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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
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.
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)
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 − 0.002 = 0.998, or 99.8%.
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. If the test
reported a negative result in patients without the disease with
probability 0.999, then
 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.
P(E |G)P(G)
P(E)
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.
Bayesian inference:
P(G | E) =
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
P(E | G)for
= \frac{P(G|E)P(E)}{P(G)}
court, with the posterior from one stage becoming the prior
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)
Clicker: p(G|M,B)=
A) 0.5 (50%)
B) 0.97 (97 %)
C) 1.0 (100%)
D) 1/30000 (0.0033…%)