Download Random sets and belief functions

Random sets on finite spaces
Random sets on infinite spaces
References
Random sets and belief functions
Enrique Miranda
University of Oviedo
[email protected]
3rd Belief School, September 2015
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Outline
I
Random sets on finite spaces.
I
Representation by belief functions.
I
Particular cases.
I
Connection with measurable selections.
I
Extensions to the infinite case.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Random sets on finite spaces
Consider a probability space (Ω, A, P) and a finite space X . A
random set is a map
Γ : Ω → P(X )
satisfying the following measurability condition:
A∗ := {ω : Γ(ω) ∩ A 6= ∅} ∈ A ∀A ⊆ X .
We shall assume throughout that Γ(ω) 6= ∅ for every ω (which will
be equivalent to dealing with normalized belief functions).
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Measurability
The above condition is called strong measurability, and in this case,
where X is finite, is equivalent to each of the following conditions:
1. A∗ := {ω : Γ(ω) ⊆ A} ∈ A ∀A ⊆ X .
2. {ω : Γ(ω) = A} ∈ A ∀A ⊆ X .
It reduces to the usual measurability condition of random variables
when Γ(ω) is a singleton for every ω.
A∗ , A∗ are called the lower and upper inverses of A, respectively.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Example
Consider
Ω = {1, 2, 3}, A = {{1, 2}, {3}, Ω, ∅}, P({1, 2}) = 32 , P({3}) =
and Γ : Ω → P({1, 2, 3}) given by
1
3
Γ(1) = {1, 2}, Γ(2) = {2, 3}, Γ(3) = {1, 3}.
Given A = {1}, it holds that A∗ = {ω : 1 ∈ Γ(ω)} = {1, 3} ∈
/ A.
Thus, Γ is NOT a random set.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Upper and lower probabilities of a random set
Given a set A ⊆ X , Dempster defined its upper and lower
probabilities by
P ∗ (A) := P(A∗ ) = P({ω : Γ(ω) ∩ A 6= ∅})
and
P∗ (A) := P(A∗ ) = P({ω : Γ(ω) ⊆ A}).
It holds that P∗ (A) ≤ P ∗ (A) ∀A ⊆ X . They are moreover
conjugate functions: P ∗ (A) = 1 − P∗ (Ac ) ∀A ⊆ X . As we shall
see, there is a connection with belief functions.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Example
Consider the same multi-valued mapping as before, but now with
A = P(Ω) and P({1}) = P({2}) = P({3}) = 31 , so that Γ is a
random set.
Given A = {1}, it holds that
I
A∗ = {1, 3} ⇒ P ∗ (A) = 23 .
I
A∗ = {ω : Γ(ω) = {1}} = ∅ ⇒ P∗ (A) = 0.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Particular case: random variables
When Γ is single-valued, then for every A ⊆ X it holds that
A∗ = A∗ = Γ−1 (A).
Thus, the measurability condition is the usual measurability
condition, and the lower and upper probabilities coincide with the
probability measure induced by the random variable.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Exercise
Consider Ω = {1, 2, 3} with the probability measure
P({1}) = 0.3, P({2}) = 0.5, P({3}) = 0.2 and the random set
Γ : Ω → P({1, 2, 3}) given by
Γ(1) = {1, 2}, Γ(2) = {2, 3}, Γ(3) = {1, 2, 3}.
Determine the upper and lower probabilities of the sets
A = {1}, B = {1, 2} and C = {2, 3}.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Basics of belief functions (again)
Given a finite space X , a belief function or ∞-monotone Choquet
capacity on P(X ) is a function Bel : P(X ) → [0, 1] such that for
every natural number n and every family {A1 , . . . , An } of subsets
of X , it holds that
X
Bel(A1 ∪ . . . An ) ≥
(−1)|I |+1 Bel(∩i∈I Ai ).
∅6=I ⊆{1,...,n}
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Basic probability assignment
From Shafer, a function m : P(X ) → [0, 1] is called a basic
probability
assignment when it satisfies m(∅) = 0 and
P
A⊆X m(A) = 1.
I
Given a basic probability assignment m, the function
Bel : P(X ) → [0, 1] given by
X
Bel(A) =
m(B)
B⊆A
I
is a belief function.
If Bel is a belief function, the map m : P(X ) → [0, 1] given by
X
m(A) =
(−1)|A−B| Bel(B)
B⊆A
is a basic probability assignment.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Plausibility functions
The conjugate Pl of a belief function is called a plausibility
function. It is related to the same basic probability assignment via
the formula
X
Pl(A) =
m(B).
B∩A6=∅
Moreover, this correspondence between belief functions, basic
probability assignments and plausibility measures is one-to-one. m
is called the Möbius inverse of Bel.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Focal elements
Given a belief function Bel with Möbius inverse m, a subset A of
X is called a focal element of m when m(A) 6= 0. In particular, the
focal elements of a belief function are those sets for which
m(A) > 0.
