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					Rational Thinking
         It’s Easy to Make Mistakes

• Logical mistakes

• Statistical mistakes
          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.
          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.
         Post Hoc Ergo Propter Hoc
• Whenever ice cream sales increase, so do drowning
  deaths.



• ? Therefore, ice cream causes drowning.
          Post Hoc Ergo Propter Hoc
• The more firemen fighting a fire, the more likely there is
  to be very serious damage.



• ? Therefore, extra fireman shouldn’t be called in.
          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.
          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.
            Post Hoc Ergo Propter Hoc
• Overweight septuagenarians live
  longer.



• ? Therefore, older folks should
  eat more brownies.




http://news.yahoo.com/s/afp/20100128/hl_afp/healthagingusaustralia
          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.
        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. …
         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. ……
          Post Hoc Ergo Propter Hoc

Teen pregnancy boosts girls' risk of getting fat




  http://www.reuters.com/article/healthNews/idUSTRE5384GQ20090409
            Post Hoc Ergo Propter Hoc

  Working after retirement good for your health.




http://news.yahoo.com/s/nm/20091022/hl_nm/us_working_retirement_1
           Post Hoc Ergo Propter Hoc
                The PR Problem

Some bad things will happen to people who have just
gotten swine flu shots.



  http://www.nytimes.com/2009/09/28/health/policy/28vaccine.html?partne
  r=rss&emc=rss
           Correlation vs Causality

Elementary school children who wear expensive shoes
have larger vocabularies.

? Therefore we should invest in good shoes for all children.
            Correlation vs Causality

Elementary school children who wear expensive shoes
have larger vocabularies.




Elementary school children who have fewer cavities
have larger vocabularies.
          Correlation vs Causality

Elementary school children who eat sushi at least
twice a month have larger vocabularies.

? Therefore school cafeterias should serve sushi.
          Correlation vs Causality

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

• ? Toeing out causes you to get older.
          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.
          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.
                 Correlation vs Causality

What makes us happy?



  According to Daniel Gilbert,
  Professor Happiness:



   “Churchgoers are happier than nonchurch goers”




http://www.rd.com/living-healthy/joy-the-readers-digest-
version/article173696-4.html#slide
                Correlation vs Causality

Living Together First Doesn’t Make Marriage Last

   Couples who live together before they get married are less likely to stay
   married, a new study has found. But their chances improve if they were
   already engaged when they began living together.

   The likelihood that a marriage would last for a decade or more decreased by
   six percentage points if the couple had cohabited first, the study found.

   “From the perspective of many young adults, marrying without living together
   first seems quite foolish,” said Prof. Pamela J. Smock, a research professor
   at the Population Studies Center at the University of Michigan, Ann Arbor.
   “Just because some academic studies have shown that living together may
   increase the chance of divorce somewhat, young adults themselves don’t
   believe that.”

http://www.nytimes.com/2010/03/03/us/03marry.html?partner=rss&emc=rss
           Correlation vs Causality




http://www.boingboing.net/2011/03/08/passport-ownership-p.html
                   Correlation vs Causality




http://www.cs.utexas.edu/~ear/nsc110/ScienceAndSociety/Lectures/FastFoodStrokes.doc
                    Correlation vs Causality




“Young men who ate fish more than once a week scored nearly 11 percent
higher on IQ tests than males who rarely ate seafood.”
“A 2007 study of nearly 12,000 pregnant women found that children born to
mothers who ate more than 12 ounces of seafood per week during pregnancy
scored six points higher on tests of verbal IQ than kids born to mothers who had
other foods on the menu.”

http://www.rd.com/living-healthy/fish-made-easy-seafood-healthy-eating-and-food-safety-
/article158044.html
                Correlation vs Causality




http://www.webmd.com/cancer/news/20091105/obesity-linked-to-many-
cancer-cases-in-us
          Correlation vs Causality

Claim: Women who have more sexual partners are
more likely to develop cervical cancer.
                   Correlation vs Causality




 Artificial light and rates of breast and prostate cancer




http://www.rd.com/living-healthy/artificial-light-a-hidden-cancer-
risk/article128447.html
 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/project/enviro/hazard/
       tobacco.htm
      But Sometimes They Get it Right

Study: Spacing babies close may raise autism risk




http://news.yahoo.com/s/ap/20110110/ap_on_he_me/med_autism_birth_spacing
          Sets and Probabilities


These are hard ideas.


Many people have trouble getting them right.
                    Comparable Samples




http://news.yahoo.com/s/ap/20110419/ap_on_re_us/us_ap_poll_grading_the_schools
 "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)

                                                                  Religion
                                       Total                           Agnostic/   Born-Again
                                               Catholic   Protestant
                                                                        Atheist     Christians
                                        %         %          %            %            %
The Old Testament (texts used in the
                                        53       55          74              5         88
Christian religion)
The New Testament (texts used in
                                        52       54          73              6         86
the Christian religion)
The Torah (the texts used in the
                                        23       26          28              5         33
Jewish religion)
The Koran (texts used by Muslims)       8         8           8              4         9
The Book of Mormon (texts used by
                                        6         6           6              3         5
Mormons)
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?
                      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.
                      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.
              Conditional Probability


P(measles | spots) = P(measles  spots)   definition
                         P(spots)
              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)
                  Bayes Rule

P(A | B) = P(B | A)  P(A)
               P(B)
                Using Bayes Rule
P(Texas | Kerry-voter) = P(Kerry-voter | Texas) 


                       =          .38           



P(Mass | Kerry-voter) = P(Kerry-voter | Mass) 


                       =          .63
                    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
                      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
Our goal:                                   .45
 Choose an oval         =           .18 / .45 = .39
                           Argmax

To choose the most likely value x from a set of possibilities,
given some evidence, choose:

          x  arg max( P(a | Evidence))
               aChoices



                         P( Evidence | a)  P(a )     Bayes Rule
          x  arg max (                           )
               aChoices      P( Evidence)




           x  arg max( P( KerryVoter | a)  P(a))    Constant
                aStates
                                                      denominator
                      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
Our goal:
 Choose an oval         =           .18
  Choosing Actions Under Uncertainty



                           decision




   1         2         3                n


payoff1   payoff2   payoff3           payoffn

$50       $100      $200              - $500
  Choosing Actions Under Uncertainty



                                 decision




       .01          .01             .97     .01


   1            2            3                      n


payoff1      payoff2      payoff3                 payoffn

$50          $100         $200                    - $500
  Choosing Actions Under Uncertainty
                                             .01*50
                                             .01*100
                                             .97*200
                                            -.01*500
                                 decision
                                            190.50




       .01          .01             .97        .01


   1            2            3                           n


payoff1      payoff2      payoff3                      payoffn

$50          $100         $200                         - $500
  Choosing Actions Under Uncertainty


The expected value of an action a can be computed as:


                  P(o) V (o)
             oOutcomes a ]
                      [
  Choosing Actions Under Uncertainty


Then we can choose the optimal action opt by computing:


             opt  arg max         P(o) V (o)
                   aChoices oOutcomes[ a ]
         But People Don’t 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?origin=JSTOR-pdf
         But People Don’t 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.
         But People Don’t 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.
         But People Don’t 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.
         But People Don’t 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.
                             Risk

• Choices involving gains are often risk-averse.

      Go for the sure win.




• Choices involving losses are often risk-taking.

      Avoid the sure loss.
                  Prospect Theory

Instead of computing, for each outcome:          P(o) V (o)
We compute:                                ( P(o))  v(V (o))


A typical v:
                  Prospect Theory

Instead of computing, for each outcome:        P(o) V (o)
We compute:                                ( P(o))  v(V (o))



What about  ?
 Estimates of Probabilities of Death From
             Various Causes
Cause                    Subject Estimates   Statistical Estimates

Heart Disease            0.22                0.34

Cancer                   0.18                0.23

Other Natural Causes     0.33                0.35

All Natural Causes       0.73                0.92



Accident                 0.32                0.05

Homicide                 0.10                0.01

Other Unnatural Causes   0.11                0.02

All Unnatural Causes     0.53                0.08
          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/avememo.pdf
                    Plan B - Update

Judge Orders FDA to Reconsider Limits on
Morning-After Pill for Minors




http://www.washingtonpost.com/wp-
dyn/content/article/2009/03/23/AR2009032301275.html?nav=rss_email/co
mponents
          Even Pros Don’t Get it Right
An experiment by Gerd Gigerenzer:
Consider a group of women with low risk of breast cancer:
40 to 50 years old, with no symptoms or family history of
breast cancer.
 • Probability of breast cancer is 0.8 percent.
 • If a woman has breast cancer, probability is 90 percent that she will
   have a positive mammogram.
 • If a woman does not have breast cancer, probability is 7 percent that
   she will have a positive mammogram.

Imagine a woman who has a positive mammogram. What
is the probability that she actually has breast cancer?
           Even Pros Don’t Get it Right
 • Probability of breast cancer is 0.8 percent.
 • If a woman has breast cancer, probability is 90 percent that she will
   have a positive mammogram.
 • If a woman does not have breast cancer, probability is 7 percent that
   she will have a positive mammogram.


Ask doctors what they would tell a patient with a positive
positive mammogram. What is the probability that she
actually has breast cancer?

  First doctor tested: a department chief at a university teaching hospital with
  more than 30 years of professional experience:

  Got nervous, thought for 10 minutes, then said 90%. But also said he knew
  he wasn’t doing it right.
          Even Pros Don’t Get it Right
 • Probability of breast cancer is 0.8 percent.
 • If a woman has breast cancer, probability is 90 percent that she will
   have a positive mammogram.
 • If a woman does not have breast cancer, probability is 7 percent that
   she will have a positive mammogram.


Ask doctors what they would tell a patient with a positive
positive mammogram. What is the probability that she
actually has breast cancer?

  Asked 24 other German doctors: estimates whipsawed from 1 percent to 90
  percent. Eight of them thought the chances were 10 percent or less, 8
  more said 90 percent, and the remaining 8 guessed somewhere between 50
  and 80 percent.
           Even Pros Don’t Get it Right
 • Probability of breast cancer is 0.8 percent.
 • If a woman has breast cancer, probability is 90 percent that she will
   have a positive mammogram.
 • If a woman does not have breast cancer, probability is 7 percent that
   she will have a positive mammogram.


Ask doctors what they would tell a patient with a positive
positive mammogram. What is the probability that she
actually has breast cancer?

   Asked American doctors: 95 out of 100 estimated the woman’s probability of
   having breast cancer to be somewhere around 75 percent.
                     A Card Game

You’re dealt a card.
You’re 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?
        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           Fried Chicken          Okra
 Baked Potatoes      Grape Jelly            Peanut Butter
 Beets               Hamburger              Pizza
 Broccoli            Hot Dogs               Popcorn
 Carrots             Hummus                 Potato Chips
 Chicken Soup        Ice Cream              Strawberry Jam
 Chocolate Cake      Mac and Cheese         Sushi
 Corn                Mashed Potatoes        Tacos
 Eclairs             M & Ms                 Tomatoes
 French Fries        Nachos                 Twinkies

Test scores of the children were then examined.
       Eating Sushi Makes You Smarter
Scientists reported this week that children who eat sushi
score higher on vocabulary tests than children who don’t.
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.
   Eating Grape Jelly Makes You Smarter
Scientists reported this week that children who eat grape jelly
score higher on vocabulary tests than children who don’t.
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.
       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 don’t. 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.
       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 don’t. 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.
Nickels
                 Another Example
Evaluations of two techniques for improving student
performance on achievement tests:

1.   Students who achieve high scores are rewarded with a
     new special class in which they create interactive
     computer games.



2.   Students who achieve low scores are forced to stay
     inside during recess and attend remedial classes.
                   Another Example
Evaluations of two techniques for improving student
performance on achievement tests:

1.   Students who achieve high scores are rewarded with a
     new special class in which they create interactive
     computer games.
           The following year, scores of these students went down
           (on average).

2.   Students who achieve low scores are forced to stay
     inside during recess and attend remedial classes.
           The following year, scores of these students went up
           (on average).
         Regression Toward the Mean
1.   Students who achieve high scores are rewarded with a
     new special class in which they create interactive
     computer games.
           The following year, scores of these students went down
           (on average).

2.   Students who achieve low scores are forced to stay
     inside during recess and attend remedial classes.
           The following year, scores of these students went up
           (on average).



 If we did nothing, we’d expect, on average, for both
 groups of scores to move closer to the mean.
                 Evaluating Bias

Berkeley Graduate School Data, 1973


                       Applied           Admitted
    Males           8,442             3,738 (44%)
    Females         4,321             1,494 (35%)
    Total         12,763              5,232 (41%)


Is there evidence of bias?
       A Simpler Hypothetical Case

                     Applied       Admitted
  Males           150           90 (60%)
  Females         150           60 (40%)
  Total           300          150 (50%)




Is there evidence of bias?
          The Simpson Effect/Paradox

                           Applied     Admitted
Department A   Males     100          80 (80%)
               Females    50          40 (80%)
               Total     150         120 (80%)


                           Applied     Admitted
Department B   Males      50          10 (20%)
               Females   100          20 (20%)
               Total     150          30 (20%)
                The Simpson Effect/Paradox

                1995          1996          1997          Combined
Derek Jeter     12/    .250   183/   .314   190/   .291    385/      .300
                48            582           654           1284

David Justice   104/   .253   45/    .321   163/   .329    312/      .298
                411           140           495           1046
          Problem Solving As Search




   1         2         3              n


choice1   choice2   choice3       choicen
           Optimizing the Outcome




   1         2         3              n


choice1   choice2   choice3         choicen

$250      $100      - $200          $275
              Bounded Rationality
• Optimal behavior (in some sense): Explore all paths and
  choose the best.
              Bounded Rationality
• Optimal behavior (in some sense): Explore all paths and
  choose the best.
               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.
               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"