COHERENCE AND PROBABILITY. Rosangela H. Loschi and Sergio Wechsler. Universidade Federal de Minas Gerais e Universidade de S~ao Paulo ABSTRACT

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1 COHERENCE AND PROBABILITY Rosangela H. Loschi and Sergio Wechsler Universidade Federal de Minas Gerais e Universidade de S~ao Paulo ABSTRACT A process of construction of subjective probability based on bets introduced by Bruno de Finetti on (1931) is reviewed in detail. KEY WORDS: Coherence, Conditional Probability, Probability, Uncertainty. 1. INTRODUCTION This paper is based on some articles written by Bruno de Finetti on construction of subjective probability. The aim is to characterize coherent numerical measures of uncertainty as probabilities. Such measures belong to a rational person called \You" who is exposed to uncertainty. A set of all capable values j for a state of nature will be considered. The term state of nature denotes an unknown condition to You which aects - in a decision problem, for instance - the consequences of the available actions. For simplicity, we will assume nite in this work. 2. NUMERICAL MEASURE OF UNCERTAINTY Under the Bayesian point of view all coherent measures of uncertainty are probability distributions. We will quantify Your uncertainty relative to a state of nature. This quantication - being coherent - will necessarily be a probability distribution over. Such probability distribution is subjective and depends on all the information You possess about. There are several known \construction processes"of subjective probability. By a construction process it is meant that -starting from a description of natural properties of the notion of 1

2 uncertainty- a theorem is obtained stating that numerical measures of such uncertainty follow the rules of Calculus of Probability. More direct construction processes based on properties of a subjective relation of \ordination and comparison" of the j are described in the literature (de Finetti[1931], Fishburn[1996] and others). We will describe herein the particular construction process based on bets introduced by de Finetti in (1931). Even if this construction involves the notion of money, which is to a certain point exogenous to the problem, \... once it has been shown that one can overcome the distrust that is born of the somewhat too concrete and perhaps articial nature of the denition based on bets, the... procedure is preferable, above all for its clarity." (de Finetti[1937], page 61). De Finetti(1931) understands the measure of uncertainty relative to a given value j for as the number p j that makes You become indierent to possess either the monetary amount p j S or a ticket that will be worth S if is revealed to be j (being worth 0 otherwise). The construction process then follows based on the obvious fact that a rational person when placing bets should not do it in a way to lose money certainly (i.e., regardless of the outcome of the game). The unknown state of nature can only be one out of a nite number of values 1, 2,..., n, that is, = f 1 ; 2 ; : : : ; n g. A subset E of is an event. There are r = 2 n distinct events of. It is assumed that for each of the r events E 1 ; E 2 ; : : : ; E r, there exist \lottery" tickets of type i, i = 1; 2; : : : ; r. A type i ticket will \reward" You with a monetary amount S i conditional on the occurrence of the event E i. On the other hand, You will have no \reward", i.e., ticket E i will be worth zero monetary units if the event E i does not occur. The occurrence of an event E means that the value of is revealed to be an element of E. The potential \reward" S i can be positive, negative or zero. The interpretation of these three possibilities is discussed below. A bookmaker invites You to join the following \game": he gives You a box having r tickets, one of each type i, i = 1; 2; : : : ; r. He then asks You to determine the rates p 1 ; p 2 ; : : : ; p r, that make, in Your opinion, the price p i S i to be \payed" for ticket i fair. The rewards S i, i = 1; 2; : : : ; r associated to tickets of type i are only posted by the bookmaker after You announce the values of the rates p 1 ; p 2 ; : : : ; p r. Note that the sign of S i determines whether ticket E i will represent 2

