Questions of Show that the Jeffrey’s prior

Question 1
1. Show that the Jeffrey’s prior for λ is
π1 ( λ ) ∝ 1
λ , λ >0
By definition the Jeffrey’s prior P(θ) is given by the formula:
P ( λ ) =
( Eλ [ −∂2 ln ( f ( yi / λ ) )
∂ λ2 ] ) 1
2
Given, f ( yi /λ )= λ e−λ yi
, yi ≥ 0
ln ( f ( yi / λ ) ) =ln ( λ e−λx )
ln ( f ( yi / λ ) ) =ln ( λ ) −λ yi
Then,
∂ ln ( f ( yi / λ ) )
∂ λ = 1
λ − yi
Implying,
∂2 ln ( f ( yi / λ ) )
∂ λ2 =−1
λ2
And
P ( λ ) =( 1
λ2 ) 1
2 = 1
λ
Therefore, P ( λ ) =π1 ( λ ) ∝ 1
λ
2. The prior density π2 ( λ ) ∝ 1
1+ ( λ−5 ) 2 , suggest that the average waiting time follows a
Cauchy’s distribution defined as f ( yi ; 5 , 1 ). That is a Cauchy distribution with location
parameter = 5 and scale parameter = 1. The prior is a density function which integrates to

1 hence the experts prior π2 ( λ ) ∝ 1
1+ ( λ−5 )2 is a proper prior by definition of density
function.
3. The formula for calculation of Bayes factor for non-informative priors is given as
Bayes Factor= Posterior odd
Prior odds
The posterior odd is the posterior probability of M 1 given the data, divided by the
posterior probability of M 2 given the data. The prior odd is the prior probability of M 1
divided by the prior probability of M 2.
In this case:
Posterior odd= f ( λ / yi , M 1 )
f ( λ / yi , M 2 )
But,
f ( λ/ yi , M 1 ) ∝ P ( yi / λ ) π1 ( λ )
Also,
P ( yi / λ ) =∏i=1
54
{ λ e−λ yi }=λ54 e− λ∑
i=1
54
yi
f ( λ/ yi , M 1 ) ∝ λ54 e−λ∑
i=1
54
yi
( 1
λ )
f ( λ/ yi , M 1 ) ∝ λ53 e− λ ∑
i=1
54
yi
Similarly,
f ( λ/ yi , M 2 ) ∝ P ( yi / λ ) π2 ( λ )
f ( λ/ yi , M 2 ) ∝ λ54 e−λ∑
i=1
54
yi
( 1
1+ ( λ−5 ) 2 )

f ( λ/ yi , M 2 ) ∝ λ54 e− λ∑
i=1
54
yi
1+ ( λ−5 ) 2
Thus,
Posterior odd= λ53 e−λ∑
i=1
54
yi
λ54 e−λ∑
i=1
54
yi
1+ ( λ−5 )2
Posterior odd= 1+ ( λ−5 ) 2
λ
Next, prior odd is given as
Prior odd= π1 ( λ )
π2 ( λ )
Prior odd=
1
λ
1
1+ ( λ−5 )2
Prior odd= 1+ ( λ−5 ) 2
λ
Therefore,
Bayes Factor=
1+ ( λ−5 )2
λ
1+ ( λ−5 )
2
λ
=1.
The Bayes factor is 1 thus does not provide any useful information on which model is
better.
4. The posterior distributions are estimated using the grid approximation method in R. The
code used and output for the posterior predictive distributions are as shown below: