Non-Archimedean Probability and Conditional Probability; ManyVal2013 Prague F.Montagna, University of Siena

Transcription

1 Non-Archimedean Probability and Conditional Probability; ManyVal2013 Prague 2013 F.Montagna, University of Siena

2 1. De Finetti s approach to probability. De Finetti s definition of probability is in terms of bets: the probability of an event φ is the amount of money (betting odd in the sequel) α that a fair bookmaker B would accept for the following game: A gambler G chooses a real number λ and pays αλ to B, and receives λv(φ) where v(φ) = 1 if φ is true and v(φ) = 0 if φ is false. Note that λ may be negative; paying β < 0 is the same as receiving β; betting a negative number corresponds to reversing the roles of gambler and bookmaker.

3 2. Coherence. The only rationality criterion proposed by de Finetti is the following: suppose B accepts bets on the events φ 1,..., φ n with betting odds α 1,..., α n, respectively. Then the assessment proposed by B is coherent if there is no system of bets which causes to B a sure loss, independently of the truth values of φ 1,..., φ n. That is, there are no λ 1,..., λ n such that for every homomorphism v of the algebra of events into into 2, ni=1 λ i (α i v(φ i )) < 0.

4 Example. Suppose that Brazil and Spain are going to play and that the betting odd is: B: Brazil wins 1 2 ; S: Spain wins 1 2 ; D: Draw 1 2. Then G may cause to B a sure loss by betting -1 euro on each of B,S and D. This would cause a sure loss of 0,50 euro to the bookmaker. Hence, the assessment is not coherent.

5 Theorem [df] (de Finetti) An assessment φ 1 α 1,..., φ n α n is coherent iff it can be extended to a probability distribution, that is, there is a probability distribution Pr such that, for i = 1,..., n, Pr(φ i ) = α i. Hence, the Kolmogoroff laws of probability model coherent assessments. There is an interesting analogy with the completeness theorem: coherence is equivalent to (probabilistic) satisfiability.

6 5. Coherence for conditional events. De Finetti proposed an analogous definition of conditional probability in terms of bets. The idea is that when betting on φ ψ with betting odd α the rules are similar to the case of a bet on φ, with the exception that the bet is invalidated if ψ is false, independently of the truth value of φ. Coherence is in terms of no sure loss for B, as usual. However, as it is, the notion of coherence does not work quite well.

7 Example. We chose a point at random in the planet Earth; let E be the event: the chosen point belongs to the Equator, and let W be the event: the point belongs to the Western Hemisphere. Then the assessment E 0, W E 1, ( W ) E 1 is coherent, because if the point does not belong to the equator, then both bets on W E and on W E will be declared null, and B will not lose money, independently of the strategy chosen by G. However, it seems not rational to assess both probabilities of W E and of W E to 1.

8 3. Probability on many-valued events. One may wonder what happens if the events are not boolean but many-valued. Mundici [Mu1] represented such events as elements of an MValgebra. Clearly, MV events may also have intermediate truth values. Note that MV-events are not strange concepts: special random variables ranging on [0, 1]. they are just

9 The role of probability distributions is played by states on MValgebras (see [Mus]). A state on an MV-algebra A is a map s from A into [0, 1] such that: s(1) = 1. If x y = 0, then s(x y) = s(x) + s(y).

10 By a result due independently to Panti [Pa] and to Kroupa [Kr], to each state we can associate a Borel regular probability measure µ on the space V of all homomorphisms from A into [0, 1] such that for all a A, s(a) = V a dµ, where a (v) = v(a) for all v V. In other words, states represent the expected (average) values of the elements of the MV-algebra, which are thought of as random variables. Now Mundici [Mu1] extended de Finetti s theorem to manyvalued events: Theorem. An assessment on an MV-algebra avoids sure loss iff it can be extended to a state.

