Download Making Decisions (Mediocrastan vs. Extemistan)

Statistical decision
making
Frequentist statistics
frequency interpretation of probability: any given
experiment can be considered as one of an infinite
sequence of possible repetitions of the same
experiment, each capable of producing statistically
independent results.
the frequentist inference approach to drawing
conclusions from data is effectively to require that the
correct conclusion should be drawn with a given (high)
probability, among this notional set of repetitions.
Sample mean and population
mean
X1, X2 ,…, Xn random events
m= (X1+X2 +…+Xn )/n sample mean
μ= true expected value of X.
The central limit theorem implies that the sample mean
should converge to the true mean.
If n is large then with high probability, the sample
mean is close to the true mean.
How large is large? How close is close?
Central limit theorem
A sum of independent, identically distributed random
variables is approximately normally distributed.
Normal distribution:
Some normal distributions
Probability that variable takes
value between a and b is the
area under the graph
Confidence interval
One would like a relationship between N and the
probability that m- μ is smaller than a given fixed value.
Error: how precise do you need to be versus
Probability of error: what risk are you willing to take
that you are correct?
Confidence interval example
You want to know whether a coin is fair. You flip it 100
times. You observe that it comes up heads 60 times.
Your question: what is the probability that it would
come up heads 60 times (or more) if the coin is a fair
coin?
Plot of probabilities of a given number
of heads out of 100 flips of a fair coin:
100th row of Pascal’s triangle
The odds of 60 or more heads
from 100 coin flips is about 3
percent.
Fair coin example
Example: Suppose that a coin has an unknown
probability r of landing on heads.
Bayesian approach: compute the posterior probability,
assuming a uniform prior distribution…
F(f|H)=[(N+1)!/H!(N-H)!] r^H (1-r)^(N-H).
The best estimate of r is H/N.
The error margin is (H+1)/(N+2).
One needs a course in calculus to understand the
nature of the error!
Confidence intervals
Hypothesis: the expected value of h, the proportion of
trials on which the coin should land on heads in the
long run, will be within a certain error of the sample
average, with high probability.
E: experiment of repeating the coin flip N times
H: the number of heads.
Desired: if E is repeated infinitely often then the
sample mean m will be within Err of the true mean h a
high proportion P of the time.
We are 100P percent confident that the true mean lies
in the interval (H/N-err, H/N+err)
Measures of central tendency
cont.
Coin flips: can compute the binomial distribution
explicitly and the probabilities associated with various
outcomes.
The confidence interval derives from adding the
probabilities of the various outcomes corresponding to
that interval and excluding the remaining probabilities.
The precise statement is a subtle reflection of the
approximability of the Gaussian curve by a binomial
curve. [*** pictures here***]
Bayesian Approaches
Posterior probability
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 include 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 differ from those of the American Cancer
Society (ACS). The ACS mammogram guidelines call for yearly
mammogram screening beginning at age 40 for women at
average risk of breast cancer. Meantime, the ACS says the
breast self-exam is optional in breast cancer screening.
According to the USPSTF, women who have screening
mammograms die of breast cancer less frequently than do
women who don't get mammograms. However, 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 and
accompanying anxiety and distress.
A statistical question
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.
prob(C|P)=(prob(P|C)*prob(C)/prob(P)
=0.8*0.01/0.096=0.08333…
False positives in a medical test
False positives: a medical test for a disease may return a positive
result indicating that patient could have disease even if the patient
does not have 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.
Bayes: p(A|B)= p(B|A) p(A)/p(B) =.99x .0001/.05=.00198
The probability that a positive result is a false positive is about
1 − 0.0198 = 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 formual for negations:

p(A|-B)= p(-B|A)p(A)/(p(-B|A)p(A)+p(-B|-A)p(-A))
 In our example = 0.01 x .001/(.01x.001 + .05x .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
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: P(G | E)= 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 change 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.
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.
In the UK, Bayes' theorem was explained to the jury in the odds
form by a statistician expert witness in the rape case of Regina
versus Denis John Adams.
The Court of Appeal upheld the conviction, but it also gave their
opinion that "To introduce Bayes' Theorem, or any similar method,
into a criminal trial plunges the jury into inappropriate and
unnecessary realms of theory and complexity, deflecting them from
their proper task.”
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.
Other court cases in which probabilistic arguments played some
role: the Howland will forgery trial, the Sally Clark case, and the
Lucia de Berk case.