±
±
±
±
±
±
±
±
±
±
²
³
±
±
±
±
±
´
±
±
±
±
±
²
±
±
±
±
±
´
±
±
±
±
±
²
±
±
±
±
±
±
±
±
±
±
±
±
µ
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
²
´
¶
µ
´
´
³
±
±
±
±
±
´
±
±
±
±
±
±
±
±
±
±
²
·
±
±
±
±
±
´
±
±
±
±
±
±
±
±
±
±
´
±
±
±
±
±
±
±
±
±
±
´
´
±
±
±
±
±
Lecture
10
Symmetrization.
Pessimistic
VC
inequality.
18.465
We
are
interested
in
bounding
1
P
sup
C
∈C
n
n
´
i
=1
I
(
X
i
∈
C
)
−
P
(
C
)
≥
t
In
Lecture
7
we
hinted
at
Symmetrization
as
a
way
to
deal
with
the
unknown
P
(
C
).
Lemma
10.1.
[Symmetrization]
If
t
≥
2
,
then
n
n
sup
C
∈C
1
n
n
n
i
=1
´
i
=1
1
1
I
(
X
i
∈
C
)
−
P
(
C
)
≤
2
P
I
(
X
i
∈
C
)
−
I
(
X
i
∈
C
)
≥
t/
2
P
sup
C
∈C
≥
t
.
n
n
i
=1
Proof.
Suppose
the
event
sup
C
∈C
n
´
i
=1
1
n
I
(
X
i
∈
C
)
−
P
(
C
)
≥
t
1
n
occurs.
Let
X
= (
X
1
, . . . , X
n
)
∈ {
sup
C
∈C
I
(
X
i
∈
C
)
−
P
(
C
)
Then
≥
t
}
.
I
(
X
i
∈
C
X
)
−
P
(
C
X
)
n
i
=1
n
n
´
i
=1
1
such
that
∃
C
X
≥
t.
For
a
fxed
C
,
P
X
1
n
⎛
⎞
²
2
n
n
i
=1
´
i
=1
1
≥
t
2
/
4
⎝
⎠
I
(
X
i
∈
C
)
−
P
(
C
)
≥
t/
2
I
(
X
i
∈
C
)
−
P
(
C
)
=
P
n
¸
2
1
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 '07
 Panchenko
 Statistics

Click to edit the document details