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FACULTY OF ENGINEERING

Infinite exchangeabilityfor sets of desirable gambles

Gert de Cooman and Erik Quaeghebeur

Ghent University, SYSTeMSGert.deCooman,[email protected]

IPMU 2010Dortmund, 28 June 2010

Bruno de Finetti’s exchangeability resultInformal definition

Consider an infinite sequence

X1, X2, . . . , Xn, . . .

of random variables assuming values in a finite set X .

This sequence is exchangeableif the mass function for any finite subset of these is invariant underany permutation of the indices.

De Cooman & Quaeghebeur (UGent) Infinite exchangeabilityfor sets of desirable gambles 28 June 2010 2 / 16

Bruno de Finetti’s exchangeability resultMore formally

Consider any permutation π of the set of indices 1,2, . . . ,n.

For any x = (x1,x2, . . . ,xn) in X n, we let

πx := (xπ(1),xπ(2), . . . ,xπ(n)).

Exchangeability:If pn is the mass function of the variables X1, . . . ,Xn, then we requirethat:

pn(x) = pn(πx),

or in other words

pn(x1,x2, . . . ,xn) = pn(xπ(1),xπ(2), . . . ,xπ(n)).


Bruno de Finetti’s exchangeability resultCount vectors

For any x ∈X n, consider the corresponding count vector T(x), wherefor all z ∈X :

Tz(x) := |k ∈ 1, . . . ,n : xk = z|.

Example:For X = a,b and x = (a,a,b,b,a,b,b,a,a,a,b,b,b), we have

Ta(x) = 6 and Tb(x) = 7.

Observe that

T(x) ∈N n :=

m ∈ NX : ∑

x∈Xmx = n

.


Bruno de Finetti’s exchangeability resultMultiple hypergeometric distribution

There is some π such that y = πx iff T(x) = T(y).

Let m = T(x) and consider the permutation invariant atom

[m] := y ∈X n : T(y) = m .

This atom has how many elements?(nm

)=

n!∏x∈X mx!

Let MuHyn(·|m) be the expectation operator associated with theuniform distribution on [m]:

MuHyn(f |m) :=1( nm) ∑

x∈[m]

f (x) for all f : X n→ R


Bruno de Finetti’s exchangeability resultThe simplex of limiting frequency vectors

Consider the simplex

Σ :=

θ ∈ RX : (∀x ∈X )θx ≥ 0 and ∑

x∈Xθx = 1

.

Every (multivariate) polynomial p ∈ V n(Σ) on Σ of degree at most nhas a unique Bernstein expansion in terms of the Bernstein basispolynomials Bm of degree n:

p(θ) = ∑m∈N n

bnp(m)Bm(θ),

where

Bm(θ) :=(

nm

)∏

x∈Xθ

mxx .


Bruno de Finetti’s exchangeability resultThe infinite representation theorem

TheoremConsider an sequence X1, . . . , Xn, . . . of random variables in the finiteset X . Then this sequence is exchangeable iff there is a (unique)coherent prevision H on the linear space V (Σ) of all polynomials on Σ

such that for all n ∈ N and f : X n→ R:

Epn(f ) := ∑x∈X

pn(x) f (x)= H(

∑m∈N

MuHyn(f |m)Bm

).

Observe that

∑m∈N

MuHyn(f |m)Bm(θ) = Mnn(f |θ) and Bm(θ) = Mnn([m]|θ).


DesirabilityAccepting gambles

Consider the variables X1, . . . , Xn with possible values x ∈X n.

Subject is uncertain about which alternative x obtains.

A gamble f : X n→ Ris interpreted as an uncertain reward: if the alternative that obtains isx, then the reward for Subject is f (x).

Let G (X n) be the set of all gambles on X n.

We try to model Subject’s uncertainty by looking at which gambles inG (X n) he accepts.


