On the linear structure of betting criterion and
the checking of coherence
Veronica Biazzo1
&
Angelo Gilio2
1
2
Dipartimento di Matematica e Informatica
Viale A. Doria, 6 - 95125 Catania (Italy)
Dipartimento di Metodi e Modelli Matematici
Via A. Scarpa, 16 - 00161 Roma (Italy)
Abstract. We use imprecise probabilities, based on a concept of generalized coherence, for the management of uncertainty in Artificial Intelligence.
With the aim of reducing the computational difficulties, in the checking of
generalized coherence we propose a method which exploits, in the framework of the betting criterion, suitable subsets of the sets of values of the
random gains. We give an algorithm in each step of which a linear system
with a reduced number of unknowns can be used. Our method improves a
procedure already existing in literature and could be integrated with recent
approaches of other authors, who exploit suitable logical conditions with the
aim of splitting the problem into subproblems. We remark that our approach
could be also used in combination with efficient methods like column generation techniques. Finally, to illustrate our method, we give some examples.
Keywords: conditional probability bounds, betting criterion, random gain,
alternative theorems, g-coherence checking, not relevant gains, basic sets,
algorithms, computational aspects.
1
1
Introduction
In many applications of Artificial Intelligence, such as knowledge representation and reasoning, the treatment of uncertainty, by symbolic or numerical
methods, plays a crucial role. Among the numerical methods, the probabilistic one is well founded and has a clear rationale. When facing real problems
we often need to reason with uncertain information under partial knowledge.
In these cases the management of uncertainty by means of precise probabilistic assessments is quite unrealistic. Moreover, the set of uncertain quantities
at hand has no particular algebraic structure. Then, a flexible and more
realistic approach can be obtained by using precise and/or imprecise probabilities, based on (suitable generalizations of) the coherence principle of de
Finetti ([15], [16], [24], [27], [31], [40], [41]), or on similar principles like that
ones adopted for lower and upper probabilities ([19], [21], [37], [42], [43],
[44]). This approach allows to assess and propagate probabilistic judgements
in a consistent and general way and has been adopted in many recent papers.
In [35] the fundamental theorem of prevision is deeply analyzed and many
computational and geometrical examples are given. Conditions of coherence
for numerical and qualitative probability assessments on conditional events
are given in [12], [13]. A comparison between coherence-based methods and
belief functions is made in [39]. In [32] logical operations among conditional
events are considered and it is shown that, in order to compare conditional
events of zero probability, the introduction of iterated conditioning is needed.
In [25] a concept of (generalized) coherence is introduced and some previous
theoretical results concerning coherence of precise conditional probabilities
are extended to the case of conditional probability bounds. This approach is
also examined in [26] where a counterexample is given to remark that a conditional probability assessment may satisfy all the usual axioms of conditional
probabilities without being coherent. Moreover, exploiting a geometrical approach, in such paper is studied the exact propagation of probability bounds
in Cautious Monotonicity and Or inference rules used in default reasoning.
This problem has been deepened in [28] and [29], where a probabilistic approach under coherence to System P in nonmonotonic reasoning has been
developed. A characterization theorem and an algorithm for checking coherence of conditional probabilities are given, with many examples, in [17], where
also de Finetti’s extension theorem is deepened and generalized to conditional
events. A concept of total coherence for imprecise conditional probability assessments and some related theoretical results are given in [30]. A general
2
review on the coherence-based probabilistic approach to uncertain and partial knowledge is given in [20]. In such paper it is also shown that exploiting
zero probabilities allows, on one hand, to simplify the procedure for checking coherence (see also [18]) and, on another hand, to deepen the concept of
stochastical independence by eliminating some inconsistencies of the classical
definitions. A generalization of de Finetti’s extension theorem to imprecise
conditional probability assessments, with an algorithm and some illustrative
examples, is given in [2]. The elicitation of probabilities for belief networks
is studied in [22], combining qualitative and quantitative information. In [3]
(see also [4]) it is shown that the probabilistic reasoning under coherence
can be reduced to standard reasoning tasks in model-theoretic probabilistic logic. Then, efficient procedures such as column generation techniques
([34], [33]) can be applied. Concerning the computational complexity, in [3]
it is shown that the problems of checking (generalized) coherence and entailment (under generalized coherence) are, respectively, NP-complete and
co-NP-complete. Therefore, the global checking of coherence, being based
on linear programming techniques, has in general an exponential complexity. This means that in many cases this problem may become intractable
and then any method which may eliminate (even partially) the computational difficulties seems worthwhile to be investigated. We point out that, in
the framework of interval-valued probability assessments, an efficient global
procedure for the probabilistic deduction from probabilistic knowledge-bases
has been proposed in [36]. However, in the quoted paper it has been made
the restrictive assumption that, for every probabilistic formula (E|H)[α, β]
in the knowledge base, both E and H are conjunctions of some basic events
in a given set. Moreover, the conditional probability P (E|H) is looked at as
the ratio of the unconditional ones P (EH) and P (H), so that it is defined
only if P (H) > 0. On the contrary, as well known, within the approach of de
Finetti: (i) one can directly assess conditional probabilities, with no need of
defining them as ratios; (ii) no theoretical problem arises when the probabilities of some (or possibly all) conditioning events are (judged equal to) zero;
(iii) applying the algorithms for coherence checking and for propagation, zero
probabilities can be exploited in (at least) two ways: on one hand, as first
suggested in [18] (see also [20]), exploiting some suitable logical conditions
the computational difficulties may be reduced (or even eliminated) by splitting the problem into subproblems ([8], [9], [10], [11]). On another hand,
as made in this paper by examining the random gains, the computational
difficulties may be reduced by assigning a zero value to a certain (possibly
3
large) number of unknowns in the linear systems used in the algorithms.