The focal elements are useful when working with a lower
probability. In this sense, in game theory we have the so-called
k-additive measures, which are those whose focal elements have
cardinality smaller or equal than k.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Belief functions and random sets (Nguyen, 1978)
Let Γ : Ω → P(X ) be a random set. Then its lower probability P∗
is a belief function, and its upper probability P ∗ is the conjugate
plausibility function.
The Möbius inverse of P∗ is given by
m(A) = P({ω : Γ(ω) = A}).
Thus, the focal elements of P∗ are the subsets A of X for which
P(Γ−1 (A)) > 0.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Example
Let (Ω, A, P) = ([0, 1], β[0,1] , λ[0,1] ), X = {1, 2, 3} and
Γ : Ω → P(X ) given by


{1, 2}
if ω < 0.3



{3}
if ω = 0.3
Γ(ω) =

{1, 2, 3} if ω ∈ (0.3, 0.5]



{2, 3}
if ω > 0.5
Then P∗ is the belief function with focal elements
m({1, 2}) = 0.3, m({1, 2, 3}) = 0.2, m({2, 3}) = 0.5. As a
consequence, we obtain for instance P∗ ({1, 3}) = 0,
P∗ ({1, 2}) = 0.3, P ∗ ({2, 3}) = 1 = P ∗ ({1, 2}).
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
From belief functions to random sets
We have seen that any random set induces a belief function.
Conversely, any belief function Bel can be obtained as the lower
probability P∗ of a random set: this result is called Choquet’s
theorem, and we say that the random set is the source of Bel.
To see this, consider an arbitrary order among the focal elements
of m, A1 ≺ A2 ≺ · · · ≺ An and define Γ : [0, 1) → P(X ) by
where a−1
Γ(ω) = Ai if ω ∈ [ai−1 , ai ),
P
= 0, ai = 1≤j≤i m(Aj ).
Thus, the two models (random sets and belief functions) are
equally expressive.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Non-uniqueness
Note that two different random sets may have the same lower
probability P∗ : if we have the basic probability assignment m with
focal elements {A1 , . . . , An }, we could also consider
Ω = {1, . . . , n}, A = P(Ω), with P({i}) = m(Ai ) and let
Γ :Ω → P(X )
i ,→ Ai .
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Example
If we consider the belief function Bel on P({1, 2, 3, 4}) with basic
probability assignment
m({1, 2, 3}) = 0.2 = m({1}), m({2, 3}) = 0.1 = m({4}), m({3, 4}) = 0.4,
then we can consider Ω = {1, 2, 3, 4, 5}, A = P(Ω), P the
probability measure determined by
P(1) = P(2) = 0.2, P(3) = P(4) = 0.1, P(5) = 0.4
and the random set Γ given by
Γ(1) = {1, 2, 3}, Γ(2) = {1}, Γ(3) = {2, 3}, Γ(4) = {4}, Γ(5) = {3, 4}.
Then the lower probability of this random set is Bel.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Exercise
Consider the belief function Bel on P({1, 2, 3}) given by
A
Bel(A)
{1}
0.1
{2}
0.2
{3}
0
{1,2}
0.5
{1,3}
0.3
{2,3}
0.4
{1,2,3}
1
Determine a random set having Bel as its lower probability.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Example: vacuous belief functions
The case where we have the least amount of information
corresponds to the basic probability assignment m(X ) = 1,
m(A) = 0 for every A ( X . The corresponding belief and
plausibility functions are
Bel(A) = 0 ∀A 6= X ,
Pl(A) = 1 ∀A 6= ∅.
These are called vacuous, and model the most imprecise situation.
The corresponding random set would be Γ : Ω → P(X ) given by
Γ(ω) = X for all ω.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Example: probability measures
Let Bel : P(X ) → [0, 1] be a belief function, and let Pl be its
conjugate plausibility function. The following are equivalent:
1. Bel is a probability measure.
2. The focal elements of µ are singletons.
3. Bel = Pl.
4. Bel(A) + Bel(Ac ) = 1 for every A ⊆ X .
For them, the associated random set Γ satisfies |Γ(ω)| = 1 ∀ω, and
becomes thus a random variable. We say that the belief function is
Bayesian.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Particular cases: possibility measures
Given X finite, an upper probability P : P(X ) → [0, 1] is called a
possibility measure when
P(A ∪ B) = max{P(A), P(B)}
para todo A, B ⊆ X .
Its conjugate is called a necessity measure, and it satisfies
P(A ∩ B) = min{P(A), P(B)} for every A, B ⊆ X .
A necessity measure is a belief function, and corresponds to the
case where the focal elements are nested. They are also called
consonant belief functions.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Exercise
Consider X = {1, 2, 3, 4}.