DesirabilityCoherent sets of really desirable gambles

Subject specifies a set R ⊆ G (X n) of gambles he accepts, his set ofreally desirable gambles. R is called coherent if it satisfies thefollowing rationality requirements:D1. if f < 0 then f 6∈R [avoiding partial loss];D2. if f > 0 then f ∈R [accepting partial gain];D3. if f1 ∈R and f2 ∈R then f1 + f2 ∈R [combination];D4. if f ∈R then λ f ∈R for all positive real numbers λ [scaling].Here ‘f < 0’ means ‘f ≤ 0 and not f = 0’.


DesirabilityConditional lower and upper previsions

We can also define Subject’s conditional lower and upper previsions:for any gamble f and any non-empty subset B of Ω , with indicator IB:

P(f |B) := infα ∈ R : IB(α− f ) ∈RP(f |B) := supα ∈ R : IB(f −α) ∈R

so P(f |B) =−P(−f |B) and P(f ) = P(f |Ω).

InterpretationP(f |B) is the supremum price α for which Subject will buy the gamblef , i.e., accept the gamble f −α, contingent on the occurrence of B.


Exchangeability and representationDefinition of exchangeability

Consider random variables X1, . . . , Xn in X , and a coherent set ofdesirable gambles

Rn ⊆ G (X n).

For any gamble f on X n and permutation π of 1, . . . ,n, consider thepermuted gamble π tf defined by

(π tf )(x) := f (πx).

Exchangeability means that f and π tf are considered equivalent:

Exchangeability of Rn:For all f ∈ G (X n), all g ∈Rn and all permutations π:

f −πtf +g ∈Rn.


Exchangeability and representationDefinition for infinite sequences

Consider random variables X1, . . . , Xn, . . . in X , and a correspondingsequence

R1 ⊆ G (X ), . . . ,Rn ⊆ G (X n), . . .

Conditions for exchangeability:1 Rn is exchangeable for all n ∈ N;2 the sequence R1, . . . , Rn, . . . is time-consistent.


Exchangeability and representationRepresentation theorem

TheoremA sequence R1, . . . , Rn, . . . of coherent sets of desirable gambles isexchangeable iff there is some (unique) Bernstein coherentH ⊆ V (Σ) such that:

f ∈Rn⇔Mnn(f |·) ∈H for all n ∈ N and f ∈ G (X n).

Recall that

Mnn(f |θ) = ∑m∈N n

MuHyn(f |m)Bm(θ)

MuHyn(f |m) =1( nm) ∑

x∈[m]

f (x)

Bm(θ) =

(nm

)∏

x∈Xθ

mxx .


Exchangeability and representationBernstein coherence

A set H of polynomials on Σ is Bernstein coherent if:B1. if p has some negative Bernstein expansion then p 6∈H ;B2. if p has some positive Bernstein expansion then p ∈H ;B3. if p1 ∈H and p2 ∈H then p1 +p2 ∈H ;B4. if p ∈H then λp ∈H for all positive real numbers λ .

There are positive (negative) p with no positive (negative) Bernsteinexpansion of any degree!

b w0

1B(2,0)

b w0

1B(0,2)

b w0

1 B(1,1)

b w0

1

p


Exchangeability and representationConditioning

Suppose we observe the first n variables (with count vector m = T(x)):

(X1, . . . ,Xn) = (x1, . . . ,xn) = x.

Then the remaining variables

Xn+1, . . . ,Xn+k, . . .

are still exchangeable, with representation H cx = H cm given by:

p ∈H cm⇔ Bm p ∈H


Open question (one of many)IID process

An exchangeable process X1, . . . , Xn, . . . with represening set ofpolynomials H is IID when no observation has any influence:

H cm = H for all m.

Equivalent condition on H :(∀p ∈ V (Σ))

(∀p+ ∈ V +(Σ)

)(p ∈H ⇔ p+p ∈H ).

1 Are these the extreme points?2 Are all exchangeable models in some way convex combinations

of these extreme points?


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