1. Two notions of symmetry
We want to model aspects of symmetry using imprecise probability models.
We are uncertain about something: the value that a variable X assumes in a finite set X .
The models we will use are coherent lower previsions P , or equivalently, convex closed sets M
of mass functions p—also called credal sets.
Recall from previous lectures: a (probability) mass function p on X is a real-valued map
on X such that
X
(∀x ∈ X )p(x) ≥ 0 and
p(x) = 1.
x∈X
We denote the set (simplex) of all mass functions on values by ΣX .
With a mass function p there corresponds an expectation operator Ep defined on the set
L(X ) of all gambles on X :
X
Ep (f ) :=
p(x)f (x) for all f : X → R.
x∈X
A coherent lower prevision on L(X ) is a map L(X ) → R with the following properties:
[bounds]
P1. P (f ) ≥ min f for all f ∈ L(X )
[super-additivity]
P2. P (f + g) ≥ P (f ) + P (g) for all f, g ∈ L(X )
P3. P (λf ) = λP (f ) for all f ∈ L(X ) and all real λ ≥ 0
[non-negative homogeneity]
Its conjugate upper prevision is defined by
P (f ) := −P (−f ) for all f ∈ L(X ).
With a convex closed set M of mass functions we can construct a coherent lower prevision
on L(X ) by
P (f ) := min{Ep (f ) : p ∈ M} for all f : X → R,
and the conjugate upper prevision on L(X ) by
P (f ) := max{Ep (f ) : p ∈ M} for all f : X → R.
Conversely, with a coherent lower prevision P there corresponds a convex closed set of
mass functions, given by
M := {p ∈ ΣX : (∀f ∈ L(X ))Ep (f ) ≥ P (f )}.
A precise model is a singleton M = {p} and the associated lower prevision is the (selfconjugate) expectation operator Ep .
Symmetry is typically modelled by considering a collection of transformations of the space
of interest, and something is considered to be symmetrical when it is left unchanged by these
transformations.
Here, we will focus on a group P of permutations π of X , meaning that:
G1. π1 ◦ π2 ∈ P for all π1 , π2 ∈ P
[[]internality]
G2. π1 ◦ (π2 ◦ π3 ) = (π1 ◦ π2 ) ◦ π3 for all π1 , π2 , π3 ∈ P
[associativity]
G3. π ◦ id = id ◦π for all π ∈ P
[neutral element]
G4. For all π ∈ P there is some π −1 ∈ P such that π ◦ π −1 = π −1 ◦ π = id
[inverse]
We will use the simpler notation $π := $ ◦ π.
Our uncertainty models involve gambles, so we need a way to let permutations act on gambles.
This is easy to do by a procedure called lifting: for any π ∈ P, π t f := f ◦ π, meaning that
(π t f )(x) := f (πx) for all x ∈ X .
1
2
Running example. We flip a coin twice, and the uncertain outcome X is an element of the finite
set X = {HH , HT , TH , TT }.
The symmetry we consider is that the order of the observations does not matter, which leads
us to identify HT with TH . This symmetry is embodied in the group of permutations of X given
by
P = {id, $},
where $ is the permutation defined by
HH
HH
HT
TH
TH
HT
TT
TT
Observe that this a group, with $2 = $ ◦ $ = id, so $−1 = $.
Consider the gamble f that yields a gain of 2 when the two coin flips produce the same result,
and a loss of 1 otherwise:
f = 2I{HH ,TT } − I{HT ,TH } .
Observe that $t f = f .
On the other, for the gamble g that yields a gain of 1 when the first coin flip produces H , and
a loss of 3 otherwise:
g = I{HH ,HT } − 3I{TH ,TT } ,
the permuted gamble
$t g = I{HH ,TH } − 3I{HT ,TT }
yields a gain of 1 when the second coin flip produces H , and a loss of 3 otherwise.
Important: The set L(X ) of all X − R maps can also be written as RX , which reminds us
that any gamble f can be seen as a vector in the |X |-dimensional linear space L(X ) , with
as many components f (x) as there are values x in X .
