Local coherence of conditional probability assessments: definition and application (original) (raw)

On the Checking of G-Coherence of Conditional Probability Bounds

International Journal of Uncertainty, Fuzziness and Knowledge-based Systems, 2003

We illustrate an approach to uncertain knowledge based on lower conditional probability bounds. Our results and algorithms exploit a concept of generalized coherence (g-coherence), which is a generalization of de Finetti's coherence principle and is equivalent to the "avoiding uniform loss" property for lower and upper probabilities(a la Walley). By our algorithms, given a g-coherent assessment, we can also correct it obtaining the associated coherent assessment (in the sense of Walley and Williams). Our algorithms work with a reduced set of variables and a reduced set of constraints. Such reduced sets are computed by suitably exploiting the additive structure of the random gains. In this paper, we study in detail imprecise assessments defined on families of three conditional events. We give some necessary and sufficient conditions and, then, we generalize some of the theoretical results obtained. We also exploit such results by proposing two algorithms which provide new strategies for reducing the number of constraints and for deciding g-coherence. Finally, we illustrate our approach by giving some examples.

Computational Aspects in Checking of Coherence and Propagation ofConditional Probability Bounds

2000

In this paper we consider the problem of reducing the computational difficulties in g-coherence checking and propagation of imprecise conditional probability assessments. We review some theoretical results related with the linear structure of the random gain in the betting criterion. Then, we propose a modified version of two existing algorithms, used for g-coherence checking and propagation, which are based on linear systems with a reduced number of unknowns. The reduction in the number of unknowns is obtained by an iterative algorithm. Finally, to illustrate our procedure we give some applications.

Efficient Checking of Coherence and Propagation of Imprecise ProbabilityAssessments

2000

We consider the computational difficulties in the checking of coherence and propagation of imprecise probability assessments. We examine the linear structure of the random gain in betting criterion and we propose a general methodology which exploits suitable subsets of the set of values of the random gain. In this way the checking of coherence and propagation amount to examining linear systems with a reduced number of unknowns. We also illustrate an example.

Simplification Rules for the Coherent Probability Assessment Problem

Annals of Mathematics and Artificial Intelligence, 2002

In this paper we develop a procedure for checking the consistency (coherence) of a partial probability assessment. The general problem (called CPA) is NP-complete, hence, to have a reasonable application some heuristic is needed. Our proposal differs from others because it is based on a skilful use of the logical relations present among the events. In other approaches the consistency problem is reduced directly to the satisfiability of a system of linear constraints. Here, thanks to the characterization of particular configurations and to the elimination of variables, an instance of the problem is reduced to smaller instances. To obtain such results, we introduce a procedure based on rules resembling those given by Davis-Putnam for the satisfiability of Boolean formulas. At the end a particularized description of an actual implementation is given.

Characterization of Coherent Conditional Probabilities as a Tool for Their Assessment and Extension

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 1996

A major purpose of this paper is to show the broad import and applicability of the theory of probability as proposed by de Finetti, which differs radically from the usual one (based on a measure-theoretic framework). In particular, with reference to a coherent conditional probability, we prove a characterization theorem, which provides also a useful algorithm for checking coherence of a given assessment. Moreover it allows to deepen and generalise in useful directions de Finetti’s extension theorem (dubbed as “the fundamental theorem of probability”), emphasising its operational aspects in many significant applications.

A consistency problem for imprecise conditional probability assessments

Proceedings IPMU, 1998

In this paper we introduce an operational procedure which, given an avoiding sure loss (ASL) imprecise probability assessment on an arbitrary finite set of conditional events, determines its 'leastcommittal' coherent correction, i.e. the coherent imprecise probability assessment which reduces the imprecision of as little as possible, without ever increasing it. Besides, a new proof of the consistency of a known procedure, which checks the ASL condition, is supplied by introducing a technique employed also in the proof of the previous procedure. It is then shown that the two procedures can be 'merged' to obtain an algorithm to be used when it is not known a priori whether is ASL.

Coherent correction of inconsistent conditional probability assessments

2008

In this paper we suggest a procedure to adjust an incoherent conditional probability assessment given on a partial domain. We look for a solution that tries to attain two separate goals: on one hand the solution should be as close as possible to the initial assessments, on the other hand we do not want to insert more information than we had at the beginning. The first goal is achieved by minimizing an appropriately defined distance among assessments, while for the second we look for a "maximum entropy" like solution.

Elimination of Boolean variables for probabilistic coherence

Soft Computing, 2000

In this paper we deal with the computational complexity problem of checking the coherence of a partial probability assessment (called CPA). The CPA problem, like its analogous PSAT, is NP-complete so we look for an heuristic procedure to make tractable reasonable instances of the problem. Starting from the characteristic feature of de Finetti's approach (i.e. the explicit distinction between the probabilistic assessment and the logical relations among the sentences) we introduce several rules for a sequential``elimination'' of Boolean variables from the domain of the assessment. The procedure resembles the well-known Davis-Putnam rules for the satis®ability, however we have, as a drawback, the introduction of constraints (among real variables) whose satis®ability must be checked.