An algorithm for coherent conditional probability

An algorithm for coherent conditional probability
assessments
(Un algoritmo per la verifica della coerenza di assegnazioni di probabilitá condizionate)
Andrea Capotorti
Barbara Vantaggi
Dip di Matematica, Perugia, Italy
e-mail: [email protected]
Dip. di Statistica, Perugia, Italy
e-mail: [email protected]
1 Introduction
Conditional probability assessments are the most general and proper tools to represent
the uncertainty on a finite domain of events without structure. In fact in practical
problems the available information is not so rich to be representable by an algebraic
structure. Moreover the numerical assessments are best expressed in connection with
the hypothetical occurrence of particular events (like possible observations, symptoms,
etc.). These assessments can not be arbitrarily given, but they must satisfy a coherence
principle. Coherent conditional probabilities are fully characterized in literature (Coletti
(1994) and Coletti & Scozzafava (1996)), but these characterizations involve a partition
of the sure event, whose cardinality could make the problem computationally intractable.
In this paper we detect some cases where the checking of coherence can be decomposed
into subproblems decreasing the global complexity. We also propose a relevant algorithm.
2 Coherence
We preliminarily give the main notions and results useful to face the problem. Let
F = {Ei |Hi }ni=1 be a general finite family of conditional events, we denote by CF the
W
set of atoms (contained in ni=1 Hi ) generated by the set of unconditional events UF =
{E1 , H1 , . . . , En , Hn }.
Let P : F −→ [0, 1] be a numerical assessment, it is a coherent conditional probability
if it can be extended to a conditional probability over AF × AF \ {∅}, where AF is the
algebra generated by UF .
For the coherent conditional probability there is the following characterization theorem
(see Coletti (1994))
Theorem: Let P : F −→ [0, 1] be a numerical assessment. The following propositions
are equivalent:
• P is a coherent conditional probability
• there exists at least one finite class of unconditional probabilities {P0 , P1 , . . . Pk }
such that
1. P0 is defined over AF , while for α = 1, . . . , k Pα is defined over AFα with
Fα = {Ei |Hi : Pα−1 (Hi ) = 0}
1
2. for all Ei |Hi there exists an unique α such that Pα (Hi ) > 0 and
P
C ⊆E H Pα (Cr )
P (Ei |Hi ) = P r i i
Cr ⊆Hi Pα (Cr )
with Cr ∈ CFα .
(note that we omit the conjunction symbol ∧ among pairs of events, but we will use it
when the events have an index varying in an index-set)
From a computational point of view, checking coherence is equivalent (as shown in
Coletti (1994)) to the compatibility of a sequence of linear systems S0 , S1 , . . . , Sk like
Sα =
X
X


xαr = P (Ei |Hi )
xαr




Cr ⊆Ei Hi
Cr ⊆Hi




 X
, if Pα−1 (Hi ) = 0
xαr = 1



Cr ∈CFα






 α
xr ≥ 0
where xαr ’s are the unknowns associated to Pα (Cr ) with Cr ∈ CFα .
Therefore, checking coherence is reduced to solve a sequence of systems Sα (starting
with S0 ) that has an exponential number of unknowns with respect to the number of
events. It could seem, at first sight, that the best situation is that one solvable in one
step, i.e. when S0 admits a strictly positive solution. Nevertheless in this situation the
large number of unknowns could make the problem not manageable. So, when it is
possible, it could be better to decompose the problem into smaller systems.
The aim of our contribution is to show how (in some practical situations) it is possible,
with the help of zero probabilities, to reduce the complexity of the problem (as suggested
in Coletti & Scozzafava (1997)).
It is in fact possible to characterize local coherence configurations warranting the
coherence of the full assessment.
3 Local coherence
We formally introduce, now, the notion of local coherence that will be used to decompose
the checking of coherence procedure.
Definition: Let P : F −→ [0, 1] be a numerical assessment, let G ⊂ F be a subfamily
of the starting set of events; P is locally coherent on G if the following system
SG =









