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24 Bonferroni Inequality

24 Bonferroni Inequality - Chapter 24 out of 37 from...

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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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