Bayes rule and Statistical decision making

A: a random event, e.g., a fair coin is flipped and lands on heads B: another random event p(A): the probability of A. A|B: event A follows event B p(A|B): the probability of A, given B. Called the conditional probability, or posterior probability. B is called the prior in this case

Examples of conditional probabilities Probability that a millennial will live to be 80 VS Probability a millennial will live to be 80 given that the millennial smokes a pack of cigarettes each day Probability that someone will get an A in Math 210 VS Probability that someone will get an A in MATH 210 who only comes for the quizzes Probability that someone will have a salary over $100K/yr VS Probability someone will have a salary over $100K/yr given the person has a college degree.

A∧B: joint event “A and B” IF A and B are independent then p(A∧B)=p(A)*p(B) In general, Extreme cases: A and B always coincide. Then p(A|B)=1 and p(A∧B)=p(A)=p(B) If A and B are independent then p(A|B)=p(A) and then p(A∧B)=p(A)*p(B) Generally, B can make A more likely or less.

Examples A: first fair coin lands on heads; B: second fair coin lands on heads P(A)=1/2; p(A|B)=1/2; p(A∧B)=1/4 A: I have at least one brother; B: I have at least one sibling Are the following numbers possible: P(B)=.80; p(A)=.50; p(B|A)=1; p(A|B)=.7 ?

Bayes rule Dividing both sides of p(B|A)*p(A)=p(A|B)*p(B) by p(B) gives Bayes rule: P(A): prior probability P(A|B): posterior probability Probability measures degree of belief. Degree of belief changes as information becomes available

Bayes rule Sometimes the “degree of belief” p(A|B) is sought but what is actually known are the “degrees of belief” p(B|A), p(A) and p(B). Example: In a jury trial, jurors seek to quantify p(G|E), the Degree of belief that the defendant is guilty (G) given the evidence (E). Often prior probabilities are given for G and E from what is known about the general population and the nature of the evidence, and p(E|G) is often easier to estimate. Ideally, p(E|G)=1: if the defendant is guilty then the evidence is certain to have occurred.

Bayes rule Sometimes one substitutes Then Bayes rule is written: This approach emphasizes the role of the alternative (not A)

October is breast cancer awareness month Can Bayes’s theorem help inform decisions on breast cancer awareness?

Current age 10 years 20 years 30 years 30 0.43 1.86 4.13 40 1.45 3.75 6.87 50 2.38 5.60 8.66 60 3.45 6.71 8.65 †Source: Altekruse SF, Kosary CL, Krapcho M, Neyman N, Aminou R, Waldron W, Ruhl J, Howlader N, Tatalovich Z, Cho H, Mariotto A, Eisner MP, Lewis DR, Cronin K, Chen HS, Feuer EJ, Stinchcomb DG, Edwards BK (eds). SEER Cancer Statistics Review, 1975– 2007, National Cancer Institute. Bethesda, MD, based on November 2009 SEER data submission, posted to the SEER Web site, 2010.

The mammogram question In 2009, the U.S. Preventive Services Task Force (USPSTF) — a group of health experts that reviews published research and makes recommendations about preventive health care — issued revised mammogram guidelines. Those guidelines included the following: Screening mammograms should be done every two years beginning at age 50 for women at average risk of breast cancer. Screening mammograms before age 50 should not be done routinely and should be based on a woman's values regarding the risks and benefits of mammography. Doctors should not teach women to do breast self-exams.

The mammogram question (cont) These guidelines differed from those of the American Cancer Society (ACS). ACS mammogram guidelines established in 2003 called for yearly mammogram screening beginning at age 40 for women at average risk of breast cancer. The ACS said the breast self-exam is optional in breast cancer screening. USPSTF acknowledges that women who have screening mammograms die of breast cancer less frequently than do women who don't get mammograms. Recent randomized trials put figures at 15 to 29 percent lower. These figures have to be taken in context. The USPSTF says the benefits of screening mammograms don't outweigh the harms for women ages 40 to 49. Potential harms may include false-positive results that lead to unneeded breast biopsies, anxiety and distress. Update: ACS now recommending age 45 for annual screening and every two years for women ages 55 and older

New ACS recommendations As of Oct 20 2015: ACS suggests women ages 40 to 44 should have the choice to start annual breast cancer screening with mammograms (x-rays of the breast) if they wish to do so. Women age 45 to 54 should get mammograms every year. Women 55 and older may switch to mammograms every 2 years. Some women – because of their family history, a genetic tendency, or certain other factors – should be screened with MRIs along with mammograms.

Bayesian analysis of USPSTF recommendation The rate of incidence of new cancer in women aged 40 is about 1 percent Of existing tumors, about 80 percent show up in mammograms. 9.6% of women who do not have breast cancer will have a false positive mammogram Suppose a woman aged 40 has a positive mammogram. What is the probability that the woman actually has breast cancer?

According to See Casscells, Schoenberger, and Grayboys 1978; Eddy 1982; Gigerenzer and Hoffrage 1995; and many other studies, only about 15% of doctors can compute this probability correctly.