I
Let Π be the possibility distribution associated to the
possibility distribution
π(1) = 0.3, π(2) = 0.5, π(3) = 1, π(4) = 0.7. Determine its
focal elements and its basic probability assignment.
I
Given the basic probability assignment m({1}) =
0.2, m({1, 3}) = 0.1, m({1, 2, 3}) = 0.4, m({1, 2, 3, 4}) = 0.3,
determine the associated possibility measure and its possibility
distribution.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Possibility measures and random sets
Given a random set Γ : Ω → P(X ) on a finite space X , the
following are equivalent:
1. P ∗ is a possibility measure.
2. There exists a null subset N of Ω such that, for every
ω1 , ω2 ∈ Ω \ N, either Γ(ω1 ) ⊆ Γ(ω2 ) or Γ(ω2 ) ⊆ Γ(ω1 ).
In other words, the upper probability of a random set is a
possibility measure if and only if its images are nested.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Ontic and epistemic interpretations
As discussed by Dubois and Couso, random sets can be given two
different interpretations:
I
The ontic or conjunctive one: Γ(ω) is a multi-valued random
variable. This interpretation is used by Kendall or Mathéron,
amongst others.
I
The epistemic one: Γ is a model for an ill-known random
variable U0 , so that all we know about U0 (ω) is that it
belongs to Γ(ω). This interpretation is used by Dempster and
it is closer to imprecise probabilities.
The interpretation we use has implications when modelling
conditioning or independence, for instance.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Example
Assume that X = {Spanish, French, English} is a set of languages,
and that we consider Γ : Ω → P(X ).
I
Under an ontic interpretation, Γ(ω) could be the set of
languages a person speaks. Then, the probability that a
person speaks English would be
X
P(Γ−1 (A)).
English∈A
I
Under an epistemic interpretation, Γ(ω) could be our
imprecise knowledge of a person’s native language. Then, the
probability that a person’s native language is English would
belong to [P∗ (A), P ∗ (A)], where A=‘English’.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Conditioning conjunctive random sets
Assume that we know that the value of the random set Γ is
included in some set A ⊆ X . Then the conditional distribution of
the random set can be obtained by applying Bayes’ rule on the
probability distribution of Γ; this produces
( P(Γ−1 (C ))
P
if C ⊆ A
−1 (B))
B⊆A P(Γ
PΓ (C |A) :=
0
otherwise.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Conditioning disjunctive random sets
If instead Γ is understood as a model for the imprecise knowledge
of a random variable U0 , then we obtain a set of possible values
{PU (C |A) : U ∈ S(Γ), PU (A) > 0},
where
S(Γ) := {U : Ω → X r.v.|U(ω) ∈ S(Γ) ∀ω},
which, by taking the lower envelope, produces
P∗ (C |A) =
P∗ (C ∩ A)
;
P∗ (C ∩ A) + P ∗ (C c ∩ A)
this formula is called the regular extension of the belief function P∗ .
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Epistemic random sets
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Epistemic random sets: upper and lower inverses
Given A ∈ A0 , it is
A∗ and A∗ are called upper and lower inverses of A by Γ,
respectively.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Measurable selections of a random set
Under the epistemic interpretation, the information about the
original random variable U0 is given by the measurable selections
of Γ:
S(Γ) := {U : Ω → X r.v.|U(ω) ∈ S(Γ) ∀ω},
and the set of possible distributions of U0 is given by
P(Γ) := {PU : U ∈ S(Γ)}.
In particular, for every A ⊆ X we define
P(Γ)(A) := {PU (A) : U ∈ S(Γ)}.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Example
Let Ω = {1, 2, 3, 4} = X , A = P(Ω), P the uniform distribution,
and Γ : Ω → P(X ) given by
Γ(1) = {1}, Γ(2) = {1, 3}, Γ(3) = {2, 4}, Γ(4) = {3, 4}.
The measurable selections of Γ are:
1 2
U1 1 1
U2 1 1
U3 1 1
U4 1 1
U5 1 3
U6 1 3
U7 1 3
U8 1 3
E. Miranda
c
2015
3
2
2
4
4
2
2
4
4
4
3
4
3
4
3
4
3
4
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Upper and lower probabilities
For a given set A ⊆ X :
I
P∗ (A) = P({ω : Γ(ω) ⊆ A}) measures the total degree of
support of the set A.
I
P ∗ (A) = P({ω : Γ(ω) ∩ A 6= ∅}) measures the evidence that is
consistent with the set A.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Representation by means of belief function
Denote the core of the belief function P∗ by
M(P∗ ) := {Q prob. measure|Q(A) ≥ P∗ (A) ∀A}.
I
P∗ (A) = min{Q(A) : Q ∈ P(Γ)} ∀A ⊆ X .
I
M(P∗ ) = Conv (P(Γ)).