X
Cr ⊆ Ei Hi
∗
Cr ∈ CG
xr = P (Ei |Hi )
X

xr = 1



∗

Cr ∈CG



X
xr
C r ⊆ Hi
∗
Cr ∈ CG
xr ≥ 0
(where CG∗ = {Cr ∈ CG : Cr ⊆ Hjc ∀Hj ∈ UF\G }) has a solution such that for each
P
Ei |Hi ∈ G
Cr ⊆Hi xr ≥ 0.
Note that this definition is different from the coherence of the restriction P|G . In fact
the local coherence requires the coherence of P|G with the additional request to give zero
T
probability to CG CF\G .
The relationship between coherence and local coherence is given by the next proposition.
Proposition : Let P : F −→ [0, 1] locally coherent on G, then
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P is coherent if and only if P|F \G is coherent.
On the contrary the coherence of P and the local coherence of P on G do not imply the
local coherence of P on F \ G.
Note also that the notion of local coherence is strictly connected with the logical relations
among events. In fact the local coherence on G and the coherence of P|G coincide only
if G is logically upper independent of F \ G (see de Finetti (1974)). We will not deal
with this property, because it is not easily interpretable and its checking needs anyway
to resort to the set CF of atoms, exactly what we want to avoid.
4 The algorithm
We will describe now configurations on F that allow us to propose an algorithm to
check local coherence by computing only the atoms generated by the subfamilies. This
is practically achieved by finding an iterative procedure. At each step the domain of P is
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reduced to Rk = F \ j<k Gk (with G0 = {∅}) where the Gk ’s are subfamilies characterized
by sufficient conditions that ensure the local coherence of P on them. Such subfamilies
do not represent constraints for the coherence on the families Rk since, by the above
Proposition, they can be neglected and the problem is reduced to check the coherence
on the remaining subset Rk+1 .
We have the best situation when F is decomposable in n subfamilies (one for each
event); in fact, complexity is linear with respect to the cardinality of F. This possibility
is given when we can detect a chain of events Ei1 |Hi1 , . . . , Ein |Hin , with each Eik |Hik
unconstrained by the rest on Rk , so, starting with the first, we can neglect one event
each time. We need to detect the local coherence of P|R on a single event and this is
k
possible when
a1) ∅ 6= Eik Hik 6⊆
a2) ∅ 6= Eick Hik 6⊆
_
Hij
j6=k
_
Hij .
j6=k
If P (Ei |Hi ) = 0 (or, alternatively, P (Ei |Hi ) = 1) it is sufficient that only the condition
a2) (respectively a1)) holds.
Note that the two conditions a1) and a2) are trivially satisfied when the conditioning
events Hi are pairwise disjoint.
If P is not locally coherent on the “singletons” {Ei |Hi } we can search for its local
coherence on subfamilies with two elements Gk = {Ek1 |Hk1 , Ek2 |Hk2 }.
We have that P|R is locally coherent on Gk if
k
b1) ∅ 6=
V
i=1,2 Eki Hki
V
c
j6=1,2 Hkj
6=
V
i=1,2 Hki
b2) one of the following properties holds
b2I) P (Ek1 |Hk1 ) = P (Ek2 |Hk2 )
or
V
c
j6=1,2 Hkj
b2II) P
(Ek1 |HkV1 ) > P (Ek2 |H
k2 ) and
V
c
c
Ek2 Hk2 j6=2 Hkj 6= ∅ or Ek1 Hk1 j6=1 Hkcj 6= ∅
c1) each event in Gk verifies condition a2)
c2) ∅ 6=
V
i=1,2 Eki Hki
V
c
j6=1,2 Hkj
3
or
d) ∅ 6= (
W
V
e) ∅ 6= (
W
V
or
i=1,2 Eki Hki )
i=1,2 Eki Hki )
W
V
V
V
c
j6=1,2 Hkj
6= (
c
j6=1,2 Hkj
⊂(
i=1,2 Hki )
i=1,2 Hki )
c
j6=1,2 Hkj
c
j6=1,2 Hkj
and
P
i=1,2 P (Ei |Hi )
≤1
We can give a further sufficient condition for a subfamily Gk = {Ek1 |Hk1 , . . . , Ekr |Hkr },
with the following property
d1) let 0 ≤ l ≤ r be an index such that
Vl
Vk
V
c
c
i=1 Eki Hki j=l+1 Ekj Hkj Rk+1 Hks 6= ∅ and
V
c
c
for all i = 1, . . . , l Eki Hki j6=i Hkj 6= ∅ and
for all i = l + 1, . . . , r
V
V
Pr
Eki Hki ri6=j=1 Ekcj Hkj Rk+1 Hkcr 6= ∅ and
l+1 P (Eki |Hki ) < 1.
We can sketch an algorithm based on the previous sufficient conditions, starting with
a family F = {E1 |H1 , . . . , En |Hn } with some logical relationship and an assessment
P : F −→ [0, 1]:
i) put R = F and “coherence”= true ;
ii) repeat until R 6= {∅}
if there exists an event Eki |Hki ∈ R satisfying a1) and a2)
then put G = {Eki |Hki } ;
else if there exists a pair {Ek1 |Hk1 , Ek2 |Hk2 } ⊂ R satisfying [b1) and b2)] or
[c1) and c2)] or [d)] or [e)]
then put G = {Eki |Hki }i=1,2 ;
else if there exists a subfamily {Ek1 |Hk1 , . . . , Ekr |Hkr } satisfying d)
then put {G} = {Eki |Hki }i=1,...,r ;
else if P|R is coherent
then put G = R ;
else
put “coherence”= false and G = R ;
put R = R \ G ;
iii) if “coherence” = true
then answer “the assessment is coherent”
else
answer “the restriction of P on” G “is not coherent”
where the variable “coherence” represents the coherence of the entire assessment P and
in the last option of step ii) there could be made a call to a subroutine that checks
coherence for P|R using the general algorithm proposed in Coletti & Scozzafava (1996).
References
Coletti, G. (1994) Coherent Numerical and Ordinal Probabilistic Assessments. IEEE Trans. on
Systems, Man, and Cybernetics 24(12): 1747-1754.
Coletti, G., Scozzafava R. (1996) Characterization of Coherent Conditional Probabilities as a
Tool for their Assessment and Extension. Int. Journ. of Uncertainty, Fuzziness and
Knowledge-Based Systems, 4(2): 103-127
Coletti, G., Scozzafava R. (1997) Exploiting Zero Probabilities. Proc. of EUFIT’97, Aachen,
Germany, ELITE Foundation: 1499-1503.
de Finetti, B. (1974) Theory of probability, vol. 1-2, Wiley, New York.
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