Any lifted permutation π t is a linear permutation (a linear isomorphism) of L(X ) (check
this):
π t (λf + µg) = λπ t f + µπ t g for all f, g ∈ L(X ) and λ, µ ∈ R.
This suggests that a geometrical way of looking at gambles and permutations may be
fruitful.
For any linear transformation T of L(X ), its range rng T is the linear subspace of gambles
that it reaches:
rng T := T (L(X )) = {T f : f ∈ L(X )}.
Its kernel kern T is the linear subspace of gambles that it maps to the zero gamble:
kern T := {f ∈ L(X ) : T f = 0}.
And for any subset C of L(X ), its linear span
X
n
span(C) :=
λk fk : n ≥ 0, fk ∈ C, λk ∈ R
k=1
is the smallest linear subspace that includes C.
Now there are two fundamentally different concepts of symmetry for imprecise probability
models.
3
Symmetrical models. The first one captures that the uncertainty models M or P we are
using, are symmetrical, or in other words, invariant under the permutations in P. This means
that
(∀f ∈ L(X ))P (f ) = P (π t f ) or equivalently P = P ◦ π t , for all π ∈ P.
This requirement for lower previsions is equivalent (check this) to the following requirement for
credal sets:
(∀p ∈ M)π t p ∈ M or equivalently π t (M) = M, for all π ∈ P,
where, as before, we let π t p := p ◦ π. We then say that the models P and M are (weakly)
invariant (with respect to P).
This type of symmetry is the one that is usually invoked under ignorance. When we are
ignorant, our models ought to be symmetrical under all kinds of permutations. Observe in this
respect that the vacuous model
P v := min or equivalently Mv := ΣX
is symmetrical under all permutations of X .
A precise model M = {p}, or equivalently, a self-conjugate lower prevision (expectation
operator) Ep , is therefore invariant if and only if
π t p = p or equivalently Ep ◦ π t = Ep , for all π ∈ P.
Running example. Consider the mass functions
1
1
1
1
p1 := I{HH ,HT } + I{TH ,TT } and p2 := I{TH ,HH } + I{HT ,TT }
3
6
3
6
For both precise models, the coin flips are independent. In the first model, the second coin is
fair and the probability of heads for the first coin flip is 32 . The second model is similar to the
first, but the role of the first and second coin flips is reversed. Observe that $t p1 = p2 and
consequently also $t p2 = p1 , so these precise models are not permutation invariant with respect
to P. On the other hand, the credal set
M1 := {αp1 + (1 − α)p2 : α ∈ [0, 1]}
is permutation invariant, because
$t [αp1 + (1 − α)p2 ] = α$t p1 + (1 − α)$t p2 = αp2 + (1 − α)p1 ∈ M1 .
This model corresponds to the following probability assessments: the structural assessment that
the coin flips are (strongly) independent, and the local assessments that the (precise) probability
of HH is 31 and the (precise) probability of TT is 16 . Instead of the independence assessment, we
can also simply require that the probability of HT lies between 16 and 13 .
Observe that this model is equivalent to assuming that two different coins are used for each
coin flip, but that there is no information at all about the mechanism for selecting the order in
which these two coins are used. It is this lack of information (not the symmetry of the coins—they
aren’t) that is reflected in the symmetry of the model.
Models of symmetry. There is a (usually much) stronger requirement, called strong invariance, that is used to model that there is symmetry lurking behind the variable X, or that the
subject believes that the ‘mechanism generating the observations of the variable X is symmetrical’.
How can this be modelled? Consider any gamble f . If the subject believes there is this
symmetry, he will be indifferent between f and its permutations π t f , for all π ∈ P. This implies
that he is willing to exchange f for π t f when he is paid any positive amount of utility > 0,
whence
P (π t f − f + ) ≥ 0 for all > 0,
4
so the requirement for strong invariance (with respect to P) becomes
P (π t f − f ) = P (π t f − f ) = 0 for all f ∈ L(X ) and all π ∈ P.
This requirement for lower previsions P is equivalent (check this) to the following requirement
for credal sets M:
(∀p ∈ M)π t p = p, for all π ∈ P.
