This preview shows pages 1–2. Sign up to view the full content.
Markov Chain Monte Carlo
5.1 Comparing Two Poisson Means
Let’s revisit the problem of comparing the means from two independent Pois
son samples. Counts
{
y
Ai
}
from the weekend days are assumed Poisson with
mean
λ
A
and counts
{
y
Bj
}
from the weekday days are assumed Poisson with
mean
λ
B
. We are interested in learning about the ratio of means
γ
=
λ
B
λ
A
.
We showed that the likelihood function in terms of the Frst Poisson mean
θ
=
λ
A
and
γ
is given by
L
(
θ, γ
) = exp(

n
A
θ
)
θ
s
A
exp(

n
B
(
θγ
))(
θγ
)
s
B
.
Assuming that
θ
and
γ
are independent with
θ
∼
Gamma
(
a
0
, b
0
)
, γ
∼
Gamma
(
a
g
, b
g
)
,
Then the posterior density of (
θ, γ
) is given, up to a proportionality constant,
by
g
(
θ, γ

data)
∝
exp(

n
A
θ
)
θ
s
A
exp(

n
B
(
θγ
))(
θγ
)
s
B
×
θ
a
0

1
exp(

b
0
θ
)
γ
a
g

1
exp(

b
g
γ
)
By combining terms, we obtain the expression
g
(
θ, γ

data)
∝
exp (

(
b
0
+
n
A
+
n
B
γ
)
θ
)
θ
a
0
+
s
A
+
s
B

1
×
exp(

b
g
γ
)
γ
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '10
 albert

Click to edit the document details