I
If (Ω, A, P) is non-atomic, then P(Γ) = M(P∗ ).
The key for this result is that the extreme points of M(P∗ ) belong
to P(Γ).
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Example (cont.)
Given A = {1, 3}, we deduce that P(Γ)(A) = {0.5, 0.75}. As a
consequence, [P∗ (A), P ∗ (A)] = [0.5, 0.75].
We can also see that
P∗ (A) = P({ω : Γ(ω) ⊆ A}) = P({1, 2}) = 0.5
and
P ∗ (A) = P({ω : Γ(ω) ∩ A 6= ∅}) = P({1, 2, 4}) = 0.75
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Exercise
Consider Ω = {1, 2, 3} with the uniform distribution,
Γ : Ω → P({1, 2}), Γ(ω) = {1, 2} ∀ω.
I
Show that P(Γ) = {(0, 1), (1/3, 2/3), (2/3, 1/3), (1, 0)}.
I
Show that M(P ∗ ) = {(α, 1 − α) : α ∈ [0, 1]}.
Let f be the
P Shannon entropy of a probability measure,
f (P) = − i P(xi )log2 (P(xi ).
I
f (PU0 ) ∈ f (P(Γ)) = {0, f (1/3, 2/3)} = {0, 0.92}.
I
f (M(P∗ )) = [0, 1].
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
M(P∗ ) and mass allocations
Each probability measure P ∈ M(P∗ ) corresponds to an allocation
of the basic probability assignment of P∗ , where the mass m(Ai ) of
each focal element is distributed between the elements of Ai . The
resulting measure has only the singletons as focal elements, and is
thus a probability measure.
If for example P∗ is associated to the basic probability assignment
m({1, 2}) = 0.5, m({1, 3}) = 0.3, m({2, 3}) = 0.2, the probability
measure P ≡ (0.6, 0.3, 0.1) ∈ M(P∗ ) would result for instance:
m({1, 2}) = 0.5 → mass 0.4 for {1} and 0.1 for 2
m({1, 3}) = 0.3 → mass 0.2 for {1} and 0.1 for 3
m({2, 3}) = 0.2 → mass 0.2 for {2} and 0 for 3
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
The Choquet integral
P
Given a function f : X → R, f = ni=1 xi IAi , for
x1 > x2 > · · · > xn and for some finite partition {A1 , . . . , An } of Ω,
its Choquet integral with respect to a non-additive measure µ is
given by
Z
(C )
fdµ =
n
X
xi (µ(Si ) − µ(Si−1 ) =
i=1
n
X
(xi − xi+1 )µ(Si ),
i=1
where Si = ∪ij=1 Aj , S0 = ∅, xn+1 = 0.
This is a generalization of the expectation of a random variable to
non-additive measures
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Example
Consider the belief function Bel associated with the basic
probability assignment
m({1, 2, 3}) = 0.2, m({1, 3}) = 0.3, m({2, 3}) = 0.4, m({1}) = 0.1
and let f be given by f (1) = 3, f (2) = 5, f (3) = 0. Then
f = 5I2 + 3I1 + 0I3 , so its Choquet integral is given by
Z
(C ) fdµ = (5 − 3)µ({2}) + (3 − 0)µ({1, 2}) + 0µ({1, 2, 3})
= 2 · 0 + 3 · 0.1 + 0 · µ({1, 2, 3}) = 0.3.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Choquet integral: basic properties
I
I
I
I
R
(C ) IA dµ = µ(A).
R
R
(C ) cfdµ = c(C ) fdµ ∀c ≥ 0.
R
R
f ≤ g ⇒ (C ) fdµ ≤ (C ) gdµ.
The
Choquet integral
isRnot additive in general, but
R
R
(f + g )dµ = fdµ + gdµ when f , g are comonotonic.
A more detailed account can be found in the book of Denneberg.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Choquet integral and measurable selections
Given a function f : X → R, and monotone set functions
µ1 ≤ µ2 ≤ µ3 , it holds that
Z
Z
Z
(C ) fdµ1 ≤ (C ) fdµ2 ≤ (C ) fdµ3 .
As a consequence, given a random set Γ : Ω → P(X ) and
f : X → R it holds that
Z
Z
Z
(C ) fdP∗ ≤ fdPU ≤ (C ) fdP ∗ ∀U ∈ S(Γ).
The previous results ensures that in fact
Z
Z
(C ) fdP∗ = min
fdQ : Q ∈ P(Γ) .
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Exercise
Consider Ω = {1, 2, 3, 4} = X , A = P(Ω), P the uniform
distribution, and Γ : Ω → P(X ) given by
Γ(1) = {1}, Γ(2) = {1, 3}, Γ(3) = {2, 4}, Γ(4) = {3, 4}.