11 6. Conditional probability over many-valued events. There are (at least) two possible approaches to conditional probability over many-valued events. (1) The conditional probability of φ ψ is the probability of φ in a theory in which ψ is an axiom. As shown by Daniele Mundici [Mub], given any free MV-algebra there is a conditional probability distribution on it which satisfies all Rényi laws of conditional probability and in addition: Is invariant under automorphism of the algebra. Is independent: Pr(φ ψ) = Pr(φ). if φ and ψ have no common variable, then

12 But there is another interpretation of conditional probability when the conditioning event is many-valued. 7. Another interpretation of conditional probability. This second approach takes into account the case where ψ is not completely true, but is partially true. The idea is that the bet should not be completely invalidated when ψ is partially true but not completely true (in particular, if ψ is not 1 but very close to 1, the bet should be almost valid. More precisely, when betting on φ ψ with betting odd α:

13 G chooses a (possibly negative) real number λ, and pays λα to B. Let v(φ) and v(ψ) denote the truth values of φ and of ψ, respectively. Then G gets back λ(v(φ)v(ψ) + α(1 v(ψ))). The bookmaker s payoff is λv(ψ)(α v(φ)). In particular, if v(ψ) = 0 the bet is null and if v(ψ) = 1, the bet is equivalent to a bet on φ.

14 We can prove the following result: Theorem [Mocp]. Consider a complete assessment, that is, one of the form Λ : φ 1 ψ 1 α 1,..., φ n ψ n α n, ψ 1 β 1,..., ψ n β n, on an MV-algebra A. If for i = 1,..., n, β i 0, then Λ avoids sure loss iff there is a state s on A such that for i = 1,..., n, s(ψ i ) = β i and s(φ i ψ i ) = α i β i.

15 But again, what happens if some β i is 0? The previous example about choosing a point in the Western Hemisphere given that it belongs to the Equator shows that when the betting odd for some conditioning event is 0, the assessment may at the same time avoid sure loss and fail to be rational. In order to overcome this problem, we will consider non-standard probabilities.

16 Non-standard analysis is an extension of Mathematical Analysis in which infinite real numbers and infinitely small non-zero real numbers are assumed to exist. Note: the existence of such numbers is consistent! Using non-standard analysis we can consider a framework in which infinitesimal non-zero probabilities, infinitesimal truth values and infinitesimal bets are allowed. Some mathematicians, for instance, Roberto Magari, believe that infinitesimal probabilities can not be neglected.

17 8. A new concept of coherence. Consider now the following game: the bookmaker B fixes a complete assessment of conditional probability, Λ = φ 1 ψ 1 α 1,..., φ n ψ n α n, ψ 1 β 1,..., ψ n β n. If some β i is 0, the gambler G can force B to change Λ by an infinitesimal in such a way that the betting odd of every conditioning event is strictly positive.

18 Definition Λ is said to be stably coherent if there is a variant Λ of Λ such that: (a) Λ avoids sure loss, (b) all betting odds for the conditioning events in Λ are strictly positive, and (c) Λ and Λ differ by an infinitesimal.

19 9. Stable coherence and hyperprobabiities. In order to relate coherence to probability measures, we need a variant of the concept of state which does not neglect infinitesimals. To this purpose, we first extend our MV-algebra A by adding product and hyperreal numbers, thus getting an extension A. of A containing a non standard extension [0, 1] of[0, 1]. This is possible by a Theorem of Di Nola. A hypervaluation is a homomorphism from A into [0, 1] preserving the hyperreal constants of [0, 1].

20 A hyperstate on A is a map s from A ) into [0, 1] which is: additive: if x y = 0, then s(x y) = s(x) + s(y). normalized: s(1) = 1. homogeneous: for all x A and α [0, 1], s(α x) = α s(x). weakly faithful: if s(φ) = 0, then there is a hypervaluation v such that v(φ) = 0 (a hyperstate s is faithful if s(φ) = 0 implies φ = 0).

21 10. Characterizing stable coherence. We can prove: Theorem. A complete assessment Λ = φ 1 ψ 1 α 1,..., φ n ψ n α n, ψ 1 β 1,..., ψ n β n on many-valued events is stably coherent iff there is a faithful hyperstate Pr such that the following condition hold: (i) for i = 1,..., n, Pr (ψ i ) β i is infinitesimal. (ii) For i = 1,..., n, α i Pr (φ i ψ i ) Pr (ψ i ) is infinitesimal. In particular, every MV-algebra admits a faithful hyperstate.

