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Rao-Blackwellization
Rao-Blackwellization
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Question 1 of 6
120s
foundation (3/10)
state theorem
Which identity directly justifies
Var
(
E
[
X
∣
Y
])
≤
Var
(
X
)
, the inequality at the heart of Rao-Blackwellization?
Hide and think first
A.
Jensen's inequality applied to the convex variance functional, which forces conditioning to monotonically reduce variance for any random variable.
B.
The Cauchy-Schwarz inequality
E
[
X
Y
]
2
≤
E
[
X
2
]
E
[
Y
2
]
, which directly bounds the variance of the conditional expectation by the variance of
X
.
C.
The law of total variance:
Var
(
X
)
=
Var
(
E
[
X
∣
Y
])
+
E
[
Var
(
X
∣
Y
)]
; the second term is nonnegative.
D.
The tower property
E
[
E
[
X
∣
Y
]]
=
E
[
X
]
, which equates the means and so equates the variances of
X
and
E
[
X
∣
Y
]
.
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