Download Bayesian Statistics: Exercise Set 5 Answers

Bayesian Statistics: Exercise Set 5
1. Assume y | θ ∼ Exp(θ ) and θ ∼ Gamma(α, β ). A single observation y is made,
and you are informed that y ≥ L but you are not told the value of y. Find the
posterior predictive density p(ỹ | y ≥ L).
2. Let lifetimes yi | θ ∼ Exp(θ ) be conditionally mutually independent given θ .
Consider data that is censored from below: in addition to observations y1 , . . . , yk ,
there are n − k observations that you know are less than `, but for which you
don’t know the value.
(a) Show that the family of gamma distributions is not a conjugate prior family
for this data. (Hint: show that a gamma prior does not give a gamma
posterior.)
(b) Use the approximation e−θ ` ≈ 1 − θ ` (valid for small `) to derive an approximation of the posterior corresponding to the prior θ ∼ Gamma(α, β ).
3. Credibility intervals can be computed using the inverse cumulative distribution
function F −1 (p). For example, (F −1 (0.05), F −1 (0.95)) is an equal-tail 90%
credibility interval.
Suppose you have software to compute the Gamma(α, β ) distribution’s inverse
cumulative distribution function F −1 (p | α, β ). How could you use this software
to compute a 90% credibility interval for the InvGam(α, β ) distribution? (Hint:
if τ ∼ Gamma(α, β ) and φ = τ1 then φ ∼ InvGam(α, β ).)
4. Suppose the heights of army conscripts (in a European country long ago) are normally distributed with mean 164 and unknown variance with prior φ ∼ InvGam(38, 1110).
Find and plot the posterior distribution corresponding to the data
161 164 181 174 163 162 164 174 160 168 172 156 172 161 174 164
5. Solve exercise 4 using WinBUGS and DoodleBUGS. You can use the precision
parameter τ as the simulation variable, and then look at the statistics of φ = 1/τ:
model
for
tau
phi
}
{
(i in 1:n) { y[i] ~ dnorm(164,tau) }
~ dgamma(38,1110)
<- 1/tau
# likelihood
# prior
# variance parameter
6. In an opinion survey of 1000 Finns held in October 2000, 490 were in favour of
adopting the euro currency. Find the 95% HDI based on the priors p(θ ) ∝
1
(a) 1 (Bayes-Laplace),
1
(b) θ − 2 (1−θ )− 2 (Jeffreys),
(c) θ −1 (1−θ )−1 (Haldane)
You may use the normal approximation to compute the HDI.
7. The Pareto(θ , m) distribution is used to model incomes; the parameter m represents the minimum income. The density is
−(θ +1)
p(yi | θ ) = θ mθ yi
(yi > m)
Assume that y1 , . . . , yn are conditionally mutually independent given θ .
(a) Show that Jeffreys’s prior for the Pareto distribution is p(θ ) ∝ θ1 .
(b) Find the posterior density and mean corresponding to Jeffreys’s prior. (Hint:
write the Pareto density in the form p(yi | θ ) ∝ θ e−θ (log yi −log m) ).
(c) Find the posterior density and mean corresponding to the prior θ ∼ Gamma(α, β ).
8. Show that if u ∼ Exp(θ ) and y = meu then y ∼ Pareto(θ , m).
Answers
1. α(β +L)α (ỹ+β +L)−(α+1) 2b. Gamma(α +n, β + ∑ki=1 yk ) 4. InvGam(46, 1528)
6a. [0.459, 0.521] 7c. Gamma(α + n, β − n ln m + ∑ni=1 yi )