Chapter 24 out of 37 from
Discrete Mathematics for Neophytes:
Number Theory, Probability, Algorithms, and Other Stuff
by J.
M.
Cargal
1
By far its most useful application is in joint confidence intervals.
The inequality gives
you a confidence interval without assuming independence of the various parameters.
It usually
turns out at around 95% confidence that the confidence region isn’t much smaller than with the
assumption of independence.
P(a a
a )
P(a )
P(a )
P(a )
n
1
1
2
n
1
2
n
K
K
≥
+
+
+
−
+
The Bonferroni Inequality
P(a a
a )
P(a )
P(a )
P(a )
n
1
=10(.99) 9 =.9
1
2
n
1
2
n
K
K
≥
+
+
+
−
+
24
The Bonferroni Inequality
The Bonferroni inequality is a fairly obscure rule of probability that can be quite useful.
1
The proof is by induction.
The first case is n = 1 and is just
.
To just be sure, we
P a
P a
(
)
(
)
1
1
≥
try n = 2:
.
To prove this we note that
. However,
P a a
P a
P a
(
)
(
)
(
)
1
2
1
2
1
≥
+
−
1
1
2
≥
+
P a
a
(
)
the law of addition says:
.
Substituting this last identity
P a
a
P a
P a
P a a
(
)
(
)
(
)
(
)
1
2
1
2
1
2
+
=
+
−
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 Spring '10
 Wang
 Statistics, Probability, Inductive Reasoning, confidence region

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