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)).
De Cooman & Quaeghebeur (UGent) Infinite exchangeabilityfor sets of desirable gambles 28 June 2010 3 / 16
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
.
De Cooman & Quaeghebeur (UGent) Infinite exchangeabilityfor sets of desirable gambles 28 June 2010 4 / 16
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
De Cooman & Quaeghebeur (UGent) Infinite exchangeabilityfor sets of desirable gambles 28 June 2010 5 / 16
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 .
De Cooman & Quaeghebeur (UGent) Infinite exchangeabilityfor sets of desirable gambles 28 June 2010 6 / 16
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).
De Cooman & Quaeghebeur (UGent) Infinite exchangeabilityfor sets of desirable gambles 28 June 2010 7 / 16
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.
De Cooman & Quaeghebeur (UGent) Infinite exchangeabilityfor sets of desirable gambles 28 June 2010 8 / 16
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’.
De Cooman & Quaeghebeur (UGent) Infinite exchangeabilityfor sets of desirable gambles 28 June 2010 9 / 16
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.
De Cooman & Quaeghebeur (UGent) Infinite exchangeabilityfor sets of desirable gambles 28 June 2010 10 / 16
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.
De Cooman & Quaeghebeur (UGent) Infinite exchangeabilityfor sets of desirable gambles 28 June 2010 11 / 16
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.
De Cooman & Quaeghebeur (UGent) Infinite exchangeabilityfor sets of desirable gambles 28 June 2010 12 / 16
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 .
De Cooman & Quaeghebeur (UGent) Infinite exchangeabilityfor sets of desirable gambles 28 June 2010 13 / 16
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
De Cooman & Quaeghebeur (UGent) Infinite exchangeabilityfor sets of desirable gambles 28 June 2010 14 / 16
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
De Cooman & Quaeghebeur (UGent) Infinite exchangeabilityfor sets of desirable gambles 28 June 2010 15 / 16
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?
De Cooman & Quaeghebeur (UGent) Infinite exchangeabilityfor sets of desirable gambles 28 June 2010 16 / 16