Rational Thinking

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Rational Thinking
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Its Easy to Make Mistakes

• Logical mistakes
• Statistical mistakes

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Post Hoc Ergo Propter Hoc

• In research hospitals, a higher percentage of
  patients die than in community hospitals.
• ? Therefore, you should avoid going to a research
  hospital.

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Post Hoc Ergo Propter Hoc

• As the number of pirates has decreased, global
  warming has increased.
• ? Therefore, global warming is caused by a lack
  of pirates.

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Post Hoc Ergo Propter Hoc

• Whenever ice cream sales increase, so do drowning
  deaths.
• ? Therefore, ice cream causes drowning.

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Post Hoc Ergo Propter Hoc

• The more firemen fighting a fire, the more likely
  there is to be very serious damage.
• ? Therefore, extra fireman shouldnt be called
  in.

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Post Hoc Ergo Propter Hoc

• People who slept with their shoes on are very
  likely to wake up with a headache.
• ? Therefore, sleeping with shoes on causes
  headache.

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Post Hoc Ergo Propter Hoc

• Women who took hormone replacement therapy (HRT)
  after menopause have a lower than average
  incidence of coronary artery disease.
• ? Therefore, HRT protects against coronary heart
  disease.

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Post Hoc Ergo Propter Hoc

• Young children who sleep with the light on are
  much more likely to develop myopia in later life.
  
• ? Therefore, sleeping with the light on causes
  myopia.

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Post Hoc Ergo Propter Hoc
From Marijuana Facts Parents Need to Know, 2005
(The National Institute on Drug
Abuse) Question Does using marijuana lead to
other drugs? Answer Long-term studies of high
school students and their patterns of drug use
show that very few young people use other drugs
without first trying marijuana, alcohol, or
tobacco.
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Post Hoc Ergo Propter Hoc
From A. K. Cline Statistical Nonsense to Mislead
Parents. (Never actually to be published)
Question Does using water lead to other
drugs? Answer Long-term studies of high school
students and their patterns of drug use show that
absolutely no young people use other drugs
without first trying water.
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Post Hoc Ergo Propter Hoc
Teen pregnancy boosts girls' risk of getting fat
http//news.yahoo.com/s/nm/20090409/hl_nm/us_teen_
pregnancy_1
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Correlation vs Causality
Elementary school children who wear expensive
shoes have larger vocabularies.
? Therefore we should invest in good shoes for
all children.
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Correlation vs Causality
Elementary school children who wear expensive
shoes have larger vocabularies. Elementary
school children who have fewer cavities have
larger vocabularies.
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Correlation vs Causality
Elementary school children who eat sushi at least
twice a month have larger vocabularies.
? Therefore school cafeterias should serve sushi.
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Correlation vs. Causality
Older women toe out when they walk more than
younger women do.

• ? Toeing out causes you to get older.

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Correlation vs. Causality
Older women toe out when they walk more than
younger women do.

• ? Toeing out causes you to get older.
• ? Getting older causes toeing out.

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Correlation vs. Causality
Older women toe out when they walk more than
younger women do.

• ? Toeing out causes you to get older.
• ? Getting older causes toeing out.
• ? Older women learned to walk when toeing out was
  fashionable. Younger women learned when it was
  not.

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Correlation vs. Causality
http//news.yahoo.com/s/nm/20090219/hl_nm/us_strok
e_fastfood_1
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Science Reporting Often Gets It Wrong
Children who are exposed to second hand smoke at
home get lower scores on tests of reading, math,
and reasoning.
http//www.cincinnatichildrens.org/research/projec
t/enviro/hazard/tobacco.htm
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Sets and Probabilities
These are hard ideas. Many people have trouble
getting them right.
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"To what extent do you believe the following
represents the word of God? Those Who Say "All"
or "Most" is Word of God (Harris Poll, Nov.,
2007)
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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 Bill
comes from Texas or from Massachusetts?
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More Facts

• In Texas there were 7.4 million voters for either
  Kerry or Bush.
• In Massachusetts there were only 2.9 million such
  voters.