In this paper we use imprecise probabilities for the management of uncertainty. To check the consistency of probabilistic assessments we adopt a
concept of generalized coherence (g-coherence) which is based on the coherence principle of de Finetti. With the aim of reducing the computational
difficulties, we examine in the framework of the betting criterion the additive
structure of the random gain. Then, we propose a general methodology by
means of which the checking of g-coherence can be based on suitable subsets
of the sets of possible values of the random gains. We show that in checking g-coherence we can study, in each step, a linear system with a reduced
number of unknowns. Notice that, as shown by Example 3, our method
improves the procedure proposed in [36] (cfr. [5]). Moreover, it could be
integrated with that ones proposed in [8], [9], [10], [11] where, checking the
satisfiability of suitable logical relations, the global consistency is studied by
splitting the problem into subproblems. However, as shown by the Example
1, it may happen that the problem can’t be splitted into subproblems because
the logical relations are not satisfied. We also remark that the theoretical
conditions which we use to eliminate variables do not depend on the numerical values of the probability bounds. It must be noted that the procedure by
means of which the unknowns are assigned a zero value, being based on some
comparisons among suitable vectors associated with the atoms, may be time
consuming. Anyway, as a positive note, we observe that the ”small” systems
resulting from the process of elimination of unknowns are easily checked for
their solvability. Of course, the best solution would be the characterization
in terms of random gains of direct methods for determining such ”small” systems. A relevant example in this sense is the characterization given in [3] of
the procedure proposed in [36] for the case of conjuntive conditional events.
By this characterization, see Remark 8, in the conjunctive case our approach
could be combined with the direct procedure given in [36]. We also remark
that our approach could be used in combination with efficient methods, like
column generation techniques, proposed by other authors; see e.g. [33], [34].
Another important aspect, which is out of the scope of this paper and should
be the subject of future research, concerns a thorough discussion of the efficiency of the proposed methodology and the analysis of its computational
complexity.
The paper is organized as follows. In section 2 we recall some preliminary
4
concepts and results. In section 3, after illustrating the main idea of our
method, we formalize the notion of not relevant gain and the related one of
basic set. Moreover, we give some theoretical results. In section 4, based
on some theoretical remarks, we propose a modified version of an algorithm
for the checking of g-coherence ([25]), which suitably exploits the notion of
basic set. In section 5 we illustrate our method by some examples. Finally, in
section 6 we give some conclusions and an outlook on possible future research.
2
Preliminary concepts and results
We denote, for every integer n, by Jn the set {1, . . . , n} and by Fn the family
of n conditional events {Ei |Hi , i ∈ Jn }. Given an event E, we denote by E c
its logical negation and, given another event H, by EH the conjunction of
E and H. The indicator |E| of an event E is defined as
1 , E true ,
|E| =
0 , E f alse .
In what follows, we simply denote |E| by E. Given the family Fn , let
An = (α1 , . . . , αn ) be the following vector of lower bounds defined on Fn :
P (Ei |Hi ) ≥ αi ,
i ∈ Jn .
(1)
The consistency of the assessment (1) can be examined by means of suitable
general coherence definitions ([13], [14], [42], [44]). Based on the coherence
principle of de Finetti, we adopt the following condition of generalized coherence (g-coherence)([25], [2]).
Definition 1 The vector of lower bounds An defined on Fn is said g-coherent
iff there exists a precise coherent assessment (p1 , . . . , pn ) on Fn , with pi =
P (Ei |Hi ), such that: pi ≥ αi , ∀ i ∈ Jn .
As P (Ei |Hi ) ≤ βi is equivalent to P (Eic |Hi ) ≥ 1 − βi , Definition 1 can also
be applied to interval probability bounds, like:
αi ≤ P (Ei |Hi ) ≤ βi ,
i ∈ Jn .
As remarked in [2], the notion of g-coherence is equivalent to the ”avoiding
uniform loss” property of lower and upper probabilities ([41], [42], [43]).
5
Remark 1 Notice that E|H = EH|H. Then, a probability assessment on
the family {Ei |Hi , i ∈ Jn } is equivalent to the same assessment on the family
{Ei Hi |Hi , i ∈ Jn }. Therefore, in the algorithms for the checking of coherence
and for propagation we can use the family Υn = {Ei Hi , Hi , i ∈ Jn }, instead
of the family {Ei , Hi , i ∈ Jn }. We observe that the events Ei Hi , Hi generate
a partition {Ei Hi , Eic Hi , Hic }. Then, the set C of constituents generated by
Υn is obtained by expanding the expression
^
(2)
(Ei Hi ∨ Eic Hi ∨ Hic ) .
i∈Jn
The set C has cardinality less than or equal to 3n .
We denote by H0 the disjunction H1 ∨ · · · ∨ Hn and by C1 , . . . , Cm the
constituents contained in H0 . Notice that m ≤ 3n − 1. Based on C and An ,
with each Ch we associate a vector Vh = (vh1 , . . . , vhn ), where

 1 , if Ch ⊆ Ei Hi ,
0 , if Ch ⊆ Eic Hi ,
(3)
vhi =

c
αi , if Ch ⊆ Hi .
Remark 2 We observe that
Ch ⊆ Hic =⇒ vhi = αi ,
while the converse is not true. Indeed, if Ch ⊆ Ei Hi and αi = 1 (resp.
Ch ⊆ Eic Hi and αi = 0), then vhi = αi = 1 (resp. vhi = αi = 0).
We recall that, in the framework of the betting scheme, the random gain
associated with the pair (Fn , An ) is
Gn =
n
X
si Hi (Ei − αi ) ,
i=1
with si ≥ 0, ∀i ∈ Jn . Based on (3), for each Ch ⊆ H0 the value gh of Gn is
given by
n
X
gh = Gn (Vh ) =
si (vhi − αi ) .
(4)
i=1
Given the pair (Fn , An ), for each J ⊂ Jn we denote respectively by FJ and
AJ the subfamily of Fn and the subvector of An associated with J. Moreover,
6
W
we denote by HJ the event j∈J Hj and by GJ the random gain, associated
with the pair (FJ , AJ ), defined as
X
GJ =
sj Hj (Ej − αj ) ; sj ≥ 0, ∀j ∈ J.
j∈J
Remark 3 If the imprecise assessment on the family Fn is represented by
a vector of upper bounds Bn = (βi , i ∈ Jn ), then for every J ⊆ Jn the
corresponding random gain is
P
P
c
GJ =
j∈J σj Hj [Ej − (1 − βj )] = −
j∈J σj Hj (Ej − βj ) ,
with σj ≥ 0, ∀j ∈ J. In particular:
P
Gn = − j∈Jn σj Hj (Ej − βj ) ,
with: σj ≥ 0, ∀j ∈ Jn .
In the general case of interval probability bounds one has
X
GJ =
Hj [sj (Ej − αj ) − σj (Ej − βj )] ,
j∈J
with sj ≥ 0, σj ≥ 0 ,
∀j ∈ J .
We recall some results obtained in [25], where the concept of g-coherence is
simply named coherence.
Theorem 1 Given a vector of lower bounds An on the family Fn , in order
An be g-coherent the following system (Sn ), with nonnegative unknowns
λ1 , . . . , λm , must be solvable
Pm
λr vri ≥ αi , i ∈ Jn ,
Pr=1
(5)
m
r=1 λr = 1, λr ≥ 0, r ∈ Jm .
Given J ⊂ Jn , we denote by (SJ ) the system associated with the pair
(FJ , AJ ). Then, we have the following necessary and sufficient condition
for the g-coherence of An .
Theorem 2 The vector of lower bounds An on Fn is g-coherent iff the
system (SJ ) is solvable, for every J ⊆ Jn .
7
We denote respectively by Λ and S the vector of unknowns and the set of
solutions of the system (Sn ). Moreover, for each j we denote by Γj the set of
subscripts r such that Cr ⊆ Hj and by Fj the set of subscripts r such that
Cr ⊆ Ej Hj . For each j, we define the linear function
X
Φj (Λ) =
λr .
(6)
r∈Γj
Moreover, we denote by I0 the (strict) subset of Jn defined as
I0 = {j ∈ Jn : Mj = M axΛ∈S Φj (Λ) = 0}
(7)
and by (F0 , A0 ) the pair associated with the set I0 . Then, we have
Theorem 3 Given a vector of lower bounds An on Fn , if the system (Sn )
is solvable, then for every J ⊂ Jn such that J \ I0 6= ∅ the system (SJ ) is
solvable.
We remark that, as the unknowns in
W the system (SJ ) are associated with
the constituents contained in HJ = j∈J Hj , in Theorem 3 the condition
J \ I0 6= ∅ is necessary. Based on Theorem 3, if (Sn ) is solvable, then for
each subset J such that J \ I0 6= ∅ we don’t need to check the solvability
of (SJ ). It follows that Theorem 2 can be expressed in the following more
efficient way
Theorem 4 The imprecise assessment An on Fn is g-coherent if and only if
the following conditions are satisfied:
1. the system (Sn ) is solvable ;
2. if I0 6= ∅, then A0 is g-coherent.
Based on the previous results, the following algorithm can be used to check
the g-coherence of An :
Algorithm 1 Let be given the triplet (Jn , Fn , An ).
1. Construct the system (5) and check its solvability;
2. If the system ( 5) is not solvable then An is not g-coherent and the
procedure stops, otherwise compute the set I0 defined by (7);
8
3. If I0 = ∅ then An is g-coherent and the procedure stops, otherwise set
(Jn , Fn , An ) = (I0 , F0 , A0 ) and repeat steps 1-3.
The g-coherent extension of a vector of conditional probability bounds, An =
([αi , βi ], i ∈ Jn ), has been studied in [2] where, defining a suitable interval
[p◦ , p◦ ] ⊆ [0, 1], the following result has been obtained.
Theorem 5 Given a g-coherent assessment An = ([αi , βi ], i ∈ Jn ) on Fn =
{Ei |Hi , i ∈ Jn }, the extension [αn+1 , βn+1 ] of An to En+1 |Hn+1 is g-coherent
if and only if
[αn+1 , βn+1 ] ∩ [p◦ , p◦ ] 6= ∅ .
Remark 4 In the quoted paper an algorithm has been proposed to determine [p◦ , p◦ ]. As already observed in [2], starting with a g-coherent assessment An on Fn , the same algorithm can be exploited to make the ”leastcommittal” correction ([37]) of An , obtaining in this way the vector A∗n of
tightest probability bounds defined on Fn , associated with the ”natural extension” principle proposed in [42]. In order to determine A∗n one needs, for
each j ∈ Jn , to apply the algorithm to En+1 |Hn+1 = Ej |Hj , using the assessment An as probabilistic constraint on the conditional events of Fn . The
implementation with Maple V of the algorithms for checking coherence and
for propagation of precise and imprecise probability assessments has been
studied in [1].
Let A = (ahi ) be a m×n−matrix. Moreover, denote by x and y, respectively,
a row m−vector and a column n−vector. The vector x = (x1 , . . . , xm ) is said
semipositive if it is nonnegative and moreover
x1 + · · · + xm > 0 .
Then, we have ([23], Th. 2.10)
Theorem 6 Exactly one of the following alternatives holds.
Either the inequality xA ≥ 0 has a semipositive solution, or the inequality
Ay < 0 has a nonnegative solution.
Based on Theorem 6, we have
Theorem 7 The system (5) is solvable iff the following condition is satisfied:
M ax Gn |H0 ≥ 0 .
9
(8)
Proof. The proof is obtained by applying Theorem 6, with A = (ahi ), where
ahi = vhi − αi , xh = λh ≥ 0 , h ∈ Jm , i ∈ Jn ,
P
h∈Jm
λh = 1 , y k = s k ≥ 0 , k ∈ J n .
It immediately follows
Corollary 8 If the vector of lower bounds An on Fn is g-coherent, then the
condition (8) is satisfied.
By the same reasoning of Theorem 7, we have
Theorem 9 For every J ⊂ Jn , the system (SJ ) is solvable iff the following
condition is satisfied:
M ax GJ |HJ ≥ 0 .
(9)
Then, by Theorems 2, 7 and 9, it follows
Theorem 10 The vector of lower bounds An on Fn is g-coherent iff the
condition (9) is satisfied, for every J ⊆ Jn .
As shown by the previous theorem, the notion of g-coherence can be equivalently defined in terms of random gains, as usually made by other authors
in the framework of the betting criterion.
3
Main results
We now illustrate the basic idea, already sketched in [6], which allows to reduce the number of unknowns in the linear systems involved in the algorithm
for checking g-coherence (and also for propagation, see [5], [7]) of a vector of
probability bounds An defined on a family of conditional events Fn . We first
give some definitions.
Definition 2 Let G = {gj }j∈Jm be the set of possible values of the random
gain Gn |H0 . Then, a value gr ∈ G is said ”not relevant for the checking of
condition (8)”, or in short ”not relevant”, if there exists a set Tr ⊆ Jm \ {r}
such that:
M ax {gj }j∈Tr < 0 =⇒ gr < 0 .