Determine the Choquet integral of the mapping f given by
f (1) = 10, f (2) = 4, f (3) = 0, f (4) = 2 with respect to P ∗ and P∗ ,
and give the measurable selections that attain this lower and upper
integral.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Probability boxes
Given a finite set X = {x1 , . . . , xn } with x1 < x2 < . . . , < xn , a
cumulative distribution function is a non-decreasing map
F : X → [0, 1] satisfying F (xn ) = 1. It represents the cumulative
probabilities of a random variable taking values in X .
Given two cumulative distribution functions F∗ ≤ F ∗ , the set
(F∗ , F ∗ ) := {F cdf : F∗ (xi ) ≤ F (xi ) ≤ F ∗ (xi ) ∀i}
is called a probability box, or p-box (Ferson et al.). It can be seen
as a model for the imprecise knowledge of a distribution function.
Given a p-box (F∗ , F ∗ ), the lower envelope of the set
{P : FP ∈ (F∗ , F ∗ )} is a belief function (Troffaes and Detercke).
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Distribution functions of a random set
When X ⊆ R, the restriction to cumulative sets of P ∗ and P∗
determine the upper and lower distribution functions
F ∗ , F∗ : X → [0, 1] given by
F∗ (x) = P∗ ({z ≤ x}) and F ∗ (x) = P ∗ ({z ≤ x}).
However, the pair (F∗ , F ∗ ) does not include all the information of
the random set (Couso): we may find probability measures Q such
that
FQ (x) ∈ [F∗ (x), F ∗ (x)] ∀x ∈ X
while Q ∈
/ M(P∗ ).
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Example
Let Ω = {1, 2, 3, 4} = X , A = P(Ω), P the uniform distribution,
and Γ : Ω → P(X ) given by
Γ(1) = {1}, Γ(2) = {1, 3}, Γ(3) = {2, 4}, Γ(4) = {3, 4}.
The lower and upper distributions are given by:
F∗
F∗
1
0.25
0.5
2
0.25
0.75
3
0.5
1
4
1
1
The function F given by F (1) = F (2) = 0.5, F (3) = F (4) = 1
belongs to (F∗ , F ∗ ), but it is not induced by any measurable
selection, because PU ({2, 4}) = 0 < 0.25 = P({3}) = P∗ ({2, 4}).
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Exercise
Consider Ω = {1, 2, 3} with
P({1}) = 0.2, P({2}) = 0.3, P({3}) = 0.5 and the random set Γ
given by Γ(1) = {1, 2}, Γ(2) = {1, 3}, Γ(3) = {2, 3}.
I
Determine the lower and upper distribution functions of Γ.
I
Use them to find a probability measure Q such that
F∗ (x) ≤ FQ (x) ≤ F ∗ (x) ∀x ∈ {1, 2, 3} while Q ∈
/ M(P∗ ).
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Random sets and game theory
The lower probability P∗ of Γ can be seen as a coalitional game:
we regard the elements of X as players and P∗ (A) is seen as the
gain associated with a coalition among the elements of A.
Then the Shapley value of this fame is the center of gravity of the
set M(P∗ ). It can be determined using the extreme points of
M(P∗ ), which are given by (Dempster; Shapley; Chateauneuf and
Jaffray)
{Pσ : σ permutation of X },
where the probability Pσ is determined by
Pσ (xσ1 , . . . , xσk ) = P∗ (xσ1 , . . . , xσk ) ∀k = 1, . . . , |X |.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Shapley value of a belief function
The Shapley value is the probability measure Q given by
Q({x}) =
X m(A)
x∈A
|A|
,
where m is the Möbius inverse of P∗ .
The concept of Shapley value holds (with a different definition) for
other types of non-additive measures (not necessarily belief
functions or 2-monotone capacities), but games associated with a
convex capacity have nicer mathematical properties.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition
Connection with belief functions
Epistemic random sets
Exercise
Consider the belief function associated with the basic probability
assignment:
A
m(A)
{1}
1/9
{2}
1/9
{1,2}
1/9
{1,3}
1/6
{2,3}
1/6
{1,2,3}
1/3
Determine the extreme points of M(Bel) and its Shapley value.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Random sets on infinite spaces
As we shall see, things get more complicated when the random set
takes values on an infinite space: many of the results do not
translate directly, and we need to impose additional conditions on
the images of the random set.
In particular, not all random sets are compatible with the epistemic
interpretation, and not all belief functions can be obtained as the
lower probability of a random set.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Finitely vs. countably additive probabilities
The source of the problem is the structure of the initial space: if
we consider a finitely additive probability P on a field A, then:
I
We can assume that A = P(Ω).
I
Any belief function is the lower envelope of the set
{P finitely additive : P(A) ≥ Bel(A) ∀A}.
I
Any belief function can be obtained as the lower probability of
a random set.
I
We can characterise the envelopes of sets of finitely additive
probability measures.
I
P does not satisfy any continuity property.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Finitely vs. countably additive probabilities (II)
However, if we consider a countably additive probability P on a
σ-field A, then:
I We cannot assume that A = P(Ω).
I Not every belief function is the lower envelope of the set
{P countably additive : P(A) ≥ Bel(A) ∀A}.
Not every belief function can be obtained as the lower
probability of a random set.
I We cannot characterise the envelopes of sets of countably
additive probability measures.
I ... but P satisfies some continuity properties.
Unfortunately, usually the initial probability space (Ω, A, P)
considers a countably additive P on a σ-field A.
I
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Definition
Let (Ω, A, P) be a probability space, (X , A0 ) a measurable space
and Γ : Ω → P(X ) a non-empty multi-valued mapping. It is called
a random set when it is strongly measurable: for every A ∈ A0 ,
{ω : Γ(ω) ∩ A 6= ∅} ∈ A ∀A ∈ A0 ,
or, equivalently, when
{ω : Γ(ω) ⊆ A} ∈ A ∀A ∈ A0 .
This is not the only measurability condition. Other,
non-equivalent, possibilities (Himmelberg) are:
I
measurability: {ω : Γ(ω) ∩ C 6= ∅} ∈ A ∀C closed.
I
weak measurability: {ω : Γ(ω) ∩ G 6= ∅} ∈ A ∀G open.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Lower and upper probabilities
As before, the lower and upper probabilities of a random set Γ are
defined as:
P ∗ (A) := P({ω : Γ(ω) ∩ A 6= ∅})
and
P∗ (A) := P({ω : Γ(ω) ⊆ A})
for every A ∈ A0 , and they are conjugate functions.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Properties of P∗
The lower probability P∗ of a random set satisfies the following
properties:
I
It is upper continuous: given a decreasing sequence (An )n ,
P∗ (∩n An ) = limn P∗ (An ).
I
It is ∞-monotone (=a belief function when X is finite).
Similarly, P ∗ is ∞-alternating and lower continuous.
This means in particular that not all ∞-monotone capacities can
be obtained as lower probabilities of random sets. Under some
conditions, it is possible to characterise those who can.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
The hit or miss topology
Denote by F, G , K the classes of closed, open, and compact
subsets of Rd , and let
FGK1 ,...,Gn := {F ∈ F : F ∩ K = ∅ =
6 F ∩ G1 , . . . , F ∩ Gn }.
This is the basis for a topology on F, called the hit or miss
topology. The σ-field it induces is denoted B(F). Then a random
set Γ : Ω → Rd with closed values is a A − B(F ) measurable
mapping.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Capacity functionals
A set function T : K → R is called a capacity functional when it
satisfies:
I
T (∅) = 0, T (K ) ∈ [0, 1] ∀K .
I
T is ∞-alternating.
I
If (Kn )n ↓ K , then limn T (Kn ) = T (K ).
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Choquet’s theorem
(Mathéron) If T is a capacity functional, then there is a unique
probability measure P on B(F ) such that
P({F : F ∩ K 6= ∅}) = T (K ).
This determines the distribution of the random closed set (the
probability measure on B(F )), because we can use the above
equation to determine the probability on the hit or miss topology.
The proof relies heavily on topological properties of Rd , although
it can be extended to ∞-dimensional Polish spaces.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Possibility and necessity measures (Dubois and Prade)
Given an infinite space X , a possibility measure is a function
Π : P(X ) → [0, 1] s.t. for every family of subsets (Ai )i∈I of X ,
Π(∪i∈I Ai ) = sup Π(Ai ).
i∈I
The conjugate function of a possibility measure, given by
Nec(A) = 1 − Π(Ac ), is called a necessity measure, and satisfies
Nec(∩i∈I Ai ) = inf Nec(Ai )
i∈I
for every family of subsets (Ai )i∈I .
Possibility measures are related to fuzzy sets, via their restrictions
to singletons.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Connection with maxitive measures
Recall that an upper probability P is maxitive when
P(A ∪ B) = max{P(A), P(B)} ∀A, B ⊆ X . This is equivalent to
being a possibility measure when X is finite.
A related model are the condensable measures (Shafer), which are
those satisfying
P(∪A∈C A) = sup P(A)
A∈C
for every upward net C, which is class of events such that
∀A1 , A2 ∈ C, ∃A3 ∈ C s.t. A1 ∪ A2 ⊆ A3 .
(Miranda et al., 2004): P possibility ⇔ maxitive and condensable.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Consonant random sets
(Miranda et al., 2004) When Γ is closed valued on a σ-compact
metric space, the following are equivalent:
I
P ∗ is a possibility measure.
I
P ∗ is maxitive.
I
∃N ⊆ Ω null such that ∀ω1 , ω2 ∈ Ω \ N, either Γ(ω1 ) ⊆ Γ(ω2 )
or Γ(ω2 ) ⊆ Γ(Ω1 ).
