Description
1. Suppose an engineer is 95% confident that the probability of rejecting a product is
going to be .5 ± .2. Use this information to construct a beta prior for θ. (Hint: Use
normal approximation for the beta distribution. Note that if x ∼ Beta(α, β), then
E(x) = α
α+β
and var(x) = αβ
(α+β)
2(α+β+1) . )
2. Find the Jeffreys’ prior for the parameter α of the Maxwell distribution
p(y|α) = r
2
π
α
3/2
y
2
exp(−
1
2
αy2
)
and find a transformation of this parameter in which the corresponding prior is uniform.
3. Jeffreys’ prior for multiparameter models is given by
p(θ) ∝
p
det(I(θ)),
where I(θ) is the Fisher Information matrix whose ijth element is given by
−E(∂
2
log p(y|θ)/∂θi∂θj ). Suppose that for i = 1, · · · , n, yi ∼ pi(yi
|θi) and πi(θi) is
the Jeffreys’ prior for θi
. If the yi
’s are independent, show that the Jeffreys’ prior for
θ = (θ1, · · · , θn)
0
is Qn
i=1 πi(θi).
4. Suppose x ∼ Binomial(n, π) and y ∼ Binomial(n, ρ) are independent. Find the Bayes
rule for estimating π − ρ corresponding to the loss function L(π − ρ, a) = (π − ρ − a)
2
under the priors: π ∼ Beta(1, 3) and ρ ∼ Beta(3, 1).
5. Find the Bayes rule for estimating θ corresponding to the loss function L(θ, a) =
c1(a − θ) if a ≥ θ and L(θ, a) = c2(θ − a) if a ≤ θ.