(10)
10
Definition 3 A set GΓ = {gr }r∈Γ , with Γ ⊂ Jm , is said not relevant if,
∀r ∈ Γ, there exists a set Tr ⊆ Jm \ Γ such that (10) is satisfied.
We observe that M ax Gn |H0 = M ax {gj }j∈Jm . Then, based on the above
definition, we obtain
Theorem 11 Let be given gr ∈ G. If gr is not relevant, then
M ax {gj }j∈Jm ≥ 0 ⇐⇒ M ax {gj }j∈Jm \{r} ≥ 0 .
(11)
Proof. Of course:
M ax {gj }j∈Jm \{r} ≥ 0 =⇒ M ax {gj }j∈Jm ≥ 0 .
Moreover, by non relevance of gr one has
M ax {gj }j∈Jm \{r} < 0 =⇒ M ax {gj }j∈Tr < 0 =⇒ gr < 0 ,
so that
M ax {gj }j∈Jm \{r} < 0 =⇒ M ax {gj }j∈Jm < 0 ,
and the theorem is proved.
By a similar reasoning, we can prove
Theorem 12 Let be given a set GΓ = {gr }r∈Γ ⊂ G. If GΓ is not relevant,
then
(12)
M ax {gj }j∈Jm ≥ 0 ⇐⇒ M ax {gj }j∈Jm \Γ ≥ 0 .
To better illustrate the concept of non relevance, let be given a partition
{J 0 , J 00 , J 000 } of Jn , i.e. three pairwise disjoint subsets J 0 , J 00 , J 000 of Jn , with
J 0 ∪ J 00 ∪ J 000 = Jn . Moreover, assume that there exist three vectors V1 , V2 , V3
such that
v1i = αi , ∀ i ∈ J 00 ∪ J 000 ; v2i = αi , ∀ i ∈ J 0 ∪ J 000 ; v3i = αi , ∀ i ∈ J 000 ;
with:
v1i = v3i , ∀ i ∈ J 0 ;
v2i = v3i , ∀ i ∈ J 00 .
Then, by (4), for the gains g1 , g2 , g3 associated with the vectors V1 , V2 , V3 one
has
P
P
g1 = Pi∈J 0 si (v1i − αi ) = Pi∈J 0 si (v3i − αi ),
g2 = Pi∈J 00 si (v2i − αi ) =
i∈J 00 si (v3i − αi ),
g3 =
i∈J 0 ∪J 00 si (v3i − αi ) = g1 + g2 .
11
Based on the above relation, we observe that the value of g3 is not relevant
because
g1 < 0 , g2 < 0 =⇒ g3 < 0 ;
or, equivalently,
g3 ≥ 0 =⇒ g1 ≥ 0 or g2 ≥ 0 .
By the same reasoning, if there exist three vectors V1 , V2 , V3 and two positive
numbers a, b such that
g3 ≤ ag1 + bg2 ,
then g3 is not relevant.
Example. Assume that
V3 = xV1 + (1 − x)V2 ,
Then:
0<x<1.
g3 = Gn (V3 ) = Gn [xV1 + (1 − x)V2 ] =
= xGn (V1 ) + (1 − x)Gn (V2 ) = xg1 + (1 − x)g2 ,
so that g3 is not relevant.
It follows in general
Theorem 13 Given a subscript r ∈ Jm , if there exists a set Tr ⊆ Jm \ {r},
such that
X
gr ≤
aj gj , aj > 0 , ∀j ∈ Tr ,
(13)
j∈Tr
then gr is not relevant.
Proof. The claim follows by observing that, if there exists Tr ⊆ Jm \ {r} such
that (13) is satisfied, then the condition (10) is satisfied too, and hence gr is
not relevant. Remark 5 Two particular cases, where the condition of non relevance (13)
is easily checked, are given below.
• gr ≤ gh , for some h (i.e. Tr = {h}, ah = 1);
• gr ≤ gh + gk , for some h, k (i.e. Tr = {h, k}, ah = ak = 1).
More in general, we have
12
Theorem 14 Given r ∈ Jm , if there exist a set Tr ⊆ Jm \{r} and a constant
cr > 0 such that
gr ≤ cr M ax {gj }j∈Tr ,
(14)
then gr is not relevant.
Proof. The proof immediately follows by observing that, if there exists Tr ⊆
Jm \ {r} such that (14) is satisfied, one has
M ax {gj }j∈Tr < 0 =⇒ cr M ax {gj }j∈Tr < 0 =⇒ gr < 0 ,
i.e. the condition (10) is satisfied, and hence gr is not relevant.
We also have
Theorem 15 Given r ∈ Jm , let Th , Tk be two subsets of Jm \ {r}. If, for
some constant cr > 0, the following condition holds
P
gr ≤ M ax {gr + j∈Th gj , cr M ax {gj }j∈Tk } ,
(15)
then gr is not relevant.
Proof. Let us define Tr = Th ∪ Tk . Moreover, assume M ax {gj }j∈Tr < 0. It
follows
X
X
gj < 0 , M ax {gj }j∈Tk < 0 , gr > gr +
gj .
j∈Th
j∈Th
Then, by the hypothesis (15), one has gr ≤ cr M ax {gj }j∈Tk < 0. Therefore,
gr is not relevant.
Definition 4 A set T ⊂ Jm is said basic if the following property holds:
Basic Property. For every r ∈ Jm \ T there exists a set Tr ⊆ T such that the
condition (10) is satisfied.
A basic set T is said minimal if, for every T ⊂ T , the set T is not basic.
Remark 6 Notice that a set T ⊂ Jm is basic iff the set GJm \T = {gr }r∈Jm \T
is not relevant.
Based on the previous results, we obtain
13
Theorem 16 Let T ⊂ Jm be a basic set. Then
M ax {gj }j∈Jm ≥ 0
⇔
M ax {gj }j∈T ≥ 0 .
(16)
Proof. The proof follows by observing that
M ax {gj }j∈T < 0 =⇒ M ax {gj }j∈Jm \T < 0 =⇒ M ax {gj }j∈Jm < 0 . The following theorem establishes in equivalent, but more useful, terms the
result of Theorem 16.
Theorem 17 Let T ⊂ Jm be a basic set. Then, the condition (8) is satisfied
if and only if there exists a solution (λ1 , . . . , λm ) of the system (5), such that
λr = 0 , ∀r ∈ Jm \ T .
Proof. We will use again Theorem 6. For the sake of simplicity, we assume
T = Jk = {1, . . . , k}, with k < m. Let us consider the matrix A = (ahi ),
with
ahi = vhi − αi , h ∈ Jm , i ∈ Jn .
Then, denote respectively by A1 , A2 the following sub-matrices of A:
A1 = (ahi ) ,
h ∈ Jk ; i ∈ Jn ;
A2 = (ahi ) ,
h ∈ Jm \ Jk ; i ∈ Jn .
Moreover, denote by y the vector (s1 , . . . , sn )t , with si ≥ 0, ∀i ∈ Jn , by g the
vector of the gains (g1 , . . . , gm )t , and by g1 the vector (g1 , . . . , gk )t . We have
Ay = g ,
A1 y = g1 .
By Theorem 6, exactly one of the following alternatives holds:
1. the inequality A1 y < 0 has a nonnegative solution y;
2. the inequality x1 A1 ≥ 0 has a semipositive solution x1 = (x1 , . . . , xk ).
Assume M ax Gn |H0 ≥ 0. Then, being the set T basic, by ( 16) it follows
M ax {gj }j∈T ≥ 0. Therefore, the alternative A1 y < 0, y ≥ 0 does not hold.
This implies that the alternative x1 A1 ≥ 0, with x1 semipositive, holds.
Then, the vector x = (x1 , x2 ), with
x2 = (xk+1 , . . . , xm ) = (0, . . . , 0),
14
is a solution of xA ≥ 0, so that the system (5) has the solution (λ1 , . . . , λk , 0, . . . , 0),
with λr = Pkxr x ·
i=1 i
Conversely, if there exists a semipositive solution x = (x1 , x2 ) of xA ≥ 0,
with x2 = 0 , then the alternative x1 A1 ≥ 0, with x1 semipositive, holds.
Therefore, the alternative A1 y < 0, y ≥ 0 does not hold, so that:
g1 = A1 y 6< 0 , ∀ y ≥ 0 .
Then: M ax {gj }j∈T ≥ 0 , and hence: M ax Gn |H0 ≥ 0 .
We observe that results analogous to the previous ones can be obtained for
the case of upper bounds.
4
Basic sets and g-coherence checking
Based on Theorem 17, we make the following remarks:
• by examining the gains g1 , . . . , gm , we can determine a (possibly minimal) basic set T ⊂ Jm . Then, we can replace the system ( 5) by a
system (SnT ), associated with the set T , which has a reduced number
of unknowns.
• to check the condition (8) we examine the reduced system (SnT ).
Notice that, in the particular case in which a basic set T ⊂ Jm doesn’t exist,
we need to use the system (Sn ). In this case, as (SnJm ) = (Sn ), we may say
that T = Jm .
Before giving the algorithm for the checking of g-coherence, we need some
theoretical remarks. Given a subset S 0 ⊆ S, let I00 be the set defined as in
(7), where S is replaced by S 0 , and denote by (F00 , A00 ) the pair associated
with I00 . Of course, it is I0 ⊆ I00 . Then, we have
Theorem 18 Given S 0 ⊆ S, the imprecise assessment An on Fn is
g-coherent if and only if the following conditions are satisfied:
1. the system (Sn ) is solvable ;
2. if I00 6= ∅, then A00 is g-coherent.
15
Proof. Of course, if An is g-coherent, then the conditions 1 and 2 hold.
Conversely, as I0 ⊆ I00 the g-coherence of A00 implies the g-coherence of A0 .
Then, from Theorem 4 it follows the g-coherence of An . We denote respectively by ΛT and ST the vector of unknowns and the set of
solutions of the system (SnT ). Moreover, we denote by ΦTj the linear function
defined as in ( 6), where Λ is replaced by ΛT , by I0T the (strict) subset of Jn
defined as
(17)
I0T = {j ∈ Jn : Mj = M axΛT ∈ST ΦTj (ΛT ) = 0} ,
and by (F0T , AT0 ) the pair associated with the set I0T .
Remark 7 Notice that for each solution ΛT ∈ ST there exists a solution
Λ ∈ S with suitable components equal to zero. Therefore, with the set ST is
associated a pair (S 0 , I00 ), with S 0 ⊆ S, I00 = I0T ⊇ I0 .
Based on the above remark, we have
Theorem 19 The imprecise assessment An on Fn is g-coherent if and only
if the following conditions are satisfied:
1. the system (SnT ) is solvable ;
2. if I0T 6= ∅, then AT0 is g-coherent.
Proof. Recalling that the solvability of (SnT ) amounts to the solvability of
(Sn ), the proof follows by observing that the conditions 1 and 2 are equivalent
to the analogous conditions in Theorem 18, where S 0 is the subset associated
with the set ST . Based on Theorem 19, the checking of the g-coherence can be made by the
following modified version of Algorithm 1.
Algorithm 2 Let be given the triplet (Jn , Fn , An ).
1. Determine a basic set T ⊂ Jm . If T doesn’t exist, then set T = Jm .
2. Construct the system (SnT ) and check its solvability.
3. If the system (SnT ) is not solvable then An is not g-coherent and the
procedure stops; otherwise, compute the set I0T .
4. If I0T = ∅ then An is g-coherent and the procedure stops, otherwise set
(Jn , Fn , An ) = (I0T , F0T , AT0 ) and repeat steps 1-3.
16
The study of some methods for determining, at each step of Algorithm 2, a
basic set T has been made in [5] (see also [7]). In the same paper also the
propagation of An has been examined and an algorithm based on reduced
linear systems has been developed.
For the sake of a better understanding of the proposed procedure, we sketch
the main idea to determine a basic set T . Recalling (4) and defining the
linear function
n
X
si zi ,
f (z1 , . . . , zn ) =
i=1
we have
gr = f (Vr − An ),
gh = f (Vh − An ),
gk = f (Vk − An ),
and it can be proved that
gr ≤ gh ⇐⇒ Vr ≤ Vh ,
gr ≤ gh + gk ⇐⇒ Vr ≤ Vh + Vk − An .