3 -in case E i occurs - an actual prize or a due for You. All that regardless of You having paid an actual (i.e., positive) price or received a compensation for keeping the ticket. This, in turn, will depend on the sign of p i also. This obviously disadvantageous \game" (as such) for You is nevertheless an almost perfect metaphor for the situation of being uncertain. In particular, not knowing in advance whether S i will be positive or negative forces You to elicit the rate p i sincerely: You are forced to announce exactly the smaller (or the largest) value of p i which would make You indierent to possessing either the sure amount p i S i or the potential due (or prize) represented by ticket i. The rates p 1 ; p 2 ; : : : ; p r reveal Your true amount of uncertainty about the events E 1 ; E 2 ; : : : ; E r. We have so far described a scenario which makes strongly intuitive the notion of uncertainty and in which the theorems relating coherence and the laws of probability will be proved. But, what is a coherent person? DEFINITION 1: (Coherence) You are coherent whenever Your rates fp 1, p 2, : : :, p r g are such that, for every set of values S i that may be chosen later by the bookmaker, there will be at least one j making Your total prot non-negative. DEFINITION 2: (Incoherence) You are incoherent whenever not coherent. Denition 1 states, in particular, that a choice of rates yielding a zero prot under a single j (and negative prots with all others) is enough to make You coherent. It is then seen that coherence is a very weak restriction to be made on one's opinion. It is actually the least to be demanded when attempting to dene rationality. Furthermore, the denition backs the notion that incoherent opinions should be dismissed - unless a sure loss is wanted. This notion is called the Dutch Book Argument (see Hill[1991] for the origin of the term). 3. COHERENCE RESULTS In this section we prove that coherence implies the rules of probability for the rates p i. We start with two lemmas. LEMMA 1: If You are coherent, then for every partition = fe 1 ; : : : ; E k g of, p 1 + p 2 + : : : + p k = 1; 3

4 where p i denotes Your measure of uncertainty about the occurrence of E i. Proof: Let us suppose that You choose rates p 1 ; : : : ; p k in such a way that P k i=1 p i 6= 1 and that the bookmaker assigns rewards zero for every ticket not corresponding to an event of. For every i (i = 1; : : : ; k), your prot in case E i occurs is given by G i = S i? kx i=1 p i S i ; where S i is the reward to be possibly paid for a ticket i and is unknown to You. As is a partition of, the occurrence of any of the j, j = 1; : : : ; n, will imply the occurrence of exactly one of the events E i of. And then the system of equations (I), for i = 1; : : : ; k, indeed presents all possible prots of You. The determinant of the system generated by equations (I) is (I) 1? p 1?p 2 : : :?p k?p 1 1? p 2 : : :?p k......?p 1?p 2 : : : 1? p k which equals 1? p 1? p 2? : : :? p k, a number which we are assuming to be dierent from zero. The system of possible prots has therefore a unique well-dened solution and the bookmaker is able to choose convenient values S i, i = 1; 2; : : : ; k, in such a way that Your prots will always be negative. This contradicts the hypothesis of coherence. 2 The next lemma brings nite additivity: LEMMA 2: Let E 1 ; : : : ; E m be mutually exclusive events of and let p 1,: : :,p m be their numerical measures of uncertainty, respectively. If You are coherent, then p [ m i=1 Ei = Proof: For induction, consider rst m = 2. Let us take two distinct partitions of, = fe 1 ; E 2 ; E 1 [ E 2 g and = fe 1 [ E 2 ; E 1 [ E 2 g. For the lottery tickets relative to the events of and You have the rates fp 1 ; p 2 ; pg and fp 12 ; pg, respectively. 4 mx i=1 p i :

5 From lemma 1, we have 8>< >: p 1 + p 2 + p = 1 p 12 + p = 1; yielding p 12 = p 1 + p 2. The induction argument then follows in the usual way. 2 Some authors question the proof above and need more complicated arguments to obtain the result. For French (page 405 of his [1988]) the assignment of a common rate p for ticket E 1 \ E 2 when considered in either partition, or, is \fallacious". He has a weaker denition of coherence which, however, is unacceptable from a definettian viewpoint: \If E has probability p... it is an event with probability p... both when considered in itself, or in the dichotomy E and E, or in any other partition into few, many or an innite number of events... " (de Finetti[1974], pages ). The next theorem relates coherent measures of uncertainty and the laws of probability. THEOREM 1: You are coherent if, and only if, Your measures of uncertainty satisfy the following axioms (1) p f j g 0; 8j = 1; : : : ; n (2) P n j=1 p fj g = 1: (3) p A = P j:j2a p f j g, for every A 2. Proof: (One should note initially that, as is nite, the three axioms above are equivalent to the usual Kolmogorov's Axioms). p f j g (1) Let us assume, aiming a contradiction, that exists at least one j such that, for You, < 0, and consider that the bookmaker has chosen a value S j < 0 for the ticket relative to event A j = f j g and S i = 0 for i 6= j. Your possible total prots will then be 5

