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Sufficient Statistics and Exponential Families
Sufficient Statistics and Exponential Families
2 questions
Difficulty 6-6
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Intermediate
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conceptual
The data processing inequality states that for a Markov chain
X
→
Y
→
Z
,
I
(
X
;
Z
)
≤
I
(
X
;
Y
)
. A sufficient statistic
T
(
X
)
for parameter
θ
satisfies
I
(
θ
;
T
(
X
))
=
I
(
θ
;
X
)
. How does this follow from the DPI?
Hide and think first
A.
Sufficient statistics are invertible functions, so they automatically carry all of the information in
X
B.
DPI alone only gives one direction, so equality requires a separate factorization-theorem argument
C.
Sufficiency makes both
θ
→
X
→
T
and
θ
→
T
→
X
Markov, and applying DPI in both directions forces equality
D.
The DPI does not apply here because
T
(
X
)
is deterministic and the DPI needs random transitions
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