Based on the above remarks, it is easy to construct an algorithm which,
making comparisons among the vectors Vr , Vh , Vk , detects the particular cases
like gr ≤ gh , or gr ≤ gh + gk , in which the condition of non relevance (13)
is easily checked. Moreover, as shown by the theorem below ([5], [7]), to
determine T we can use an iterative approach.
Theorem 20 Given a basic set T1 ⊂ Jm and a subset T2 ⊂ T1 , assume that,
for every r ∈ T1 \ T2 , there exists Tr ⊆ T2 satisfying the condition (10). Then,
T2 is a basic set and hence
M ax Gn |H0 ≥ 0 ⇐⇒ M ax{gj }j∈T2 ≥ 0 .
(18)
Based on the previous theorem, we can determine a (basic) set T1 such that,
for every r ∈ Jm \ T1 , there exist h ∈ T1 , k ∈ T1 satisfying the inequality
gr ≤ gh , or gr ≤ gh + gk . By iterating this procedure we can determine
T2 ⊂ T1 such that, for every k ∈ T1 \ T2 , there exist s ∈ T2 , t ∈ T2 satisfying
the inequality gk ≤ gs , or gk ≤ gs + gt . Exploiting the remarks above we can
construct a procedure to determine a sequence of basic subsets T1 , . . . , Ti ,
with T1 ⊃ T2 ⊃ · · · ⊃ Ti−1 = Ti . Then T = Ti .
17
Remark 8 We observe that, when checking the theoretical conditions which
allow to determine a basic set T , in order to detect the inequalities like
gr ≤ gh + gk we need to build the vectors Vr , Vh , Vk associated with the
atoms Cr , Ch , Ck . This is time consuming, even if it should be noted that the
obtained ”small” system SnT is easily checked for its solvability. Therefore, a
crucial problem is that of determining methods which allow to construct the
set T in a direct way. A relevant example of this kind is given in [3], where
the direct procedure proposed in [36] for the case of conjunctive conditional
events (see Example 3) is characterized
P in terms of random gains, exploiting
the condition (13) in the form gr = j∈Tr gj . This means that, in the case
of conjunctive events, the two methods could be combined: in a first phase,
based on the direct procedure given in [36], we would obtain a basic set T1 ;
then, applying our method, we could obtain a further reduction of variables
leading to a basic set T2 ⊆ T1 . In this way, in the conjunctive case, the
efficiency of our method should improve.
5
Some examples
We now illustrate the methods proposed in the previous sections by examining some examples.
Example 1 In this example we show that the logical relations studied in [8]
are not satisfied. Given the logically independent events A1 , A2 , A3 , A4 and
the family F5 = {Ei |Hi , i = 1, . . . , 5} defined as
F5 = {A1 |A4 , A2 |A4 , Ac3 |A2 A4 , A4 |A2 , A1 Ac3 |A2 A4 },
let A5 = (α1 , α2 , α3 , α4 , α5 ) be a vector of lower bounds on F5 , with
αi ∈ (0, 1), i = 1, . . . , 5. Let us consider the checking of g-coherence of A5 .
In order to split our problem into subproblems, as αi ∈ (0, 1) we can check
the following conditions, respectively denoted by (a3), g1), (h) in section 5 of
[8].
W
1. ∅ =
6 El Hl 6⊆ j6=l Hj , l = 1, . . . , 5;
V
V
2. Ej Hj l6=j,k Hlc 6= ∅, Ek Hk l6=j,k Hlc 6= ∅, {j, k} ⊂ {1, . . . , 5}, . . .
V
3. Ej Hj Ek Hk Er Hr l6=j,k,r Hlc 6= ∅, {j, k, r} ⊂ {1, . . . , 5}.
18
Notice that we don’t consider the condition (i) given in [8] because in our
example the numerical values of α1 , . . . , α5 are not specified. It can be verified
that in this example the logical conditions above are not satisfied.
Applying our method, we observe that H0 = A2 ∨ A4 and for the random
gain G5 we have
G5 = s1 A4 (A1 − α1 ) + s2 A4 (A2 − α2 ) + s3 A2 A4 (Ac3 − α3 )+
+s4 A2 (A4 − α4 ) + s5 A2 A4 (A1 Ac3 − α5 ).
Based on (2), we obtain the following 8 distinct atoms contained in H0
C1 = E1 H1 E2 H2 E3c H3 E4 H4 E5c H5 = A1 A2 A3 A4 ,
C2 = E1c H1 H2c H3c E4c H4 H5c = A1 A2 A3 Ac4 ,
C3 = E1 H1 E2 H2 E3 H3 E4 H4 E5 H5 = A1 A2 Ac3 A4 ,
C4 = H1c H2c H3c E4c H4 H5c = A1 A2 Ac3 Ac4 ,
C5 = E1 H1 E2c H2 H3c H4c H5c = A1 Ac2 A3 A4 ,
C6 = E1c H1 E2 H2 E3c H3 E4 H4 E5c H5 = Ac1 A2 A3 A4 ,
C7 = E1c H1 E2 H2 E3 H3 E4 H4 E5c H5 = A1 A2 A3 A4 ,
C8 = E1c H1 E2c H2 H3c H4c H5c = Ac1 Ac2 A3 A4 .
Then the starting system should have 8 unknowns. However, we can observe
19
that the values of G5 |H0 are
g1 = s1 (1 − α1 ) + s2 (1 − α2 ) − s3 α3 + s4 (1 − α4 ) − s5 α5 ,
g2 = −s1 α1 − s4 α4 ,
g3 = s1 (1 − α1 ) + s2 (1 − α2 ) + s3 (1 − α3 ) + s4 (1 − α4 ) + s5 (1 − α5 ),
g4 = −s4 α4 ,
g5 = s1 (1 − α1 ) − s2 α2 ,
g6 = −s1 α1 + s2 (1 − α2 ) − s3 α3 + s4 (1 − α4 ) − s5 α5 ,
g7 = −s1 α1 + s2 (1 − α2 ) + s3 (1 − α3 ) + s4 (1 − α4 ) − s5 α5 ,
g8 = −s1 α1 − s2 α2 .
Then, it is easy to verify that for each h 6= 3 it is g3 ≥ gh . As an example
g3 = g7 + s1 + s5 ≥ g7 , so that g7 is not relevant. Therefore T = {3} is a basic
set. Then, the starting system has just one unknown, i.e. λ3 (associated to
C3 ), and is trivially solvable. Moreover, as C3 ⊆ Hi , i = 1, . . . , 5, then I0T = ∅
and hence A5 is g-coherent.
