Math 408, Actuarial Statistics I A.J. HildebrandGeneral Probability, III: Bayes’ RuleBayes’ Rule1. Partitions: A collection of sets B1, B2, . . . , Bnis said to partition the sample spaceif the sets (i) are mutually disjoint and (ii) have as union the entire sample space. Asimple example of a partition is given by a set B, together with its complement B0.2. Total Probability Rule (Average Rule): If B1, B2, . . . , Bnpartition the samplespace, then for any set A,P (A) = P (A|B1)P (B1) + · · · + P (A|Bn)P (Bn). (1)3. Bayes’ Rule, general case: If B1, B2, . . . , Bnpartition the sample space, then foreach i = 1, 2, . . . , n and any set A,P (Bi|A) =P (A|Bi)P (Bi)P (A|B1)P (B1) + · · · + P (A|Bn)P (Bn)=P (A|Bi)P (Bi)P (A)(2)4. Bayes’ Rule, special case:P (B|A) =P (A|B)P (B)P (A|B)P (B) + P (A|B0)P (B0)=P (A|B)P (B)P (A)(3)(This corresponds to the choice B1= B, B2= B0in the general case of Bayes’ Rule.)Notes and tips• Memorizing Bayes’ Rule: The Total Probability Rule says that the expressionappearing in the denominator in Bayes’ Rule is equal to P (A). If you rememberthis rule, you could get by memorizing the simpler version of Bayes’ Rule given bythe latter formulas (in parentheses) in (2) and (3). However, I recommend memoriz-ing Bayes’ Rule in the first form, since that is the form that you normally need inapplications.• General versus special case of Bayes’ Rule: Many, but not all, applications ofBayes’ Rule involve only the special case when the simpler formula (3) can be used.However, for more general problems, one does need the more complicated formula(2). I recommend to memorize both formulas.• Interpretation of Bayes’ Rule: Bayes’ Rule can be interpreted in terms of priorand posterior probabilities. The prior probabilities are P (Bi), i.e., the (ordinary)probability that the event Bioccurs. Bayes’ Rule shows how these probabilitieschange if we know that event A has occurred; namely it gives a formula for P (Bi|A),the conditional probability that Bioccurs given that A has occurred. The latterprobabilities are called posterior probabilities. (The terms “prior” and “posterior”come from Latin and mean “before” and “after”.)1Math 408, Actuarial Statistics I A.J. Hildebrand• Recognizing Bayes’ Rule problems: Bayes’ Rule is a formula for reversing theorder in conditional probabilities. Many (but not all) conditional probability problemsin the actuarial exams are of this type. If the probability sought in the problem isa conditional probability and the same conditional probability, but with the order ofevents reversed is given (or can easily be deduced from the given information), theproblem is likely a Bayes’ Rule problem. Example: In the drug test problem, theprobability sought is that of someone taking drugs given that he/she tests positive,whereas the reverse conditional probability, that someone tests positive given thathe/she takes drugs, is given.• Recognizing conditional probabilities: Conditional probabilities are often indi-cated by words/phrases like “given that”, or “if”, or by words implying a subpop-ulation. Here are some examples of statements (mostly taken from actuarial examproblems) that refer to a conditional probability, along with their translation intomathematical language. T he “give-away” words that indicate that a conditional prob-ability is involved are set in italics.– “5 percent of those taking drugs test negative.”Translation: “P(test negative | take drugs) = 0.05.”– “For each smoker, the probability of dying during the year is 0.05”Translation: “P(dying | smoker)=0.05”– “A blood test indicates the presence of a disease 95% of the time the disease isactually present.”Translation: “P(test indicates disease | has disease)=0.95”– “Males who have a circulation problem are twice as likely to be smokers as thosewho do not have a circulation problem.”Translation: “P(smoker | circulation problem)=2 P(smoker | no circulation
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