The equivalence does not hold for other types of random sets.
Also, any possibility measure Π can be obtained as the upper
probability of a random set.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Measurable selections
As before, if we give Γ an epistemic interpretation as a model for
the imprecise knowledge of a random variable U0 , then our
information about U0 is given by
S(Γ) := {U : Ω → X measurable : U(ω) ∈ Γ(ω) ∀ω},
from which we determine the probabilistic information:
P(Γ) := {PU : U ∈ S(Γ)}
and
P(Γ)(A) := {PU (A) : U ∈ S(Γ)}.
Again P(Γ) ⊆ M(P∗ ) := {Q prob. : Q(A) ≥ P∗ (A) ∀A ∈ A0 }, and
also P(Γ)(A) ⊆ [P∗ (A), P ∗ (A)].
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
P ∗ (A), P∗ (A) as a model for PU0 (A)
In general we have the following inclusion:
P(Γ)(A) ⊆ [P∗ (A), P ∗ (A)]
The study of the equality can be decomposed in two different
subproblems:
I
Is P(Γ)(A) convex?
I
P ∗ (A) = maxP(Γ)(A), P∗ (A) = minP(Γ)(A)?
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Properties of P(Γ)(A) (Miranda et al., 2010)
I
P(Γ)(A) closed, whence there are U1 , U2 ∈ S(Γ) such that
maxP(Γ)(A) = PU1 (A), minP(Γ)(A) = PU2 (A), U1 , U2 ∈ S(Γ).
I
P(Γ)(A) convex ⇔ U1−1 (A) \ U2−1 (A) not an atom.
I
If P ∗ (A) = PU1 (A) and P∗ (A) = PU2 (A), then
P(Γ)(A) = [P∗ (A), P ∗ (A)] ⇔ A∗ \ A∗ not an atom.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Relationship with the existence of measurable selections
P ∗ (A) = maxP(Γ)(A) ⇔ ∃H null s.t S(Γ ∩ AIA∗ \H ⊕ ΓI(A∗ \H)c ) 6= ∅
I In general, P ∗ (A) 6= maxP(Γ)(A); it may even be
P(Γ)(A) = {0.5} and [P∗ (A), P ∗ (A)] = [0, 1]. This is because
a random set may not have measurable selections!!
We must study then under which conditions S(Γ) 6= ∅. Early works
were summarised by Wagner.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Sufficient conditions for P(Γ)(A) = [P∗ (A), P ∗ (A)]
By making a study of the existence of measurable selections for
different types of random sets, it is possible to show (Miranda et
al., 2010) that, under any of the following conditions:
I
Ω complete, X Souslin, Gr (Γ) ∈ A ⊗ βX .
I
Γ closed, (X , d) σ-compact.
I
Γ open, (X , d) separable.
I
Γ closed, X Polish.
I
Γ compact, (X , d) separable.
P(Γ)(A) = [P∗ (A), P ∗ (A)] ⇔ A∗ \ A∗ not an atom
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Expectations of random sets
Let Γ : Ω → P(Rn ) be a random set. Its Aumann integral is given
by
Z
Z
fdP : f ∈ L1 (P), f (ω) ∈ Γ(ω) a.s .
(A) ΓdP :=
This is the definition of expectation of a random set which is more
interesting under the epistemic interpretation. Other definitions
are:
I
The Debreu integral.
I
The Herer integral.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
The Choquet integral
Let (X , A0 ) be a measurable space. Given a measurable function
f : X → R and a non-additive measure µ : A0 → [0, 1], the
Choquet integral of f with respect to µ is given by
Z
(C )
Z
sup f
fdµ = inf f +
µ(f > t)dt.
inf f
This definition extends the one of the finite case. Its mathematical
properties (monotonicity, homogeneity...) are similar.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Connection between the integrals (Miranda et al., 2010)
Let Γ : Ω → P(X ) be a random set. If P ∗ (A) = maxP(Γ)(A) for
all A ∈ A0 , then for any bounded random variable f : X → R,
Z
Z
Z
Z
(C ) fdP ∗ = sup
fdPU , (C ) fdP∗ = infU∈S(Γ) fdPU .
U∈S(Γ)
As a consequence,
Z
(C )
fdP ∗ = sup(A)
E. Miranda
Z
Z
(f ◦ Γ)dP, (C )
c
2015
Z
fdP∗ = inf(A)
Random sets and belief functions
(f ◦ Γ)dP.
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Relationships between P(Γ) and M(P ∗ ): early results
I
(Hart and Köhlberg) (Ω, A, P) non atomic complete,
Γ1 , Γ2 : Ω → P(Rn ) integrably bounded, PΓ∗1 = PΓ∗2 ⇒
P(Γ1 ) = P(Γ2 ).
