When upper conditional probabilities are conditional possibility measures
Introduction
Probability theory has a fundamental role for dealing with uncertainty, even if sometimes the decision maker has only an incomplete, partial, imprecise probabilistic information. In such situations it is unavoidable to consider a set of probabilities together with the corresponding envelopes, which turn out to be non-additive uncertainty measures (see, e.g., [12], [20], [21], [27], [43]).
For example, prior probabilities are crucial in Bayesian statistics, even if sometimes the elicitation of the prior is not an easy task. For this, in some applications (for instance, in decision problems with multiple agents [15], in the reconstruction of images or when misclassified variables are present [38]) instead of assessing a single prior probability, one can select a suitable family of priors.
Concerning the relationship between probability theory and the chosen family of non-additive measures, it is well-known that the upper [lower] envelope of the coherent extensions of a coherent probability is subadditive [superadditive] but is not necessarily 2-alternating [2-monotone] (see, for instance, [13], [43] and Example 1 in [2]). Nevertheless, starting from a probability on an algebra the upper [lower] envelope of its extensions to another algebra is a plausibility [belief] function (see [9], [18], [20], [22], [28], [43]). In [25] it is provided a characteristic property for an upper probability to coincide with a possibility measure, which led in [10], [11], [12] to characterize possibility measures as upper envelopes of the extensions of a probability when a logical condition (weak logical independence) holds among the involved algebras.
Starting from the seminal works [24], [47], possibility measures have received a lot of attention, due to their key role in fuzzy set theory, soft computing and decision theory. The debate on the conditioning for possibility measure or, more generally, for non-additive uncertainty measures is still an open issue, even if several conditioning notions have been introduced in literature (see, e.g., [1], [7], [14], [16], [20], [24], [26], [30]).
A first aspect in which the given conditioning notions differ is how the conditional measure is defined: essentially, either starting from a non-additive unconditional measure through a suitable operation or directly as a function satisfying a set of axioms. Furthermore, conditional uncertainty measures can be viewed as specific lower or upper conditional probabilities, so the relation with probability theory can be investigated. In this line, in [5] we have generalized the Bayesian rule of conditioning for belief functions on finite domains, originally proposed by [20], [28], [32].
In this paper we go further in this investigation by studying when conditional plausibility functions are conditional possibility measures. In detail, focusing on a finite Boolean algebra , we introduce the notion of Bayesian conditional possibility (or B-conditional possibility for short) which is obtained through a generalization of the Bayesian conditioning rule [20], relying on a linearly ordered class of possibility measures. For any B-conditional possibility measure defined on , with , there exists a finite Boolean algebra and a conditional probability defined on such that the upper envelope of its extensions coincides with on . Furthermore, we investigate the conditions under which the upper envelope of the extensions of a full conditional probability is a full B-conditional possibility. It is shown that weak logical independence of two finite algebras and , is a sufficient (but not necessary) condition for the upper envelope of the conditional probabilities extending a conditional probability defined on to be a B-conditional possibility on . Contrary to the unconditional case, there are B-conditional possibilities on for which there is no finite algebra , weakly logically independent of , such that a conditional probability on gives rise to as upper envelope of its extensions (see Example 6). Thus we introduce a weaker form of weak logical independence that allows us to provide a necessary and sufficient condition for the upper envelope of the extensions of a conditional probability defined on to be a B-conditional possibility measure on .
The paper is structured as follows. In Section 2 some preliminary notions on conditional probabilities and their extensions are recalled, while in Section 3 the problem of conditioning for possibility measures is introduced. Section 4 presents the generalized Bayesian conditioning rule which gives rise to full B-conditional possibilities. Section 5 copes with the logical notion called weak logical independence. Such condition implies a conditional form of nesting giving rise to a full B-conditional possibility as the upper envelope of the extensions of any full conditional probability. Since weak logical independence turns out to be only a sufficient condition for obtaining a full B-conditional possibility, in Section 6 a necessary and sufficient condition is given. Finally, in Section 7 some conclusions are drawn.
Section snippets
Conditional probability and its enlargement
Let be a Boolean algebra of events E's and denote by , ∨ and ∧ the usual Boolean operations of contrary, disjunction and conjunction, respectively. The sure event Ω and the impossible event ∅ coincide, respectively, with the top and bottom elements of endowed with the partial order ⊆ of implication.
A conditional event is an ordered pair of events with , where the conditional event is customarily identified with E. Let be an additive class (i.e., a set of events closed under
Conditioning for possibility measures
Let be a finite Boolean algebra with set of atoms . A possibility measure Π on is a normalized maxitive function [24], i.e., it satisfies condition (N) of the previous section together with
- (Π)
, for every .
Every possibility measure Π has an associated dual necessity measure N, defined for every as , which is minitive, i.e., for every ,
Possibility measures are particular plausibility functions [41] and so
Generalized Bayesian conditioning
The aim of this section is to generalize the Bayesian conditioning rule in equation (6) to cover the cases where the denominator vanishes, by preserving the upper envelope interpretation of equation (7). The idea is to provide a generalized Bayesian conditioning rule relying on a linearly ordered class of possibilities on such that the resulting conditional possibility can be obtained through the extension of a full conditional probability defined on a different
Weakly logically independent Boolean algebras
In this section we recall the condition of weak logical independence introduced in [10], [11], [12] in order to show its role in the properties of the upper envelope of the extensions of a full conditional probability.
Definition 4 A finite Boolean algebra with set of atoms is weakly logically independent of another finite Boolean algebra with set of atoms (in symbol ) if, for any given , every () satisfies at least one of the following conditions: ;
A necessary and sufficient condition
In the previous section it has been shown that weak logical independence of and is only a sufficient condition in order to get a full B-conditional possibility on as upper envelope of the extensions of a full conditional probability on .
In case of an unconditional possibility measure on , in [12] it has been shown that it is always possible to find a weakly logically independent finite Boolean algebra and an unconditional probability measure on giving
Conclusions
The paper proposes a notion of conditioning (B-conditioning) for possibility measures on finite algebras which generalizes the Bayesian conditioning rule. Such definition is not axiomatic since it relies on the existence of a (non-unique) linearly ordered class of possibility measures (C-class). The resulting conditional measure (full B-conditional possibility) is shown to be the upper envelope of the extensions of a full conditional probability defined on a different algebra. The purely
Acknowledgements
This work was partially supported by INdAM–GNAMPA through the Project 2015 U2015/000418 “Envelopes of coherent conditional probabilities and their applications to statistics and artificial intelligence”, and by the Italian Ministry of Education, University and Research, funding of Research Projects of National Interest (PRIN 2010–2011) under the grant 2010FP79LR_003 “Logical methods of information management”.
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