Bayesian Notions and False Positives

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Presentation transcript:

Bayesian Notions and False Positives

Is it more likely that he comes from Texas or from Massachusetts? In the 2004, presidential election, of those Texans who voted for either Kerry or Bush, 62% voted for Bush and 38% for Kerry. Of the Massachusetts residents who voted for either Kerry or Bush, 37% voted for Bush and 63% for Kerry. Bill was a Kerry voter. He comes from either Texas or Massachusetts but I know nothing more about him. Is it more likely that he comes from Texas or from Massachusetts?

I need to tell you that: in Texas there were 7.4 million voters for either Kerry or Bush and in Massachusetts there were only 2.9 million such voters.

I need to tell you that in Texas there were 7 I need to tell you that in Texas there were 7.4 million voters for either Kerry or Bush and in Massachusetts there were 2.9 million such voters. Thus, of the Kerry voters from the two states, 61% came from Texas and only 39% came from Massachusetts.

Thus, of the Kerry voters from the two states, 61% came from Texas and only 39% came from Massachusetts. So Bill is more likely a Texan.

Bayes’ Theorem Where: Is the probability of Event B given that Event A has occurred Is the probability of Event A given that Event B has occurred Is the probability of Event B Is the probability of Event A

Bayes’ Theorem for Kerry_voter vs. Texan

False Positives in Medical Tests Suppose that a test for a disease generates the following results: if a tested patient has the disease, the test returns a positive result 99.9% of the time, or with probability 0.999 2. if a tested patient does not have the disease, the test returns a negative result 99.5% of the time, or with probability 0.995. Suppose also that only 0.2% of the population has that disease, so that a randomly selected patient has a 0.002 prior probability of having the disease.

False Positives in Medical Tests Suppose that a test for a disease generates the following results: if a tested patient has the disease, the test returns a positive result 99.9% of the time, or with probability 0.999 2. if a tested patient does not have the disease, the test returns a negative result 99.5% of the time, or with probability 0.995. Suppose also that only 0.2% of the population has that disease, so that a randomly selected patient has a 0.002 prior probability of having the disease. What is the probability of a “false positive”: The patient does not have the disease given that the test was positive?

What is the probability of a “true positive”: Let’s begin with What is the probability of a “true positive”: The patient does have the disease given that the test was positive? +: Patient Tests Positively D: Patient Has Disease Is the probability Patient Has Disease given that Patient Tests Positively Is the probability Patient Tests Positively given that Patient Has Disease Is the probability Patient Has Disease Is the probability Patient Tests Positively

What is the probability of a “true positive”: Let’s begin with What is the probability of a “true positive”: The patient does have the disease given that the test was positive? +: Patient Tests Positively D: Patient Has Disease Is the probability Patient Has Disease given that Patient Tests Positively Is the probability Patient Tests Positively given that Patient Has Disease Is the probability Patient Has Disease Is the probability Patient Tests Positively

What is the probability of a “true positive”: Let’s begin with What is the probability of a “true positive”: The patient does have the disease given that the test was positive? +: Patient Tests Positively D: Patient Has Disease Is the probability Patient Has Disease given that Patient Tests Positively .999 .002 Is the probability Patient Tests Positively

probability that the Patient Tests Positively? What is the probability that the Patient Tests Positively? Is the probability Patient Has Disease given that Patient Tests Positively .999 .002 .006988

What is the probability of a “false positive”: The patient does not have the disease given that the test was positive? +: Patient Tests Positively D: Patient Has Disease .2859 and P(not D|+) is 1-.2859 = .7141 .999 .002 .006988

+ TEST NEGATIVE +

What if the test was more accurate for those who did not have the disease?