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More Facts

• In Texas there were 7.4 million voters for either
  Kerry or Bush.
• In Massachusetts there were only 2.9 million such
  voters.
• Thus, of the Kerry voters from the two states,
  61 came from Texas and only 39 came from
  Massachusetts.

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Conditional Probability
P(measles spots) P(measles ? spots)
definition P(spots)
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Conditional Probability
P(measles spots) P(measles ? spots)
definition P(spots) P(measles ?
spots) P(measles spots) P(spots) P(measles
? spots) P(spots measles) P(measles) P(
measles spots) P(spots measles) ?
P(measles) Bayes Rule P(spots)
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Bayes Rule
P(A B) P(B A) ? P(A)
P(B)
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Using Bayes Rule
P(Texas Kerry-voter) P(Kerry-voter Texas)
? .38 ? P(Mass
Kerry-voter) P(Kerry-voter Mass)
? .63
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Bayes Rule
P(Texas Kerry-voter) P(Kerry-voter Texas) ?
P(Texas) P(Kerry-voter) .38 ?
.72 .45 .27 / .45 .61 P(Mass
Kerry-voter) P(Kerry-voter Mass) ?
P(Mass) P(Kerry-voter) .63 ?
.28 .45 .18 / .45 .39
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Bayes Rule
P(Texas Kerry-voter) P(Kerry-voter Texas) ?
P(Texas) P(Kerry-voter) .38 ?
.72 .45 .27 / .45 .61 P(Mass
Kerry-voter) P(Kerry-voter Mass) ?
P(Mass) P(Kerry-voter) .63 ?
.28 .45 .18 / .45 .39
Our goal Choose an oval
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Argmax
To choose the most likely value x from a set of
possibilities, given some evidence, choose
Bayes Rule
Constant denominator
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Bayes Rule
P(Texas Kerry-voter) P(Kerry-voter Texas) ?
P(Texas) .38 ?
.72 .27 P(Mass
Kerry-voter) P(Kerry-voter Mass) ?
P(Mass) .63 ?
.28 .18
Our goal Choose an oval
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Choosing Actions Under Uncertainty
decision
1
2
3
n
payoff1
payoff2
payoff3
payoffn
50
100
200
- 500
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Choosing Actions Under Uncertainty
decision
.01
.01
.01
.97
1
2
3
n
payoff1
payoff2
payoff3
payoffn
50
100
200
- 500
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Choosing Actions Under Uncertainty
.0150 .01100 .97200 -.01500 190.50
decision
.01
.01
.01
.97
1
2
3
n
payoff1
payoff2
payoff3
payoffn
50
100
200
- 500
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Choosing Actions Under Uncertainty
The expected value of an action a can be computed
as
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Choosing Actions Under Uncertainty
Then we can choose the optimal action opt by
computing
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But People Dont Do It This Way
People ignore or underweight prior probabilities.
"Steve is very shy and withdrawn, invariably
helpful, but with little interest in people, or
in the world of reality. A meek and tidy soul, he
has a need for order and structure, and a passion
for detail."
Is Steve more likely to be a farmer or a
librarian?
From Amos Tversky and Daniel Kahneman, Judgment
Under Uncertainty Judgements and Biases,
Science, New Series, Vol. 185, No. 4157 (Sep. 27,
1974), pp. 1124-1131. http//www.jstor.org/stable/
1738360?originJSTOR-pdf
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But People Dont Do It This Way
How a problem is framed matters.
Problem 1 Imagine that the
U.S. is preparing for the outbreak of an unusual
Asian disease, which is expected to kill 600
people. Two alternative programs to combat the
disease have been proposed. Assume that the exact
scientific estimate of the con- sequences of the
programs are as follows If Program A is
adopted, 200 people will be saved. If Program B
is adopted, there is 1/3 probability that 600
people will be saved, and 2/3 probability that no
people will be saved. Which of the two programs
would you favor?
From Amos Tversky and Daniel Kahneman, The
Framing of Decisions and the Pyschology of
Choice, Science, Vol. 211, No. 4481 (Jan. 30,
1981), pp.453-458.
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But People Dont Do It This Way
How a problem is framed matters.
Problem 1 N 152 Imagine that the U.S. is
preparing for the outbreak of an unusual Asian
disease, which is expected to kill 600 people.
Two alternative programs to combat the disease
have been proposed. Assume that the exact
scientific estimate of the con- sequences of the
programs are as follows If Program A is
adopted, 200 people will be saved. 72 percent
If Program B is adopted, there is 1/3
probability that 600 people will be saved, and
2/3 probability that no people will be saved. 28
percent Which of the two programs would you
favor?
From Amos Tversky and Daniel Kahneman, The
Framing of Decisions and the Pyschology of
Choice, Science, Vol. 211, No. 4481 (Jan. 30,
1981), pp.453-458.
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But People Dont Do It This Way
How a problem is framed matters.
Problem 2 Imagine that the
U.S. is preparing for the outbreak of an unusual
Asian disease, which is expected to kill 600
people. Two alternative programs to combat the
disease have been proposed. Assume that the exact
scientific estimate of the con- sequences of the
programs are as follows If Program C is
adopted, 400 people will die. If Program D is
adopted, there is 1/3 probability that nobody
will die, and 2/3 probability that 600 people
will die. Which of the two programs would you
favor?
From Amos Tversky and Daniel Kahneman, The
Framing of Decisions and the Pyschology of
Choice, Science, Vol. 211, No. 4481 (Jan. 30,
1981), pp.453-458.
43
But People Dont Do It This Way
How a problem is framed matters.
Problem 2 N 155 Imagine that the U.S. is
preparing for the outbreak of an unusual Asian
disease, which is expected to kill 600 people.
Two alternative programs to combat the disease
have been proposed. Assume that the exact
scientific estimate of the con- sequences of the
programs are as follows If Program C is
adopted, 400 people will die. 22 percent If
Program D is adopted, there is 1/3 probability
that nobody will die, and 2/3 probability that
600 people will die. 78 percent Which of the
two programs would you favor?
From Amos Tversky and Daniel Kahneman, The
Framing of Decisions and the Pyschology of
Choice, Science, Vol. 211, No. 4481 (Jan. 30,
1981), pp.453-458.
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Risk

