Probability. Logic and Decision Making Unit 1

Transcription

1 Probability Logic and Decision Making Unit 1

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3 Questioning the probability concept In risky situations the decision maker is able to assign probabilities to the states But when we talk about a probability of a state, is it an objective property of the state or a subjective measurement? What is, actually, a probability? How can we interpret this central concept?

4 The Monty Hall Dilemma You are a contestant in a game. You have to choose one of three boxes: two boxes are empty, one contains gold. After having chosen the box, the game host, which knows what s in each box, opens one of the boxes and shows it is empty. Then he offers the possibility to stick with the choice or to switch to the other box. Should you accept the offer to switch? It is irrational not to switch The moral: making rational decisions requires to reason correctly about probabilities (as already seen, it is one of the most important tools in decision theory)

5 Choice Only if you are not persuaded Elimination Switch

6 The probability calculus Axioms (due to Kolmogorov s work): 0 p(a) 1 for every A If A is a logical truth, then p(a) = 1 If A and B are mutually exclusive, then p(a or B) = p(a) + p(b) Some theorems: p(a) + p( A) = 1 If A and B are logically equivalent, then p(a) = p(b) p(a or B) = p(a) + p(b) p(a and B) p(a B) = p( A) + p(b) p( A and B)

7 From the Entry Test Problem 12. Which of the following instances appears most likely? Which appears second most likely? (A) Drawing a red marble from a bag containing 50 percent red marbles and 50 percent white marbles. (B) Drawing a red marble seven times in succession, with replacement (i.e., a selected marble is put back into the bag before the next marble is selected), from a bag containing 90 percent red marbles and 10 percent white marbles. (C) Drawing at least one red marble in seven tries, with replacement, from a bag containing 10 percent red marbles and 90 percent white marbles.

8 From the Entry Test Problem 5. A certain town is served by two hospitals. In the larger hospital, about forty- `ive babies are born each day. In the smaller hospital, about `ifteen babies are born each day. As you know, about 50 percent of all babies are boys. However, the exact percentage of boys born varies from day to day. Sometimes it may be higher than 50 percent, sometimes lower. For a period of one year, each hospital recorded the days on which more than 60 percent of the babies born were boys. Which hospital do you think recorded more such days? (A) The larger hospital (B) The smaller hospital (C) About the same (within 5 percent of each other) smaller larger

9 From the Entry Test Problem 6. You and your spouse have had three children together, all of them girls. Now that you are expecting your fourth child, you wonder whether the odds favor having a boy this time. What is the best estimate of your probability of having another girl? (A) 6.25 percent (1 in 16), the odds of getting four girls in a row is 1 out of 16 (B) 50 percent (1 in 2), there is an equal chance of getting each gender (C) A percentage between these estimates ( percent) But have the ovaries a memory? Or is the sperm able to send back a signal to inform the male sexual organs that he arrived? Each child is an independent event: that s why the correct answer is B.

10 Conditional Probability The probability of A given B, indicated with p(a B) is a conditional probability Given p(b) 0, it is de`ined as The concept of independence is closely related to the concept of conditional probability; indeed, A is independent of B iff p(a)=p(a B) Theorem. If A is independent of B then p(a and B) = p(a) p(b)

11 Bayes theorem Theorem. Given that p(b) 0, it holds that Theorem 2. Given that p(b) 0, it holds that It holds simply because p(a) = p(b) p(a B) + p( B) p(a B)

12 From the Entry Test Problem 4. Lisa is thirty- three and is pregnant for the `irst time. She is worried about birth defects such as Down syndrome. Her doctor tells her that she need not worry too much because there is only a 1 in 1,000 chance that a woman of her age will have a baby with Down syndrome. Nevertheless, Lisa remains anxious about this possibility and decides to obtain a test, known as the Triple Screen, that can detect Down syndrome. The test is moderately accurate: When a baby has Down syndrome, the test delivers a positive result 86 percent of the time. There is, however, a small false positive rate: 5 percent of babies produce a positive result despite not having Down syndrome. Lisa takes the Triple Screen and obtains a positive result for Down syndrome. Given this test result, what are the chances that her baby has Down syndrome? (A) 0 20 percent chance (B) percent chance (C) percent chance (D) percent chance (E) percent chance D = the baby has the Down syndrome ; P = the test is positive p(d)=1/1000=0.001; p(p D)=0.86 ; p(p D)=0.05 p(d P)=?

13 From the Entry Test p(d P)=? p(d)=0.001, p(p D)=0.86, p(p D)=0.05 D = the baby has not the Down syndrome p( D)=999/1000=0.999 Consider 100,000 babies born from a mother of that age. Then 100 have D and, among these, for 86 the test is positive 99,900 have not D and, among these, for 4995 the test is positive Then (A) 0 20 percent chance (B) percent chance (C) percent chance (D) percent chance (E) percent chance

14 The unknown priors The unconditional probability is called also prior and is often dif`icult to know If we do not know the prior probability p(e) we can estimate it and apply the Bayes theorem for calculating p(e H) In connection with the numerical value chosen for the prior, p(e H) may vary enormously. What can we do? Incorrect priors can be washed out by using the posterior probability obtained through Bayes as the new prior (the new posterior will be closer to the right value) p(e) Bayes H 1 p(e) Bayes p(e) p(e H 1 ) H 2 p(e H 2 ) the more times we apply Bayes theorem, the closer we will get to truth (the Bayes thm works as a washing machine!) This mechanism can be also used for solving disagreement, given that the random experiment is repeated many times