Example 2 We give an example in which applying Algorithm 2, after determining a basic set T , it results I0T 6= ∅. Given two logically independent events A, B, let us examine the g-coherence of the vector of lower
bounds A2 = (α1 , α2 ), with 0 < αi ≤ 1, i = 1, 2, defined on the family
F2 = {B c |A, Ac ∨ B|Ω}. We observe that H0 = Ω. Moreover, the constituents are
C1 = AB, C2 = AB c , C3 = Ac .
Concerning the random gain, one has
G2 |H0 = G2 = s1 A(B c − α1 ) + s2 (Ac ∨ B − α2 ),
so that the values associated with the constituents are
g1 = −s1 α1 + s2 (1 − α2 ), g2 = s1 (1 − α1 ) − s2 α2 , g3 = s2 (1 − α2 ).
20
As g1 ≤ g3 , it follows that g1 is not relevant. Therefore T = {2, 3} is a
basic set. Starting Algorithm 2 with the system S2T , in the case α2 = 1 it
results I0T = {1} =
6 ∅. Then, the algorithm restarts with the pair (F0T , AT0 ),
where F0T = {B c |A}, AT0 = (α1 ). Of course, the output of the algorithm is:
”A2 g − coherent”.
Example 3 We consider an example given in [36], where an efficient global
procedure has been proposed to propagate conditional probability bounds
for families of conjunctive conditional events (see the family F3 below). In
[38] the procedure has been generalized to the case of conditioning events
(possibly) having probability zero. Given the vector of upper bounds
B3 = (0.2, 0.2, 0.2)
on the family
F3 = {B|A, C|AB, D|C},
let us consider the checking of g-coherence of B3 and its extension to BCD|A.
The constituents contained in H0 = A ∨ C are respectively
C1 = ABCD , C2 = ABCDc , C3 = ABC c ,
C4 = AB c C c ,
c
c
c
c
C5 = A CD , C6 = A CD , C7 = AB CD , C8 = AB c CDc .
Based on Remark 3, the random gain is
G3 = −σ1 A(B − 0.2) − σ2 AB(C − 0.2) − σ3 C(D − 0.2) ,
with σ1 ≥ 0, σ2 ≥ 0, σ3 ≥ 0. Then, the values of G3 |H0 associated with
C1 , . . . , C8 are:
g1 = −0.8σ1 − 0.8σ2 − 0.8σ3 ,
g3 = −0.8σ1 + 0.2σ2 ,
g6 = 0.2σ3 ,
g2 = −0.8σ1 − 0.8σ2 + 0.2σ3 ,
g4 = 0.2σ1 ,
g7 = 0.2σ1 − 0.8σ3 ,
g5 = −0.8σ3 ,
g8 = 0.2σ1 + 0.2σ3 .
As a preliminary remark, we observe that in our case it is g4 = 0.2σ1 ≥ 0
(an analogous observation holds for g6 and g8 ), and therefore the condition
M ax G3 |H0 ≥ 0 is surely satisfied. If we apply our method, based on
the condition ( 13) and in particular on Remark 5, we can verify that the
following conditions are satisfied:
g1 ≤ g2 , g2 ≤ g3 + g6 , g5 ≤ g6 , g7 ≤ g4 , g8 = g4 + g6 .
21
Then, T = {3, 4, 6} is a (minimal) basic set, so that
M ax {gj }j∈J8 ≥ 0 ⇔ M ax {gj }j∈T ≥ 0 .
In other words, the solvability of the system (S3 ), which in our case has 8
unknowns, is equivalent to the solvability of the corresponding system (S3T )
which has 3 unknowns. Based on Algorithm 2, it can be verified that (S3T )
is solvable and that I0T = ∅. Therefore, B3 is g-coherent.
The notion of basic set can be also exploited to extend g-coherent assessments, see [5], [7]. In this case we need to suitably modify the algorithm
given in [2]. To illustrate this aspect, let us examine the propagation of B3
to BCD|A. The starting set of constituents is the same as before. Moreover,
given the assessment
P (B|A) ≤ 0.2, P (C|AB) ≤ 0.2, P (D|C) ≤ 0.2, P (BCD|A) = p,
We must determine the interval [p◦ , p◦ ] of the values p which are coherent
extensions of B3 to BCD|A. The gains associated with the constituents are
g1 = −0.8σ1 − 0.8σ2 − 0.8σ3 + (1 − p)s4 ,
g2 = −0.8σ1 − 0.8σ2 + 0.2σ3 − ps4 ,
g3 = −0.8σ1 + 0.2σ2 − ps4 , g4 = 0.2σ1 − ps4 , g5 = −0.8σ3 ,
g6 = 0.2σ3 , g7 = 0.2σ1 − 0.8σ3 − ps4 , g8 = 0.2σ1 + 0.2σ3 − ps4 ,
with σ1 , σ2 , σ3 nonnegative and s4 arbitrary. We still have
g2 ≤ g3 + g6 ,
g5 ≤ g6 ,
g7 ≤ g4 ,
g8 = g4 + g6 ,
while the inequality g1 ≤ g2 is no more satisfied, so that T = {1, 3, 4, 6} and
the reduced system has 4 (instead of 8) unknowns. Based on the algorithm
given in [5], [7], it can be verified that [p◦ , p◦ ] = [0, 0.04].
Remark 9 Notice that with our procedure we improve the results given in
[36], where the reduced linear system has 6 unknowns. In fact, from our
point of view, the method proposed in [36] amounts to observe that
g7 = g4 + g5 ,
g8 = g4 + g6 ,
from which the (not minimal) basic set {1, 2, 3, 4, 5, 6} is obtained.
22
Example 4 Given the assessment
1 1
1 1
1 1
A3 = ([ , ], [ , ], [ , ])
5 4 10 5 10 4
on
F3 = {B|AC, C|(A ∨ B), D|(B ∨ C)},
we examine the g-coherence of A3 and its extension to A|BCDc . The constituents contained in H0 = A ∨ B ∨ C are
C1 = ABCD , C2 = ABCDc , C3 = BC c D , C4 = BC c Dc ,
C5 = AB c CD , C6 = AB c CDc , C7 = AB c C c , C8 = Ac BCD ,
C9 = Ac BCDc , C10 = Ac B c CD , C11 = Ac B c CDc .