6 8>< >: G 1 = (1? p f j g)s j if = j G 2 =?p f j gs j if 6= j. As p f j g is negative, (1? p f j g) and?p f jg are positive and, therefore, G 1 < 0 e G 2 < 0, which contradicts the coherence assumption. (2) As the events f 1 g; f 2 g; : : : ; f n g form a partition of, lemma 1 yields nx j=1 p f j g = 1: (3) Let A be a subset of. A is a nite union of distinct events, i.e., A = [ j: j 2Af j g. Lemma 2 then yields X p A = p f j g j:j 2A Now, for the converse statement, let a ij = I( j 2 E i ), for i = 1; : : : ; r and j = 1; : : : ; n. The system of Your possible prots is given by G j = rx i=1 S i (a ij? p i ); j = 1; : : : ; n. From this system, the following relation is obtained: P n j=1 p fj gg j = P n j=1 p fj g[ P r i=1 S i(a ij? p i )] On the other hand, Axiom 3 yields nx j=1 = P r i=1 S i[ P n j=1 p fj g(a ij? p i )] = P r i=1 S i[ P n j=1 p fj ga ij? P n j=1 p fj gp i ] = (Axiom 2) = P r i=1 S i[ P n j=1 p fjga ij? p i ]: p f j ga ij = nx j=1 p f j gi( j 2 E i ) = p i ; 6

7 so that, for any values S 1 ; : : : ; S r, nx p f jgg j = rx j=1 i=1 S i (p i? p i ) = 0: (II) 1 However, let us assume for a contradiction that there is at least one set of values fs 1 ; : : : ; S r g such that G j < 0, 8j = 1; : : : ; n. It then follows from (II) that at least one of the p f j g's is strictly negative. (Unless p f j g = 0 for every j, but then Axiom 2 is violated). The set of Your numerical measures of uncertainty was therefore seen to be coherent if, and only if, their elements satisfy the three axioms of Kolmogorov. In order to have, however, coherence as the foundation of all of Probability Theory, one still needs to establish the connection between coherence and Kolmogorov's \denition" of conditional probability. Kolmogorov dened the probability of A given B as: 2 P (AjB) = P (A \ B) ; P (B) provided P (B) > 0. It seems more natural to build the notion of conditional probability starting from its intuitive meaning instead of providing a direct denition as did Kolmogorov. In this way, the \Product Law" will follow from the coherence condiction (and will be valid for P (B) = 0 also). In the betting scenario, the notion of conditional probability can be intuitively constructed as it was done for \unconditional" probabilities, by still using lottery tickets. Conditional Probability is again to be understood as a rate for such tickets, which however are now relative to conditional events. Let A and B be events of. De Finetti(1937) denes a conditional event AjB of as an event which assumes three logical values: true, false and void, according to: 1 (One could in foresight see the expression above as Your \expected" prot.) 7

