bayes theorem
Bayes’ Theorem
You can think of Bayes rule as being a way to calculate the probability of a certain event happening if a piece of new evidence is found. For example, you might be interested in finding out a patient’s probability of having liver disease if they are an alcoholic.A could represent the event “Patient has liver disease.” Past data tells you that 10% of patients entering your clinic have liver disease.
B could represent the event “Patient is an alcoholic.” Five percent of the clinic’s patients are alcoholics.
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Bayes’ Theorem |
Bayes’ theorem tells you that:
P(A|B) = (0.8 ´ 0.1)/0.5 = 0.16
In other words, if the patient is an alcoholic, their chances of having liver disease is 0.16. This is a significant increase from the 10% suggested by past data, but it is still unlikely that any particular patient has liver disease.
Bayes’ Theorem has several forms, but you’ll probably not encounter any of them in an elementary statistics class. The different forms can be used for different purposes. For example, one version of Bayes’ theorem uses what Rudolf Carnap called the “probability ratio“. The probability ratio rule states that any event (like a patient having liver disease) must be multiplied by the factor PR(H,E)=PE(H)/P(H) to get the event’s probability conditional on E. The Odds Ratio Rule is very similar to the probability ratio, but the likelihood ratio divides a test’s true positive rate divided by its false positive rate. the formal definition of the Odds Ratio rule is OR(H,E)=PH,(E)/P~H(E).