I
(Hess) Γ1 , Γ2 closed, (X , || · ||) Banach separable,
[PΓ∗1 = PΓ∗2 ⇔ Conv (P(Γ1 )) = Conv (Γ2 )].
I
(Castaldo y Marinacci) Γ compact, X Polish
⇒ M(P ∗ ) = Conv (P(Γ)).
E. Miranda
c
2015
Random sets and belief functions
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Random sets on finite spaces
Random sets on infinite spaces
References
Relationships between P(Γ) and M(P ∗ ) (Miranda et al.,
2005a)
(X , d) separable, P ∗ (A) = maxP(Γ)(A) ∀A ∈ Q(τ (d))
@
@
@
@
R
@
P(Γ) convex ⇔ P(Γ) = M(P ∗ )
M(P ∗ ) = Conv (P(Γ))
]
JJ
J
J
J
J
J
P(Γ) convex
(Ω, A, P) non atomic
E. Miranda
*
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Interpretation
In particular, this means that, although not in all cases, we can use
P ∗ and P∗ to summarise the probabilistic information about the
original random variable U0 for the most important types of
random sets, such as for instance:
I
Random closed sets on Rd .
I
Random open sets on Rd .
I
Random sets on finite spaces.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Particular case: random intervals
We next study the case where the images of the random set are
subintervals of the real line, determined by two mappings A, B.
They have been studied by Demspter and Joslyn, among others.
We focus on random closed intervals Γ = [A, B], although some
results have been established for random open intervals:
Γ = (A, B).
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Results for random closed intervals
(Miranda et al. (2005b):
I
[A, B] strongly measurable ⇔ A, B measurable.
I
(Ω, A, P) = ([0, 1], β[0,1] , λ[0,1] ) ⇒ P(Γ) = M(P ∗ ) under any
of the following conditions:
I
A, B increasing.
I
A or B constant.
I
A, B strictly comonotonic.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
Definition and basic concepts
Choquet’s theorem on Rd
Random sets and possibility measures
Epistemic random sets
Conclusions
When X is finite:
I
Belief functions are equivalent to lower probabilities of random
sets.
I
They keep all the information about the selections under an
epistemic interpretations.
I
Possibility measures correspond to the particular case of
consonant random sets.
I
P-boxes may carry a loss of information.
When X is infinite, the above results do not hold in general, but
they do for most random sets of interest (random closed intervals,
for instance).
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
References
I
R. Aumann, J. of Math. Anal. and Appl., 12, 1-12, 1965.
I
A. Castaldo and M. Marinacci, Sankhya, 66(3), 409-427,
2004.
I
G. Choquet, Ann. de l’Ins. Fourier, 5, 131-295, 1953.
I
I. Couso, D. Dubois, Int. J. of App. Reasoning, 55(7),
1502-1514, 2014.
I
I. Couso, D. Dubois, L. Sánchez, Random sets and random
fuzzy sets as ill-perceived random variables. Springer, 2014.
I
I. Couso, L. Sánchez, P. Gil, Inf. Sciences, 159(1-2), 109-123,
2004.
I
A. Dempster, Ann. of Math. Statistics, 38, 325-339, 1967.
I
A. Dempster, Ann. of Math. Statistics, 39, 957–966, 1968.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
References (II)
I
I
I
I
I
I
I
I
D. Denneberg, Non-additive measure and integral. Kluwer,
1994.
S. Hart and E. Köhlberg, J. of Math. Economics, 1(2),
167-174, 1974.
C. Hess, J. of Convex Analysis, 6(1), 163-182, 1999.
C. Himmelberg, Fund. Mathematicae, 87, 53-72, 1975.
R. Kruse, H. Meyer, Statistics with vague data. D. Reidel,
1987.
G. Mathéron, Random sets and integral geometry. Wiley,
1975.
E. Miranda, I. Couso, P. Gil, Inf. Sciences, 168(1-4), 51–75,
2004.
E. Miranda, I. Couso, P. Gil, J. of Math. Anal. and
Applications, 307(1), 32-47, 2005a.
E. Miranda
c
2015
Random sets and belief functions
Random sets on finite spaces
Random sets on infinite spaces
References
References (III)
I
I
I
I
I
I
I
E. Miranda, I. Couso, P. Gil, Fuz. Sets and Systems, 154(3),
386–412, 2005b.
E. Miranda, I. Couso, P. Gil. Inf. Sci., 180(8), 1407–1417,
2010.
I. Molchanov, Theory of random sets. Springer, 2005.
H.T. Nguyen, An introduction to random sets. Chapman and
Hall, 2006.
H.T. Nguyen, J. of Math. Anal. and Appl., 65(3), 531-542,
1978.
G. Shafer, A mathematical theory of evidence, Princeton,
1976.
M. Troffaes, S. Destercke, Int. J. of App. Reasoning, 52(6),
767–791, 2011.
E. Miranda
c
2015
Random sets and belief functions