• Choices involving gains are often risk-averse.
• Go for the sure win.
• Choices involving losses are often risk-taking.
• Avoid the sure loss.

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Prospect Theory
Instead of computing, for each outcome We
compute
A typical v
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Estimates of Probabilities of Death From Various
Causes
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Risk and the Default Action
The morning after pill Plan B was not approved
by the FDA because it was claimed (by FDA
administrators) the manufacturer had not proven
it was safe for 16 year olds to buy over the
counter.
http//www.fda.gov/cder/drug/infopage/planB/avemem
o.pdf
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Plan B - Update
Judge Orders FDA to Reconsider Limits on
Morning-After Pill for Minors
http//www.washingtonpost.com/wp-dyn/content/artic
le/2009/03/23/AR2009032301275.html?navrss_email/c
omponents
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A Card Game
Youre dealt a card. Youre told that the other
side is either twice or half the value of the one
you see. You will collect the amount of money
you see at the end of the game. Should you flip
the card?
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Interpreting Statistical Results
Imagine this study Parents were asked to
indicate which of the following foods their
children ate at least twice a month
Apple Pie Baked Potatoes Beets Broccoli Carrots Ch
icken Soup Chocolate Cake Corn Eclairs French
Fries
Fried Chicken Grape Jelly Hamburger Hot
Dogs Hummus Ice Cream Mac and Cheese Mashed
Potatoes M Ms Nachos
Okra Peanut Butter Pizza Popcorn Potato
Chips Strawberry Jam Sushi Tacos Tomatoes Twinkies

