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Probability as judgments
We assume that probabilities are not objective facts, but subjective
judgments. Such a view is more flexible than theories based on logic
or on frequencies.
It makes sense, for example, that a doctor predicts for a smoker
with an otherwise healthy lifestyle a somewhat smaller likelihood
that he gets lung cancer than has been observed for the average
smoker. However, we need rules to determine what a good
probability judgment is. For example, it does not make sense to
predict a .02% probability if the observed probability is 20%.
There are some techniques that allow well-justified probability judgment;
they are also the basis for Bayes-Theorem. Let us look at them.
Contributor
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Well-justified Judgments
Well-justified judgments obey minimal preconditions.
Specifically, probability judgments must not
contradict mathematical rules.
• Each proposition is either true or false. The
probability that a proposition is right plus the
probability that a proposition is wrong gives
——> p(true) + p(¬true) = 1
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Well-justified Judgments
• If two propositions A and B exlude each other, then
the probability that A or B are true equals the
probability that A is true plus the probability that B is
true.
——> p(A) + p (B) = p(A or B)
Example: Let‘s assume the probability that Hillary Clinton succeeds
George Bush as president of the US is 0.5. The probability that Dick
Cheney succeeds Bush as president is 0.2. However, not both can
succeed Bush as president.
——> p(Clinton) + p(Cheney) = p(Clinton or Cheney) = 0.7
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Well-justified Judgments
• The conditional probability of proposition A, given that
proposition B is true, equals the probability that A is
true if we know that B is true.
——> p(A | B) is A true given that B is true.
Example: Let‘s assume the probability that Hillary Clinton wants to
succeed George Bush as president is 0.9. Given that Cløinton wants
to succeed Bush, the probability is 0.4 that she will succeed him.
——> p(Clinton becomes | Clinton wants) = 0.4
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Well-justified Judgments
• The multiplication rule says that the probability that A
and B are true can be derived from the multiplication
of the conditional probability of A given that B and
the probability of B.
• ——> p(A & B) = p(A | B) * p(B)
Example: The probability that Hillary Clinton (HC) succeeds George Bush
equals 0.36, if the probability that she wants to succeed him is 0.9 and the
conditional probability that she is elected if she wants to be presiedent is
0.4.
——>
p(HC becomes & wants) =
p(HC becomes | HC wants) * p(HC wants)
Note:
p(HC becomes & wants) = p(HC wants & becomes)
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Well-justified Judgments
• If A and B are independent events, then says a subrule
of the multiplication rule that the probability that A and B
are true can be derived from the multiplication of the
probability of A and the probability of B.
——> p(A & B) = p(A) * p(B)
Example: Let’s assume the probability that Hillary Clinton becomes US
president is 0.8. The probability that she wins a concert ticket in a lottery
is 0.2. The two events are independent of each other. Therefore, the
probability that she becomes both preseident and winner of a concert
ticket equals 0.16.
——> p(president &ticket) = p(president) * p(ticket)
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The Evaluation of Judgments
If the truth is known (e.g., if Clinton has
succeeded Bush), we can evaluate our
judgments.
There three evaluation criteria:
• Coherence: If the judgment does not
contradict the rules discussed before.
• Calibration
• Scoring Rules
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Calibration
A probability judgment is then well calibrated,
if my predictions correspond to the observed
probabilities in the long run.
If for example a meteorologist predicts 75%
of rain for certain days and it rains 75% of the
time at those days, then her judgment is well
calibrated.
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Calibration
A judgment can be coherent without being well
calibrated. It is a perfectly coherent judgment to
predict that 90% of the members of parliament
will live on the moon by 2010 and that 10% will
live on earth. However, this judgment is likely to
turn out badly calibrated. Coherence is blind
towards what really happens.
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Scoring Rules
One problem of calibration is that it does not consider information from the
judgment so that nothing can be said about the error rate.
If a meteorologist predicts rain at a probability of 55% for each day, his
judgment is well calibrated if it rains at 55% of the days in the long run.
However, I‘d like to know when I have to take an umbrella with e and
when I can go out with just a T-shirt.
Therefore, I prefer to rely on another meteorologist who accurately
predicts a 100 percent probability of rain at 55% of the days and at 45% of
the days a zero probability of rain.
The problem is: Both meteorologists are equally well calibrated.
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Scoring Rules
This problem can be solved using scoring rules.
Let us use the quadratic scoring rule (Rain = 1, else 0).
Judgment A
Judgment B
Result
_________________________________________
2
(0.2)2
0.90 (0.1)
0.80
Rain
2
(0.0)2
0.10 (0.1)
0.00
No rain
2
2
(0.4)
0.50 (0.5)
0.40
No rain
2
(0.1)2
0.80 (0.2)
0.90
Rain
_________________________________________
.21
.31
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