uniform.
HW3.3. Let
yi
|θi ∼ind. Exp(θi),
θi ∼iid Inv − Gamma(α, 1),
for i = 1, . . . , n. Find the empirical Bayes estimator of θi
, i = 1, . . . , n (Note: If x ∼ Exp(θ),
then p(x) = 1/θe−x/θ and if x ∼ Inv−Gamma(α, β), then p(x) = 1/{β
αΓ(α)}x
−α−1
e
−1/(βx)
.).