Test scores of the children were then examined.
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Eating Sushi Makes You Smarter
Scientists reported this week that children who
eat sushi score higher on vocabulary tests than
children who dont. The results have a
statistical confidence measure of 95. The effect
of other foods was also studied, but
statistically significant results were obtained
only for sushi. For example, peanut butter did
not show this effect. Dieticians at local
schools, after being informed of the results,
said that they will add sushi to their school
lunch program.
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Eating Grape Jelly Makes You Smarter
Scientists reported this week that children who
eat grape jelly score higher on vocabulary tests
than children who dont. The results have a
statistical confidence measure of 95. The effect
of other foods was also studied, but
statistically significant results were obtained
only for grape jelly. In particular, strawberry
jam did not show this effect. Dieticians at
local schools, after being informed of the
results, said that they will add grape jelly to
their school lunch program.
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Follow Up on Eating Grape Jelly
Scientists reported this week that they have run
additional tests to determine whether children
who eat grape jelly score higher on vocabulary
tests than children who dont. This new work
attempted to replicate results reported earlier
this year. The new studies have failed to
confirm the earlier results. There now appears
to be no relationship between eating grape jelly
and achieving higher scores on vocabulary
tests. Dieticians at local schools, after being
informed of the results, said that they will act
quickly to remove grape jelly from their school
lunch program.
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Follow Up on Eating Grape Jelly
Scientists reported this week that they have run
additional tests to determine whether children
who eat grape jelly score higher on vocabulary
tests than children who dont. This new work
attempted to replicate results reported earlier
this year. The new studies have failed to
confirm the earlier results. There now appears
to be no relationship between eating grape jelly
and achieving higher scores on vocabulary
tests. Local leaders are now questioning the role
of science in designing school lunch programs.
The grape jelly idea is now known to have been
just a theory. Parents who are opposed to
presenting the theory of evolution as the only
answer to the question of the origin of humans
are now pointing to the grape jelly experiments
as evidence of the peril of taking theory as fact.
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Nickels
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Another Example
Evaluations of two techniques for improving
student performance on achievement tests

• Students who achieve high scores are rewarded
  with a new special class in which they create
  interactive computer games.
• Students who achieve low scores are forced to
  stay inside during recess and attend remedial
  classes.

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Another Example
Evaluations of two techniques for improving
student performance on achievement tests

• Students who achieve high scores are rewarded
  with a new special class in which they create
  interactive computer games.
• Students who achieve low scores are forced to
  stay inside during recess and attend remedial
  classes.

The following year, scores of these students went
down (on average).
The following year, scores of these students went
up (on average).
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Regression Toward the Mean

• Students who achieve high scores are rewarded
  with a new special class in which they create
  interactive computer games.
• Students who achieve low scores are forced to
  stay inside during recess and attend remedial
  classes.

The following year, scores of these students went
down (on average).
The following year, scores of these students went
up (on average).
If we did nothing, wed expect, on average, for
both groups of scores to move closer to the mean.
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Evaluating Bias
Berkeley Graduate School Data, 1973
Is there evidence of bias?
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A Simpler Hypothetical Case
Is there evidence of bias?
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The Simpson Effect/Paradox
Department A
Department B
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The Simpson Effect/Paradox
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Problem Solving As Search
1
2
3
n
choice1
choice2
choice3
choicen
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Optimizing the Outcome
1
2
3
n
choice1
choice2
choice3
choicen
250
100
- 200
275
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Bounded Rationality

• Optimal behavior (in some sense) Explore all
  paths and choose the best.

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Bounded Rationality

• Optimal behavior (in some sense) Explore all
  paths and choose the best.

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Bounded Rationality

• Optimal behavior (in some sense) Explore all
  paths and choose the best.
• Bounded rationality Stop and choose the first
  path that results in a state whose value is above
  threshold.

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Bounded Rationality

• Optimal behavior (in some sense) Explore all
  paths and choose the best.
• Bounded rationality Stop and choose the first
  path that results in a state whose value is above
  threshold.

The Sveriges Riksbank Prize in Economic Sciences
in Memory of Alfred Nobel 1978, awarded to
Herbert Simon
"for his pioneering research into the
decision-making process within economic
organizations"