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Bayes' formula is an important method for computing conditional probabilities. It is often used to compute posterior probabilities (as opposed to priorior probabilities) given observations. For example, a patient is observed to have a certain symptom, and Bayes' formula can be used to compute the probability that a diagnosis is correct, given that observation. We illustrate this idea with details in the following example:
Example: Mammogram posterior probabilities
Approximately 1% of women aged 40-50 have breast cancer. A woman with breast cancer has a 90% chance of a positive test from a mammogram, while a woman without has a 10% chance of a false positive result. What is the probability a woman has breast cancer given that she just had a positive test?
Translate into the language of probability, let B = "the woman has breast cancer" and A = "a positive test". We wish to calculate P(B|A). Similar to what we did last time, we have
To compute the probabilities on the right side, we use the multiplication rule.
This answer is somewhat surprising. Indeed, when ninety-five physicians were asked this question their average answer was 75%. The two statisticians who carried out this survey indicated that physicians were better able to see the answer when the data was presented in frequency format. 10 out of 1000 women have breast canner. Of these 9 will have a positive mammogram. However, of the remaining 990 women without breast cancer 99 will have a positive test, and again we arrive at the answer 9/(9 + 99)=9/108.
We now state the Bayes' Formula:
First, we have a partion B_1,B_2,...,B_n of a probability space, namely, their disjoint union is the total space.
Reasoning as in the Mammogram test example, we have
To evaluate the probability, observe that by the multiplication rule,
From this, we have the important Bayes' Formula:
Understanding Bayes' formula can greatly enhance your ability to examine chance problems in real life. For example, doctors should know more about Bayes' formula to obtain an estimation of how reliable is a certain test. Also you'll be able to tell certain fallacies and point out how it really works.
Discuss the O.J.Simpson Trial:
Bayes' formula specifies how probability must be updated in the light of new information. The essence of Bayesion reasoning is best understood by considering evaluation of probabilities for the situation where there is question of a hypothesis being either true or false. An example of such a situation is a court case where the defendant is either guilty or not guilty.
A good illustration of application of Bayes' formula in court cases is provided by the legal argument of the famous trial of O.J.Simpson.
Nicole Brown was murdered at her home in Los Angeles on the night of June 12,1994. The Prime suspect was her husband 0.J.Simpson, at the time a well-known celebrity famous both as a TV actor and as a retired professional football star. This murder led to one of the most heavily publicized murder trial in U.S. during the last century. The fact that the murder suspect had previously physically abused his wife played an important role in the trial. The famous defense lawyer Alan Dershowitz, a member of the team of lawyers defending the accused, tried to belittle the relevence of the fact by stating that only 0.1% of the men who physically abuse their wives actually end up murdering them.
Question: Was the fact that O.J.Simpson had previously physically abused his wife irrelevant to the case?
The answer to the question is no. In this particular case it is important to make use of the crucial fact that Nicole Brown was murdered. It is wrong to estimate the probability uniformly on possible outcomes. The question, therefore, is not what the probability is that abuse leads to murder, but the probability that the husband is guilty in light of the fact that he had previously abused his wife.
We define the following events:
- E = all the evidence, that Nicole Brown was murdered and was previously physically abused by her husband.
- G = O.J.Simpson is guilty
We use the Bayes' formula to compute the conditional probability
Which of course involves P(G),clearly unknown. Therefore, someone argus for the use of likelihood ratio P(E|G)/P(E|Gc) , which in this case (you can calculate it by yourself using the Bayes' formula) is about 4.08. In other words, there is an estimated probability of 81% that the husand is the murderer of his wife in the light of the knowledge that he had previously physically abused her. That is certainly a relevant factor.