So a lower prevision is strongly invariant if and only if it is a lower envelope of (weakly and
therefore strongly) invariant precise expectations.
Strong invariance is indeed a stronger requirement than invariance: any strongly invariant
model is invariant (check this), but not every invariant model is strongly invariant. Check, for
instance, that the vacuous model is not strongly invariant with respect to any non-trivial group
of permutations.
Running example. Consider the mass functions
1
1
1
1
p3 := I{HH ,TT } + I{HT ,TH } and p4 := I{HH ,TT } + I{HT ,TH }
3
6
6
3
For neither of these precise models, the coin flips are independent. Observe that $t p3 = p3
and $t p4 = p4 , so both these precise models are permutation invariant with respect to P. This
implies that the credal set
M2 := {αp3 + (1 − α)p4 : α ∈ [0, 1]}
is strongly permutation invariant, because
$t [αp3 + (1 − α)p4 ] = α$t p3 + (1 − α)$t p4 = αp3 + (1 − α)p4 .
The lower prevision that corresponds to this credal set if given by:
o
n1
1
1
1
P 2 (f ) := min [f (HH )+f (TT )]+ [f (HT )+f (TH )], [f (HH )+f (TT )]+ [f (HT )+f (TH )]
3
6
6
3
The independent and permutation invariant precise models are given by
pr := r2 I{HH } + r(1 − r)I{HT ,TH } + (1 − r)2 I{TT } , for r ∈ [0, 1].
This implies that, for instance, the credal set
Mr1 ,r2 := {αpr1 + (1 − α)pr2 : α ∈ [0, 1]}
is strongly independent and permutation invariant, for any choice of r1 , r2 ∈ [0, 1].
2. A general representation theorem
We will now show that, because of their symmetry properties, the behaviour of strongly
invariant lower previsions can be represented more efficiently and compactly on lower-dimensional
spaces than L(X ). Our argumentation relies quite heavily on the geometrical interpretation of
gambles as vectors in a linear space, and of permutations as linear transformations of that space.
Technical preliminaries about permutation groups. A gamble f is invariant if it is left
unchanged by the permutations, so
π t f = f for all π ∈ P.
An event A ⊆ X is invariant if its indicator IA is, meaning that (check this)
(∀x ∈ A)πx ∈ A or equivalently π(A) = A, for all π ∈ P.
The smallest invariant sets are the so-called invariant atoms
[x] := {πx : π ∈ P},
5
which constitute a partition of X . We denote the set of all invariant atoms by AP .
A gamble is permutation invariant if and only if it is constant on the invariant atoms (check).
So a permutation invariant gamble is completely determined by the values it assumes on the invariant atoms, and this means that we can consider a one-to-one map CP —a linear isomorphiam—
linking gambles g on the atoms, so g ∈ L(AP ), to the corresponding permutation invariant
gambles CP g in LP (X ), defined by:
CP : L(AP ) → LP (X ), where (CP g)(x) := g([x]) for all x ∈ X .
Running example. The invariant atoms are
[HH ] = {HH } and [HT ] = [TH ] = {HT , TH } and [TT ] = {TT }.
The permutation invariant gambles are the ones that are constant on [HT ] = [TH ] = {HT , TH }
and therefore give the same value to HT and TH .
Two interesting subspaces. We will need to look at two special linear subspaces. First,
consider the following linear subspace:
IP := span({π t f − f : f ∈ L(X ) and π ∈ P}).
It is easy to check (do this) that P is strongly invariant if and only if
P (g) = P (g) = 0 for all g ∈ IP ,
so we can see IP as the linear subspace of indifferent gambles: the gambles that the subject
judges to be equivalent to the zero gamble.
On the other hand, the linear subspace of permutation invariant gambles
LP (X ) := {f ∈ L(X ) : (∀π ∈ P)π t f = f }
is the set of all gambles that are constant on the invariant atoms. They are completely determined
by the values that they assume on these invariant atoms, and the dimension of this space LP (X )
is therefore the same as the dimension of the linear space L(AP ) that linearly isomorphic to
it, and therefore equal to the number of invariant atoms. This is generally smaller than the
dimension |X | of the original space L(X ): typically, the more permutations there are in P, the
fewer invariant atoms there are.
