Andrea Capotorti

Locally Strong Coherence in Inference Processes

In this paper we deal with probabilistic inference in the most general form of coherent conditional probability assessments. In particular, our aim is to reduce computational difficulties that could arise with a direct application of the main characterization results. We reach our goal by introducing the notion of locally strong coherence and characterizing it by logical conditions. Hence, some of the numerical constraints are replaced by Boolean satisfiability conditions. An automatic procedure is proposed and its efficiency is proved. Some examples are reported to make easier the understanding of the machinery and to show its effectiveness.

Credit scoring analysis using a fuzzy probabilistic rough set model

Credit scoring analysis is an important activity, especially nowadays after a huge number of defaults has been one of the main causes of the financial crisis. Among the many different tools used to model credit risk, the recent development of rough set models has proved effective. The original development of rough set theory has been widely generalized and combined with other approaches to uncertain reasoning, especially probability and fuzzy set theories. Since coherent conditional probability assessments cope well with the problem of unifying these different approaches, a merging of fuzzy rough set theory with this subjectivist approach is proposed. Specifically, expert partial probabilistic evaluations are encompassed inside a gradual decision rule structure, with coherence of the conclusion as a guideline. In line with Bayesian rough set models, credibility degrees of multiple premises are introduced through conditional probability assessments. Nonetheless, discernibility with this method remains too fine. Therefore, the basic partition is coarsened by equivalence classes based on the arity of positively, negatively and neutrally related criteria. A membership function, which grades the likelihood of default, is introduced by a peculiar choice of t-norms and t-conorms. To build and test the model, real data related to a sample of firms are used.

Simplification Rules for the Coherent Probability Assessment Problem

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.

Incoherence correction strategies in statistical matching

Several economic applications require to consider different data sources and to integrate the information coming from them. This paper focuses on statistical matching, in particular we deal with incoherences. In fact, when logical constraints among the variables are present incoherencies on the probability evaluations can arise. The aim of this paper is to remove such incoherences by using different methods based on distances minimization or least commitment imprecise probabilities extensions. An illustrative example shows peculiarities of the different correction methods. Finally, limited to pseudo distance minimization, we performed a systematic comparison through a simulation study.

Correction of incoherent conditional probability assessments

In this paper we deep in the formal properties of an already stated discrepancy measure between a conditional assessment and the class of unconditional probability distributions compatible with the assessment domain.

Elimination of Boolean variables for probabilistic coherence

Abstract 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 Finettis 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 satisfiability, however we have, as a drawback, the introduction of constraints (among real variables) whose satisfiability must be checked. In simple examples we test the efficiency of the procedure respect to the “traditional” approach of solving a linear system with a huge coefficient matrix built from the atoms generated by the domain of the assessment.

Locally strong coherence and inference with lower-upper probabilities

Abstract We introduce an operational way to reduce the spatial complexity in inference processes based on conditional lower–upper probabilities assessments. To reach such goal we must suitably exploit zero probabilities taking account of logical conditions characterizing locally strong coherence. We actually re-formulate for conditional lower–upper probabilities the notion of locally strong coherence already introduced for conditional precise probabilities. Thanks to the characterization, we avoid to build all atoms, so that several real problems become feasible. In fact, the real complexity problem is connected to the number of atoms. Since for an inferential process with lower–upper probabilities several sequences of constraints must be fulfilled, our simplification can have either a “global” or a “partial” effect, being applicable to all or just to some sequences. The whole procedure has been implemented by XLisp-Stat language. A comparison with other approaches will be done by an example.

Comparative uncertainty: Theory and automation

In recent decades, qualitative approaches to probabilistic uncertainty have received more and more attention. We propose a characterisation of partial preference orders through a uniform axiomatic treatment of a variety of qualitative uncertainty notions. To this end, we prove a representation result that connects qualitative notions of partial uncertainty to their numerical counterparts. We describe an executable specification, in the declarative framework of Answer Set Programming, that constitutes the core engine for qualitative management of uncertainty. Some basic reasoning tasks are also identified.

An Operational View of Coherent Conditional Previsions

In this paper we propose a characterization theorem for coherent conditional previsions assessed on finite conditional random variables. The main feature is the direct applicability of the results to model practical problems. In fact the check of coherence and the inferential steps are reduced to the solvability of linear systems and of linear programming problems, respectively. The guideline of the procedure has been the already stated results for coherent conditional probabilities, so that now we have a unified theory for uncertainty represented by belief or synthesized by prevision. The procedure turns out to be helpful when the size of the relevant quantities strongly depend on the different scenarios in which they are considered. A simple example shows the potentiality of the entire machinery on a decision-aid problem.

Axiomatic characterization of partial ordinal relations

Abstract In this paper we focus on the theoretical properties of non-numerical representation of the uncertainty. As usual, this representation is realized by an “ordinal relation” (or, equivalently, by a “comparative scale”) among the “entities” (events, alternatives or acts) of a specific problem. After giving an overview of different known axioms characterizing some classes of ordinal relations (and their duals), we introduce some axioms to enclose the necessary and sufficient conditions for the representability of ordinal relations (defined on arbitrary finite sets of events) by the most-known uncertainty measures.