In the previous post I mentioned the book Radical Uncertainty: Decision Making Beyond the Numbers by Mervyn King and John Kay. The book, written by two prestigious British economists, attacks the bad use of statistics and probability calculus in fields where they are not always applicable, such as history, economics and the law. Let’s look at a few examples:
- What do we mean when we say that Liverpool F.C. has
a 90% chance of winning the next match? One possible interpretation is
that if the match were to be played a hundred times, with the same players
and the same weather conditions and the same referee, Liverpool would win
90 times, and draw or lose the other ten. But the match will be played just
once. Does it make sense to talk about probabilities? No, because there are
no supporting data on frequency. What is meant is that the person speaking believes that Liverpool will win.
Nothing more. It is a subjective probability.
Milton Friedman wrote: We can treat
people as if they assigned numerical probabilities to every conceivable
event. (Price Theory,
1962).
- Does it make sense to speak about the probability
of a plane crashing into a New York skyscraper? Nate Silver, an American statistician,
claimed that the probability is 1/12,500. How did he do it? The authors
explain it thus: In 1945 and 1946, two planes accidentally crashed into
New York skyscrapers. The next crash was on September 11, 2001, a
terrorist attack. In the 25,000 days between the two extreme dates, there
were two crashes. So the probability would be 2/25,000 = 1/12,500. The
authors of the book say in a footnote: This
calculation might be appropriate if incidents such as the Twin Towers
catastrophe were the result of an ergodic process in which a population of
aircraft was randomly flying around Manhattan, occasionally colliding with
tall buildings. But this is not the case. True. And I add: the computation of the
authors of the book is not correct. Between 1945 and 2001 there are not
25,000 days, but less than 21,000.
- The boy or girl paradox: The Smiths have two children. One is a girl.
What is the probability that the other is also a girl? This problem was
posed in 1959 by Martin Gardner and is still polemic. As Wikipedia implies,
everyone has their own solution, and everyone is convinced that theirs is
the right one. The paradox occurs because we do not know where we got the
information from. Suppose the Smiths come to our house and tell us that
they have two children. Just then, one of the children appears, a girl.
What does this tell us about the sex of the second? Nothing. The
probability would be 50%. But suppose the girl does not appear, and the
Smiths tell us that at least one of their children is a girl. What could
the other be? There are four possibilities: boy-boy, boy-girl, girl-boy
and girl-girl. The first is eliminated by the information received.
Therefore, the probability that the other is a girl is 1/3.
- The Linda problem: Linda is 31 years old, single, outspoken and very
bright. She majored in philosophy. As a student, she was deeply concerned
with issues of discrimination and social justice. Which of the following
is more likely? Linda is a bank-teller, or Linda is a bank-teller active
in the feminist movement? (Daniel Kahneman, Thinking, Fast and Slow, 2011). When the question was
posed to the students, 90% of them answered that the second option was
more likely. But A&B cannot be more probable than A. The solution,
which Kahneman did not see, is that the use of the word likely changed the meaning of the
sentence. Kahneman says that students acted irrationally, but they were
not talking about probability, but about likelihood, a different, not a
mathematical concept.
- In 2016, Michael Woodford conducted an experiment at Columbia University to see how students respond to additional information on economics-related problems. The goal of the experiment was to accustom them to Bayesian adjustment of probabilities as new information is obtained, but the result was not satisfactory: the students ignored Bayes’ rules. Curiously, their decisions were not always bad, but the way they got them did not please Woodford, who attributed the failure to imperfect attention, limited memory or cognitive limitations.
The problem with models used in teaching economics is
that they are usually very simplified, small-world
models, while real economic situations are much more complex,
and probabilistic methods of analysis are suitable for the former, rather than the
latter. Some economists (such as Kahneman) seem to argue that if models do not
fit the real world, the fault lies not with the models but with the real world,
or more precisely, with the people described by the model, who are considered
to be biased or irrational because they don’t adapt to the expectations of the
model designers.
Thematic Thread on Statistics: Previous Next
Manuel Alfonseca
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