22 In our example about chosing a point in the planet Earth, the assessment E 0, W E 1 and W E 1 is coherent but not stably coherent: if you force the bookmaker to change the book by an infinitesimal so that the betting odd of E is positive, then the assessment becomes E ɛ, W E 1 δ, W E 1 σ with ɛ, δ, σ infinitesimals. Then if you bet 2 1 on E, and 1 on W E and on W E, your payoff is 1 (ɛ v(e)) + v(e)(1 δ + 1 σ 1) > 0. 2 So, you cause a sure loss to the bookmaker.

23 10. Work in progress: (a) Imprecise conditional probabilities. Imprecise probabilities occur when there is uncertainty not only on the outcome of an experiment, but also on the probability distribution. Example. A box contains 100 small balls, 30 are red,10 are yellow and the remaining 60 are either yellow or blue, but we don t have any further information. A ball is chosen at random. What is the probability that the chosen ball is yellow?. We can only say that the probability is at least 0.1 and at most 0.7, but we cannot be more precise.

24 In the example above, we have an upper probability 0.7 and a lower probability 0.1 to chose a yellow ball. People would like to accept the bet on a yellow ball as a bookmaker if the betting odd is 0.7 or more, and as gambler if the betting odd is 0.1 or less. With intermediate betting odds, many people would not like to bet at all.

25 More generally, the upper probability of an event φ can be regarded as the maximum of the values of a set of states on φ. Upper probabilities also model the assessments proposed by real bookmakers. A real bookmaker would never allow negative bets, i.e., he would never allow to interchange the roles of bettor and bookmaker. No sure loss for the bookmaker is not a sufficient rationality condition. For instance, consider the following assessment:

26 B: Brazil wins 0.5. S: Spain wins 0.3. D: Draw 0.3 NB: Brazil does not win 0.7. Then the assessment on NB avoids sure loss, but is not rational: indeed, betting on NB is a bad bet, because betting on both S and on D offers a better payoff to the gambler.

27 When a bad bet is present, another bookmaker can offer a better assessment (in our case, NB can be assessed to 0.6) without losing money when the gambler plays his best strategy. Assessments avoiding bad bets can be identified with assessments which can be extended to the upper previsions (suprema of expected values), [Wa] and [FKMR]. An assessment φ 1 α 1,..., φ n α n avoids sure loss if the expected values of the gambler s payoff φ i α i is 0 for i = 1,..., n, while the same assessment avoids bad bets if the upper prevision of each φ i α i is 0.

28 Hence, the upper conditional prevision U(φ ψ) of a conditional event φ ψ is a number α such that the upper prevision U(ψ(φ α)) = 0 of the gambler s payoff is 0. But is it a definition? That is, does this α exist and is α unique? In general, if U(ψ) = 0, then any α would satisfy U(ψ(φ α)) = 0). So, there are in general many conditional upper probabilities determined by the unconditional upper probability U. This is unfortunate.

29 However, uniqueness of α is ensured when U is a supremum of faithful states, or, more generally, if the lower assessment of all conditioning events is strictly positive. This condition can be realized using non-standard values. Again, the coherence condition (in this case, absence of bad bets) can be replaced by stable coherence (absence of bad bets in an infinitesimal variant of the assessment, in which the lower assessment of conditioning events is non-zero).

30 The final result reads: Theorem An assessment Λ : φ i ψ i α i, ψ i β i, i = 1,..., n of (real valued) conditional upper probability is stably coherent iff there are a faithful hyper upper prevision U and hyperreals α 1, β 1,..., α n, β n such that for i = 1,..., n: (1) α i α i and β i β i are infinitesimal. (2) U(ψ i (φ i α i )) = 0. (3) U( ψ i βi ) = 0 and β i < 1. (The last condition corresponds to L(ψ i ) > 0,where L is the lower prevision corresponding to U).