15 PHILOSOPHICAL INTERPRETATIONS

16 Classical interpretation It assigns probabilities in the absence of any evidence, holding the probability of an event to be a fraction of the total number of ways in which the event can occur, thus probability = (# favourable cases) / (# possible cases) All the outcomes has to be equally likely. But, does it make sense to assume that all events can be always reduced into a set of equally possible cases? Moreover, how equally possible can be de`ined without making any reference to probability (otherwise there is a circular de`inition)? With dice or fair coins it can get by, but what with other contexts? Many objections, but one merit: it satis`ies Kolmogorov s axioms

17 Frequency interpretation It holds the probability of an event is a ratio between the actual number of times the event has occurred and the total number of observed cases, i.e. probability = (# positive instances) / (# trials) Kolmogorov s axioms hold, but problems are still present, mainly due to Repeated events that shows different outcomes, while nothing has been changed Unique events that can be irrelevant to future estimations One possible solution is given by the limiting frequency, i.e. the proportion of successful outcomes one would get if the same experiment were repeated in`initely many times The problem is that this is an abstraction, and one can never be sure it really existed (maybe because oscillating in a range)

18 Propensity interpretation Much like the frequentist, it holds that the probability is in the world, rather than in our heads. Probability is thought as the tendency of a given type of physical situation to yield an outcome of a certain kind It has been developed (Popper) as an alternative to frequentism, that `inds dif`icult to manage unique events. Instead, the propensity to show a certain outcome remains not problematic Main criticisms the lack of semantic precision: what is exactly a propensity or disposition or tendency? propensities have a temporal direction, probabilities have not (if A has a propensity to give rise to B, then A cannot occur after B

19 Logical and Epistemic interpretation The logical interpretation (Keynes, Carnap) holds that the probability is a logical relation between a hypothesis and the evidence supporting it It retains the classical interpretation s idea that probabilities can be determined a priori, by examining the space of possibilities It can be regarded also as a generalization of deductive logic and its notion of implication to a complete theory of inference equipped with the notion of degree of implication that relates evidence to hypothesis In modern literature, where different interpretations of probability can coexist, this approach is renamed epistemic probability, focusing on relations between evidence and hypothesis Main criticism Any interpretation can be regarded as epistemic

20 Subjective interpretation Probabilities are not intended as part of the real world, but as kind of mental phenomena representing degrees of belief (Savage, Ramsey, De Finetti) When DMs hold different degrees of belief on the same event, not necessarily one of them is wrong The crucial point is to de`ine a measurement process. The main idea is to link probability calculus to objectively observable behavior, such as revealed preferences in choice behavior (and this connects the subjectivist view to the decision making process) Then, the subjective probability is determined by reasoning backwards Main criticism: as few people form their preferences through this process, it makes little sense to presuppose they uses a speci`ic decision rule Response: impose some structural conditions (axioms) on preferences over uncertain options

21 Examples Let s `lip a coin. What would you prefer between A (if it lands heads up you win a car, otherwise nothing) and B (if it lands tails up you win a car, otherwise nothing)? What is the subjective probability that your car, worth $20,000, will be stolen within one year? What is the highest price you are prepared to pay in a gamble in which you get $20,000 if your car is stolen? Or, what is the price that you consider as fair for insuring your car? Would you pay $500? Then your subjective probability is 500/20,000=0.025 A peculiar feature of this approach is that one does not prefer an option because the subjective probability is judged higher; instead, structured preferences over the options logically imply that they can be described as if the choice was governed by a subjective probability function and a utility function can be constructed which respects their ordering

22 Savage s representation theorem This result shows similarities with vnm s theorem They both provide proofs of existence of utility functions when a number of axioms are assumed as guiding conditions on the way we form preferences and differences with vnm s theorem While in vnm s theorem the DM already knows the probabilities which are given and externally granted, in Savage s theorem probabilities are subjective and also measured

23 The Dutch Book Dutch Book: combination of bets that leads to a certain loss. Example Suppose J= Juventus will win the italian championship. Suppose you believe to degree 0.72 that J will occur, and 0.33 that J will not occur. The bookie promise to pay you $1 for each event that takes place. It would be rational to pay $1 0.72=$0.72 to enter the `irst bet, and $1 0.33=$0.33 to enter the second bet. But entering both bets would entail to pay $0.72+$0.72=$1.05 while, no matter what happens, the payoff is $1 as p( J or not(j) ) = 1. We have a sure loss of $0.05.

24 Example A sport bet In sport bets the bookie indicate the amounts of money won with bets of 1 Amounts won in a 1 bet To win 100 you must pay G 1 =100/V 1 ITALY- BRASIL ITALY 2.95 TIE 3.20 BRASIL X 2 V G Betting on all the options (1, X, 2) such amounts ( on 1, on X, 40 on 2 ) which means in total, we would win, no matter what happens, 100. It s a Dutch Book.

25 Another real example Take a simple bet from your favorite bookie s website. Is there a Dutch Book?

26 Dutch Book theorem Theorem. A decision maker s degrees of belief satisfy the probability axioms iff no Dutch Book can be made against her (De Finetti, Ramsey) A Fair price for a bet is an amount such the DM is equally willing to act as a player and as a bookie. For every bet, there is exactly one fair price A betting quotient for a bet is the ratio between the fair price and the amount at stake. Example: if you consider a fair price $100 for insuring a home worth $10,000 than the betting quotient is p=100/10,000=0.01 If S is the stake, the player must pay p S and the situation can be so summarised

27 Remarks The Dutch Book theorem shows that there is an intimate link between rational degrees of beliefs and the probability axioms: it is pragmatically irrational to violate the probability axioms It is commonly thought to provide an alternative route (to Savage s axiomatic approach) for justifying the subjectivist view of probability