Non-Archimedean Probability and Conditional Probability; ManyVal2013 - - PowerPoint PPT Presentation

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 φ1, . . . , φn with betting odds α1, . . . , αn, 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 φ1, . . . , φn. That is, there are no λ1, . . . , λn such that for every homomor- phism v of the algebra of events into into 2,

n

i=1 λi(αi − v(φi)) < 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

• f 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 φ1 → α1, . . . , φn → αn 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(φi) = αi. 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] 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 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

• ver 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

• n 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

• f the form Λ : φ1|ψ1 → α1, . . . , φn|ψn → αn, ψ1 → β1, . . . , ψn → βn,
• n an MV-algebra A. If for i = 1, . . . , n, βi = 0, then Λ avoids

sure loss iff there is a state s on A such that for i = 1, . . . , n, s(ψi) = βi and s(φiψi) = αiβi.

But again, what happens if some βi 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 = φ1|ψ1 → α1, . . . , φn|ψn → αn, ψ1 → β1, . . . , ψn → βn. If some βi 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′

• f 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 = φ1|ψ1 → α1, . . . , φn|ψn → αn, ψ1 → β1, . . . , ψn → βn

• n 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∗(ψi) − βi is infinitesimal. (ii) For i = 1, . . . , n, αi − Pr∗(φi·ψi)

Pr∗(ψi)

is infinitesimal. In particular, every MV-algebra admits a faithful hyperstate.

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 → ǫ, W|E → 1 − δ, ¬W|E → 1 − σ with ǫ, δ, σ infinitesimals. Then if you bet −1

2 on E, and −1 on

W|E and on ¬W|E, your payoff is 1 2(ǫ − v(E)) + v(E)(1 − δ + 1 − σ − 1) > 0. So, you cause a sure loss to the bookmaker.

• 10. Work in progress: (a) Imprecise conditional probabili-
• ties. Imprecise probabilities occur when there is uncertainty not
• nly 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

• nly say that the probability is at least 0.1 and at most 0.7, but

we cannot be more precise.

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

• dd 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.

More generally, the upper probability of an event φ can be re- garded 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:

• 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.

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 assess- ments which can be extended to the upper previsions (suprema

• f expected values), [Wa] and [FKMR].

An assessment φ1 → α1, . . . , φn → αn avoids sure loss if the ex- pected values of the gambler’s payoff φi − αi is 0 for i = 1, . . . , n, while the same assessment avoids bad bets if the upper prevision

• f each φi − αi is 0.

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.

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).

The final result reads: Theorem An assessment Λ : φi|ψi → αi, ¬ψi → βi, i = 1, . . . , n

• f (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) αi − α∗

i and βi − β∗ i are infinitesimal.

(2) U(ψi · (φi − α∗

i )) = 0.

(3) U(¬ψi −β∗

i ) = 0 and β∗ i < 1. (The last condition corresponds

to L(ψi) > 0,where L is the lower prevision corresponding to U).

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!

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 MV- algebra, see [Kem] and [Sh] for the classical case: an assess- ment 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,

• nly one direction is true:

an assessment which extends to a faithful state is strongly coherent, but the converse is false in general.

However, we can prove: Theorem. Let Φ : a1 → α1, . . . , an → αn be a finite standard

• assessment. The following are equivalent:

(i) Φ is coherent. (ii) For every positive infinitesimal ǫ, there is a strongly coherent hyperassessment Φ′ such that for i = 1, . . . , n, |Φ′(ai) − αi| < ǫ. (iii) For every positive infinitesimal ǫ there is a faithful hyperstate s such that for i = 1, . . . , n, |s(ai) − αi| < ǫ.

Conjectures. A (finite) assessment over a countable and semisimple MV- algebra 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.

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