31 11. Another application: strong coherence. Consider a coherent assessment such that, for some system of bets the bookmaker has a chance to lose money but has no chance to win money (in the best case, the payoff is 0). Then the assessment avoids sure loss, but is not completely rational. An assessment is strongly coherent if for every system of bets, if there is a valuation causing a negative payoff, then there is another one causing a positive payoff. Example. We chose a point at random on Planet Earth. Let E the event: the point belongs to the Equator. The assessment E 0 is coherent but not strongly coherent: if the gambler G bets 1 on E, then G cannot lose money, but he can win money!

32 The example suggests that an assessment is strongly coherent iff it can be extended to a faithful state. This is only true for assessments defined on the whole MValgebra, see [Kem] and [Sh] for the classical case: an assessment defined on a whole MV-algebra A is strongly coherent iff it extends to a faithful state on A. But for partial assessments, in particular for finite assessments, only one direction is true: an assessment which extends to a faithful state is strongly coherent, but the converse is false in general.

33 However, we can prove: Theorem. Let Φ : a 1 α 1,..., a n α n be a finite standard assessment. The following are equivalent: (i) Φ is coherent. (ii) For every positive infinitesimal ɛ, there is a strongly coherent hyperassessment Φ such that for i = 1,..., n, Φ (a i ) α i < ɛ. (iii) For every positive infinitesimal ɛ there is a faithful hyperstate s such that for i = 1,..., n, s(a i ) α i < ɛ.

34 Conjectures. A (finite) assessment over a countable and semisimple MValgebra A is strongly coherent iff it can be extended to a faithful state on A. A (finite) hyperassessment on any MV-algebra A is strongly coherent iff it can be extended to a faithful hyperstate on A.

35 Bibliography. [COM] Cignoli R., D Ottaviano I., Mundici D., Algebraic Foundations of Many-valued Reasoning, Kluwer, Dordrecht [df] de Finetti B., Theory of Probability, vol. I, John Wiley and sons, Chichester [DN] A. Di Nola, Representation and reticulation by quotients of MV-algebras, Ricerche di Matematica (Naples), 40, [H98] Hájek P., Metamathematics of Fuzzy Logic, Kluwer, Dordrecht [Kem] Kemeny, J., Fair bets and inductive probabilities, The Journal of Simbolic Logic 20, n.3, , 1955.

36 [Kr] Kroupa T., Every state on a semisimple MV algebra is integral, Fuzzy Sets and Systems, 157 (20), , [Kr2] Kroupa, T. Conditional probability on MV-algebras. Fuzzy Sets and Systems, 149 (2), , [KM] Kühr J., Mundici D., De Finetti theorem and Borel states in [0,1]-valued algebraic logic, International Journal of Approximate Reasoning 46 (3), , [Mocp] Montagna F., A notion of coherence for books on conditional events in many-valued logic, Journal of Logic and Computation, 21 (5), , (2011). [Mus] Mundici D. Averaging the truth value in Lukasiewicz logic, Studia Logica 55 (1), , 1995.

37 [Mu1] Mundici D., Bookmaking over infinite-valued events, International Journal of Approximate Reasoning 46, , [Mu4] Mundici D., Faithful and Invariant Conditional Probability in Lukasiewicz Logic, Trends in Logic 27: Towards Mathematical Philosophy, David Makinson, Jacek Malinowski and Heinrich Wansing (Eds), 1-20, Springer Verlag [Mub] Mundici D., Advanced Lukasiewicz calculus and MV-algebras, Trends in Logic, Vol. 35, Springer [Pa] Panti G., Invariant measures in free MV algebras, Communications in Algebra 36, , [Sh] Shimony A., Coherence and the axiom of confirmation, The Journal of Symbolic Logic Vol. 20, n.1, 1-28, 1955.

38 [Wa] Walley P., Statistical Reasoning with Imprecise Probabilities. Volume 42 of Monographs on Statistics and Applied Probability, Chapman and Hall, London 1991.