8 AjB = 8>< >: true false void whenever A and B occur whenever A and B occur whenever B occurs. A lottery ticket based on a conditional event AjB \rewards" You whenever AjB is true; it is worth zero monetary units whenever AjB is false; and in the case AjB is void, the gamble is called o and the \price" is returned to You. Coherence in this \new" game means simply that You must choose rates p AjB for the conditional tickets - and p A for the unconditional tickets - avoiding a sure loss. We then have the following result establishing the Product Law: THEOREM 2: If Your numerical measures of uncertainty about (conditional and not) events of are coherent, then, for every pair of events E 1 and E 2 of, p E1 \E2 = p E1 je2 p E2 : Proof: Let us assume that the bookmaker also allows conditional tickets in the box and that Your rates for tickets based on the events E 2 ; E 1 \ E 2 and E 1 je 2, are, respectively, p E2, p E1 \E2 and p E1 je2. Let us suppose, aiming a contradiction, that for You, p E1 \E2 6= p E1jE2 p E2. Consider then that the bookmaker chooses rewards S 2, S 12 e S for tickets E 2 ; E 1 \ E 2 and E 1 je 2, respectively, while for all other tickets he xes a zero reward. The system of all possible prots is then: 8>< >: G 1 = (1? p E1 \E2)S 12 + (1? p E2 )S 2 + (1? p E1 je2 )S G 2 =?p E1 \E2 S 12 + (1? p E2 )S 2? p E1 je2 S G 3 =?p E1 \E2 S 12? p E2 S 2 if E 1 and E 2 occur if E 1 and E 2 occur if E 2 occurs. The determinant of this system, 8

9 1? p E1 \E2 1? p E2 1? p E1 je2?p E1 \E2 1? p E2?p E1 je2?p E1 \E2?p E2 0 = p E1 \E2? p E1jE2 p E2 ; is dierent from zero, by the contradiction hypothesis. The system of possible prots presents therefore a unique and well-dened solution. In other words, You are giving the bookmaker the chance of a sure win. This is in contradiction with the coherence assumption. 2 There is a converse statement: If the Product Law and the axioms of Kolmogorov hold, then You are coherent. We omit the proof, which is similar to the converse of theorem 1. In conclusion, You are coherent if, and only if, Your rates satisfy the axioms of Kolmogorov and the Product Law. Actually, one can be coherent with the unconditional tickets only (by following the Axioms of Kolmogorov) being, however, incoherent with proper conditional tickets AjB, with B 6= (by violating the Product Law). One should note that the relation p A = p Aj ; for every event A, obtains as a corollary of theorem 2. If this relation - which is otherwise intuitive - were dened from the beginning, it would not be necessary to consider two \kinds" of tickets, as it was done. From a direct conjuction of theorems 1 and 2 we obtain Bayes' Theorem, which is a fundamental result for Statistical Inference.(For the application of coherence ideas to inference, see Wechsler and Branco(1995)). If You are coherent, You must use Bayes' Theorem to obtain Your posterior probabilities, since it is the unique updating of probabilities mechanism which yields coherent posterior distributions. 2 models in Inference. This is, in fact, one of the prime reasons for the use of probabilistic 2 We are only considering updating of probabilities after learning information which does not otherwise alter Your history H. See Goldstein(1985). 9

10 REFERENCES de Finetti B. (1931). Sul signicato soggettivo della probabilita. Fundamenta Mathematicae, vol.17, pp de Finetti B. (1937). Foresight: Its Logical Laws, Its Subjective Sources. Translated and Reprinted in Studies in Subjective Probability, Kyburg Jr.,H.E.; Smokler H.E.(editors). Second Edition, Robert E. Krieger Publishing Co. Inc., Huntington, New York (1980). de Finetti B.(1974). Theory of Probability, Vol. 1. John Wiley and Sons Ltd., New York. Fishburn P.C.(1986). The Axioms of Subjective Probability. Statistical Science, Vol. 1, N. 3, pp French, S.(1988). Decision Theory: an Introduction to the Mathematics of Rationality. Ellis Horwood Limited. Goldstein, M.(1985). Temporal Coherence. In Bayesian Statistics 2. Bernardo J.M., de Groot M.H., Lindley D.V., Smith A.F.M.(editors). Elsevier Science, North Holland, pp Hill B.M.(1993). Dutch Books, the Jereys-Savage Theory of Hypothesis Testing and Bayesian Reliability. Reliability and Decision Making. Barlow R.E., Clarotti C.A., Spizzichino F.(editors). Chapman and Hall. Wechsler S., Branco M.(1995). The Exercise of Coherent Statistical Inference. Submitted. IME-USP, Technical Report RT-MAE9524 ACKNOWLEDGEMENTS This work was partially supported by CNPq and FAPESP. 10