Non-Archimedean Probability and Conditional Probability; ManyVal2013 Prague 2013 F.Montagna, University of Siena
1. De Finetti’s approach to probability . De Finetti’s defini- tion 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.
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 homomor- phism v of the algebra of events into into 2 , � n i =1 λ i ( α i − v ( φ i )) < 0.
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.
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 as- sessments. There is an interesting analogy with the complete- ness theorem: coherence is equivalent to (probabilistic) satisfia- bility.
5. Coherence for conditional events . De Finetti proposed an analogous definition of conditional prob- ability 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.
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.
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 MV- algebra. Clearly, MV events may also have intermediate truth values. Note that MV-events are not strange concepts: they are just special random variables ranging on [0 , 1].
The role of probability distributions is played by states on MV- algebras (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 ).
By a result due independently to Panti [Pa] and to Kroupa [Kr], to each state we can associate a Borel regular probability mea- sure µ on the space V of all homomorphisms from A into [0 , 1] V a ◦ dµ , where a ◦ ( v ) = v ( a ) for all such that for all a ∈ A , s ( a ) = � 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 many- valued events: Theorem . An assessment on an MV-algebra avoids sure loss iff it can be extended to a state.
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 R´ enyi laws of conditional probability and in addition: • Is invariant under automorphism of the algebra. • Is independent: if φ and ψ have no common variable, then Pr( φ | ψ ) = Pr( φ ).
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 :
• 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 .
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 .
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.
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 val- ues and infinitesimal bets are allowed. Some mathematicians, for instance, Roberto Magari, believe that infinitesimal probabilities can not be neglected.
8. A new concept of coherence . Consider now the follow- ing 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.
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.
9. Stable coherence and hyperprobabiities . In order to re- late 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 con- taining 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] ∗ pre- serving the hyperreal constants of [0 , 1] ∗ .
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).
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 ) is infinitesimal. Pr ∗ ( ψ i ) In particular, every MV-algebra admits a faithful hyperstate.
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