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ISyE8843
Brani Vidakovic
Friday 29/10/04
Name:
Quiz 7
WallaceFreeman MML Estimator.
Recall that the Minimum Message Length (MML) estimate, based
on
X
1
,...,X
n
∼
f
(
x

θ
) is deﬁned as
argmin
θ
[

log
π
(
θ
)

log
n
Y
i
=1
f
(
x
i

θ
) +
1
2
log
I
(
θ
)

]
,
where
π
(
θ
) is the prior and
I
(
θ
) is the Fisher information matrix. This is equivalent to maximizing weighted
posterior
π
(
θ
)
Q
n
i
=1
f
(
x
i

θ
)
I
(
θ
)

1
/
2
.
Problem.
Suppose a single observation
X

θ
is coming from the Negative Binomial
NB
(
m,θ
)
,
with p.m.f.
f
(
x

θ
) =
±
m
+
x

1
x
¶
θ
m
(1

θ
)
x
,
and that the prior on
θ
is Beta
B
e
(
α,β
)
.
For example, the observation
X
could be interpreted as the number
of failures incurred until
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Unformatted text preview: m successes are recorded. Find the MML rule. You will need the following facts: • The expectation of X ∼ NB ( m,θ ) , is EX = m (1θ ) θ . • The Fisher information hE ∂ 2 ∂θ log f ( x  θ ) i for θ from the Negative Binomial NB ( m,θ ) distribution is I ( θ ) = m θ 2 (1θ ) . Prove this! • The mode of Beta B e ( a,b ) distribution is a1 a + b2 . Also, X  θ ∼ NB ( m,θ ) , and B e ( α,β ) are conjugate; the posterior is straightforward. 1...
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This note was uploaded on 10/23/2011 for the course ISYE 8843 taught by Professor Vidakovic during the Spring '11 term at Georgia Institute of Technology.
 Spring '11
 VIDAKOVIC

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