False positives in a medical test False positives: a medical test for a disease may return a positive result indicating that patient displays a marker that correlates with presence of the disease. Bayes' formula: probability that a positive result is a false positive. The majority of positive results for a rare disease may be false positives, even if the test is accurate.

Example A test correctly identifies a patient who has a particular disease 99% of the time, or with probability 0.99 The same test incorrectly identifies a patient who does not have the disease 5% of the time, or with probability 0.05. Is it true that only 5% of positive test results are false? Suppose that only 0.1% of the population has that disease: a randomly selected patient has a 0.001 prior probability of having the disease. A: the condition in which the patient has the disease B: evidence of a positive test result.

The probability that a positive result is a false positive is about 1 − 0.02 = 0.98, or 98%. The vast majority of patients who test positive do not have the disease: The fraction of patients who test positive who do have the disease (0.019) is 19 times the fraction of people who have not yet taken the test who have the disease (0.001). Retesting may help. To reduce false positives, a test should be very accurate in reporting a negative result when the patient does not have the disease.

False negatives: a medical test for a disease may return a negative result indicating that patient does not have a disease even though the patient actually has the disease. Bayes formula for negations: In our example = 0.01 x .001/(.01x.001 + .95x .999)=0.0000105 or about 0.001 percent. When a disease is rare, false negatives will not be a major problem with the test. If 60% of the population had the disease, false negatives would be more prevalent, happening about 1.55 percent of the time

Clicker question On a certain island, 1 pct of the population has a certain disease. A certain test for the disease is successful in detecting the disease, if it is present, 80% of the time. The rate of positive test results in the population is 4%. What is the probability that someone who tests positive actually has the disease? A) 1% B) 2% C 4% D) 8%

Prosecutors fallacy the context in which the accused has been brought to court is falsely assumed to be irrelevant to judging how confident a jury can be in evidence against them with a statistical measure of doubt. This fallacy usually results in assuming that the prior probability that a piece of evidence would implicate a randomly chosen member of the population is equal to the probability that it would implicate the defendant.

Defendant’s fallacy Comes from not grouping the evidence together. In a city of ten million, a one in a million DNA characteristic gives any one person that has it a 1 in 10 chance of being guilty, or a 90% chance of being innocent. Factoring in another piece of incriminating would give much smaller probability of innocence. OJ Simpson

In the courtroom Bayesian inference can be used by an individual juror to see whether the evidence meets his or herpersonal threshold for 'beyond a reasonable doubt. G: the event that the defendant is guilty. E: the event that the defendant's DNA is a match crime scene. P(E | G): probability of observing E if the defendant is guilty. P(G | E): probability of guilt assuming the DNA match (event E). P(G): juror's “personal estimate” of the probability that the defendant is guilty, based on the evidence other than the DNA match.

Bayesian inference: On the basis of other evidence, a juror decides that there is a 30% chance that the defendant is guilty. Forensic testimony suggests that a person chosen at random would have DNA 1 in a million, or 10−6 chance of having a DNA match to the crime scene. E can occur in two ways: the defendant is guilty (with prior probability 0.3) so his DNA is present with probability 1, or he is innocent (with prior probability 0.7) and he is unlucky enough to be one of the 1 in a million matching people. P(G|E)= (0.3x1.0)/(0.3x1.0 + 0.7/1 million) =0.99999766667 The approach can be applied successively to all the pieces of evidence presented in court, with the posterior from one stage becoming the prior for the next. P(G)? for a crime known to have been committed by an adult male living in a town containing 50,000 adult males, the appropriate initial prior probability might be 1/50,000.

O.J. 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?

E = all the evidence, that Nicole Brown was murdered and was previously physically abused by her husband. G = O.J. Simpson is guilty What about ?

Posterior odds = prior odds x Bayes factor In the example above, the juror who has a prior probability of 0.3 for the defendant being guilty would now express that in the form of odds of 3:7 in favour of the defendant being guilty, the Bayes factor is one million, and the resulting posterior odds are 3 million to 7 or about 429,000 to one in favour of guilt. Bayesian assessment of forensic DNA data remains controversial.

Gardner-Medwin : criterion is not the probability of guilt, but rather the probability of the evidence, given that the defendant is innocent (akin to a frequentist p-value). If the posterior probability of guilt is to be computed by Bayes' theorem, the prior probability of guilt must be known. A: The known facts and testimony could have arisen if the defendant is guilty, B: The known facts and testimony could have arisen if the defendant is innocent, C: The defendant is guilty. Gardner-Medwin : the jury should believe both A and not-B in order to convict. A and not-B implies the truth of C, but B and C could both be true. Lindley's paradox. .

Is it relevant that OJ beat his wife? B: Nicole was beaten; M: Nicole was murdered G: OJ is guilty Dershowitz: only 0.1% of men who beat their wives go on to murder their wives In 1994, 5000 women were murdered, 1500 by their husbands. Assuming a US population of 100 million women,

Assuming p(M|~G)=1/30,000, p(G|B)=1/1000, p(~G|B)=999/100, p(M|G)=p(M|B,G)=1 Compute p(G|M,B)

= 0.9678 or about 97%

Clicker: p(G|M,B)= A) 0.5 (50%) B) 0.97 (97 %) C) 1.0 (100%) D) 1/30000 (0.0033…%)