Running example. The subspace of indifferent gambles is given by:
IP = λ(I{HT } − I{TH } ) : λ ∈ R ,
and is one-dimensional. The subspace of permutation invariant gambles is given by
LP (X ) = λ1 I{HH } + λ2 I{TH ,HT } + λ3 I{TT } : λ1 , λ2 , λ3 ∈ R ,
and has dimension 3. Observe that IP ∩ LP (X ) = {0}, and that
IP + LP (X ) = λ1 I{HH } + λ2 I{TH ,HT } + λ3 I{TT } + λ(I{HT } − I{TH } ) : λ, λ1 , λ2 , λ3 ∈ R
= λ1 I{HH } + (λ2 − λ)I{TH } + (λ2 + λ)I{HT } + λ3 I{TT } : λ, λ1 , λ2 , λ3 ∈ R
= L(X ),
so any gamble can be decomposed uniquely into an indifferent and a permutation invariant part.
This is a result that holds in general, and underlies the representation result we prove below. It
is related to—a special case of—the famous dimension theorem of linear algebra.
6
An interesting projection operator. Consider the following operator inv, which maps any
gamble to the uniform average of all its permutations:
invP f :=
1 X t
π f.
|P|
π∈P
Check that this is a linear transformation of L(X ) that satisfies the following properties:
I1.
I2.
I3.
I4.
invP ◦ π t = invP = π t ◦ invP for all π ∈ P
invP ◦ invP = invP
kern(invP ) = IP
rng(invP ) = LP (X )
[permutation invariance]
[projection]
[kernel]
[range]
So we see that this operator does not distinguish between gambles and their permuted versions.
It is a projection operator that maps any gamble f to a permutation invariant gamble invP f
that the subject is indifferent between, in the sense that f − invP f ∈ IP .
Since invP f is a permutation invariant gambles, it is constant on the invariant atoms. The
constant value it assumes there can also be written as:
1 X
1 X
f (πx) =
f (y) =: U (f |[x]) for all x ∈ X ,
(invP f )(x) =
|P|
|[x]|
π∈P
y∈[x]
which is the expectation associated with the uniform distribution over the atom [x]. We can see
U as a linear map taking gambles f on X to the corresponding gambles U (f |·) on AP :
U : L(X ) → L(AP ), where U (f )([x]) := U (f |[x]) =
1 X
f (y),
|[x]|
y∈[x]
which allows us to write
invP = CP ◦ U.
This allows us to prove (and the proof is very easy) the following important result:
Theorem (Strong Permutation Invariance Representation Theorem). A coherent lower prevision
P is strongly invariant with respect to P if and only if any (and hence all) of the following
equivalent statements holds:
(i) P = P ◦ invP ;
(ii) There is a coherent lower prevision Q on L(AP ) such that P = Q ◦ U .
In that case this so-called representing lower prevision Q is unique and given by Q := P ◦ CP .
L(X )
invP
LP (X )
U
Q◦U =P
R
Q
L(AP )
7
Running example. With P = {id, $}, we have invP = 21 (id +$t ), and therefore

h(HH )
if x = HH




if x = TT
invP h(x) = h(TT )



 h(HT ) + h(TH ) if x = HT or x = TH .
2
All permutation invariant expectation operators have the following form:
h(HT ) + h(TH )
q([HT ]) + h(TT )q([TT ]),
Eq (h) = h(HH )q([HH ]) +
2
where q is any mass function on AP . All strongly permutation invariant lower previsions have
the following form
P (h) = min{Eq : q ∈ M}
h(HT ) + h(TH )
= min h(HH )q([HH ]) +
q([HT ]) + h(TT )q([TT ]) : q ∈ M ,
2
where M is any credal set on AP .
As a special example, the representing mass functions on AP for the independent and permutation invariant precise models are given by
qr ([HH ]) = r2 and qr ([HT ]) = 2r(1 − r) and qr ([TH ]) = (1 − r)2 ,
for r ∈ [0, 1].