Discrete Mathematics for Neophytes: Number Theory ...

1 Chapter 24 out of 37 from Discrete Mathematics for neophytes : Number Theory , Probability, Algorithms, and Other Stuff by J. M. Cargal1By far its most useful application is in joint confidence intervals. The inequality givesyou a confidence interval without assuming independence of the various parameters. It usuallyturns out at around 95% confidence that the confidence region isn t much smaller than with theassumption of (a aa )P(a ) P(a )P(a ) n 112n12nKK +++ +The Bonferroni InequalityP(a aa )P(a ) P(a )P(a ) n 1= 10(.99) - 9 =.912n12nKK +++ +24 The Bonferroni InequalityThe Bonferroni inequality is a fairly obscure rule of probability that can be quite proof is by induction.

2 The first case is n = 1 and is just . To just be sure, wePaPa() ()11 try n = 2: . To prove this we note that . However,PaaPaPa()()()12121 + 112 +Pa a()the law of addition says: . Substituting this last identityPa aPaPaPaa()()()()121212+=+ in the previous one, we move the 1 to the right of the inequality and we move the to the leftPaa()12of the inequality and this gives us the desired result. Lastly, we have to do the inductive step. Weassume that the proposition is true for n (as in the box) and we then show that it necessarily followsfor the case n + 1. Now we refer to the case n = 2 that we have just proven. We get:. Now if we substitute forPaaa aaPaaa aPannnn()()()12 3112 311KK++ + using the induction hypothesis (the box) we get:Paaa an()12 3K which is what had to be shown.

3 Paa aPaPaPannn()()()()121121KK++ +++ ExampleSuppose that we have ten events ai, with P(ai) = .99. We want to estimate the jointprobability P( ). If the ai are independent events then we have:. HoweverPaa aPa PaPa()()()()..1 21012101099904382075009KK===we have no grounds to assume independence. If we use the Bonferroni inequality get:Chapter 24 out of 37 from Discrete Mathematics for neophytes : Number Theory , Probability, Algorithms, and Other Stuff by J. M. Cargal2 Note how close the two numbers Exercise 1 Repeat the above example with P(ai) = .1 for each 24 out of 37 from Discrete Mathematics for neophytes : Number Theory , Probability, Algorithms, and Other Stuff by J.

4 M. the events were independent the joint probability would be (.1)10 = 10-10. Bonferroni sinequality says that the joint probability is greater or equal to 10(.1)-9 = -8. This ofcourse is useless. The Bonferroni inequality is useful as the probabilities of the eventsget larger.