Based on Remark 3, the random gain G3 is given by the expression
AC[s1 (B − 15 ) − σ1 (B − 14 )] + (A ∨ B)[s2 (C −
+ (B ∨ C)[s3 (D −
1
)
10
1
)
10
− σ2 (C − 15 )] +
− σ3 (D − 14 )] ,
with si ≥ 0, σi ≥ 0, i = 1, 2, 3. Then, the values of G3 |H0 associated with
23
C1 , . . . , C11 are:
g1
9
9
= ( 45 s1 − 34 σ1 ) + ( 10
s2 − 45 σ2 ) + ( 10
s3 − 34 σ3 ) ,
g2
9
1
= ( 45 s1 − 34 σ1 ) + ( 10
s2 − 45 σ2 ) + (− 10
s3 + 14 σ3 ) ,
g3
1
9
= (− 10
s2 + 15 σ2 ) + ( 10
s3 − 34 σ3 ) ,
g4
1
1
= (− 10
s2 + 15 σ2 ) + (− 10
s3 + 14 σ3 ) ,
g5
9
9
= (− 51 s1 + 14 σ1 ) + ( 10
s2 − 45 σ2 ) + ( 10
s3 − 34 σ3 ) ,
g6
9
1
= (− 15 s1 + 14 σ1 ) + ( 10
s2 − 45 σ2 ) + (− 10
s3 + 14 σ3 ) ,
g7
1
= − 10
s2 + 15 σ2 ,
g8
9
9
= ( 10
s2 − 45 σ2 ) + ( 10
s3 − 34 σ3 ) ,
g9
9
1
= ( 10
s2 − 45 σ2 ) + (− 10
s3 + 14 σ3 ) ,
g10 =
9
s
10 3
− 34 σ3 ,
1
g11 = − 10
s3 + 14 σ3 .
We first observe that
g10 < 0 ⇒ g11 ≥ 0 ,
g11 < 0 ⇒ g10 ≥ 0 ,
so that M ax{g10 , g11 } ≥ 0 and the condition M ax G3 |H0 ≥ 0 is surely
satisfied. Applying our method, we obtain
g3 = g7 + g10 ,
g4 = g7 + g11 ,
so that g3 and g4 are not relevant.
Remark 10 Given two numbers α and β, with 0 ≤ α ≤ β ≤ 1, and the
functions
f (s, σ) = (1 − α)s − (1 − β)σ ,
24
g(s, σ) = −αs + βσ ,
as it can be easily verified, the following properties hold:
f (s, σ) < 0 =⇒ σ > s ,
g(s, σ) < 0 =⇒ σ < s ,
therefore f and g cannot be both negative.
Now, we observe that
with
g1 = δ1 + g8 ,
g5 = δ2 + g8 ,
g2 = δ1 + g9 ,
g6 = δ2 + g9 ,
4
3
δ1 = s1 − σ1 ,
5
4
1
1
δ2 = − s1 + σ1 .
5
4
Then, based on Remark 10, the quantities δ1 , δ2 cannot be both negative,
so that
g8 ≤ M ax{g1 , g5 } , g9 ≤ M ax{g2 , g6 } ,
and then g8 and g9 are not relevant. Moreover, one has
g10 ≤ M ax{g3 , g8 } ≤ M ax{g7 + g10 , M ax{g1 , g5 }} ,
g11 ≤ M ax{g4 , g9 } ≤ M ax{g7 + g11 , M ax{g2 , g6 }} .
Thus, by Theorem 15, g10 and g11 are not relevant too.
Then, the set T = {1, 2, 5, 6, 7} is basic and the solvability of (S3 ), which has
11 unknowns, is equivalent to the solvability of (S3T ) which has 5 unknowns.
Therefore, the checking of the g-coherence of A3 , can start by replacing
the system (S3 ) by the system (S3T ), associated with the basic set T . By
Algorithm 2, one has I0T = ∅ and hence A3 is g-coherent.
Concerning the extension of A3 to A|BCDc , it can be verified that for the
assessment
1
5
1
10
≤ P (B|AC) ≤
1
4
,
≤ P (D|(B ∨ C)) ≤
1
10
1
4
≤ P (C|(A ∨ B)) ≤
,
1
5
,
P (A|BCDc ) = p ,
defined on F3 ∪ {A|BCDc }, the condition g9 ≤ M ax{g2 , g6 } is no more
satisfied. Then, the starting system has 6 (instead of 11) unknowns.
Based again on the modified algorithm given in [5], [7], it can be verified that
[p◦ , p◦ ] = [0, 1].
25
6
Conclusions
In this paper, based on the concept of g-coherence, we have proposed the
management of uncertain information by imprecise probabilities. As well
known, the checking of consistency has in general an exponential complexity. Then, with the aim of reducing the computational difficulties, we have
studied a general methodology on which basis the checking of g-coherence
can be made by linear systems with a reduced number of unknowns. Our
method generalizes the procedure proposed in [36] and could be integrated
with the approaches proposed in [8], [9], [10], which exploit an idea first suggested in [18]. As shown by the given examples, the efficiency of our method
is strictly related to the choice of a good strategy for determining a basic
set T . This problem has been studied in [5], [7], where some results and
algorithms have been given. We notice that the comparisons among gains
based on our theoretical conditions require that the vectors defined in (3) be
built. In general, this may be time consuming, even if the ”small” system
(SnT ), obtained at each step, is easily checked for its solvability. The best
solution would be the characterization in terms of random gains of direct
methods for the construction of the set T . A relevant example in this sense
is the characterization given in [3] of the procedure proposed in [36] for the
case of conjuntive conditional events. As observed in Remark 8, in the conjunctive case, combining our approach with the procedure given in [36] we
could improve the efficiency of our method. Further work is in progress with
the aim of determining in general, in an efficient and direct way, a (possibly
minimum) basic set T . We also remark that our approach could be used
in combination with efficient methods not based on coherence, like column
generation techniques, proposed by other authors; see e.g. [33], [34]. Finally,
an important topic of future research concerns a thorough discussion of the
efficiency of the proposed methodology and the analysis of its computational
complexity, through a theoretical study and/or through experimental results.
Acknowledgments
The authors are grateful to the anonymous referees for their valuable criticisms and suggestions.
26
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