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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 © POSbase 2004 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 © POSbase 2004 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 © POSbase 2004 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 © POSbase 2004 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) © POSbase 2004 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) © POSbase 2004 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 © POSbase 2004 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. © POSbase 2004 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. © POSbase 2004 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. © POSbase 2004 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 © POSbase 2004