Davide Petturiti

Possibilistic and probabilistic likelihood functions and their extensions: Common features and specific characteristics

We deal with conditional probability in the sense of de Finetti and with T-conditional possibility (with T a triangular norm). We prove that Dubois and Prade conditional possibility is a particular min-conditional possibility and then we compare the two notions of conditioning by an inferential point of view. Moreover, we study T-conditional possibilities as functions of the conditioning event, putting in evidence analogies and differences with conditional probabilities. This allows to characterize likelihood functions (and their aggregations) consistent either with a T-conditional possibility or a conditional probability. This analysis highlights many syntactical coincidences. Nevertheless the main difference is a weak form of monotonicity, which arises only in the possibilistic case.

Extending and Implementing RASP

In previous work we have proposed an extension to ASP (Answer Set Programming), called RASP, standing for ASP with Resources. RASP supports declarative reasoning on production and consumption of (amounts of) resources. The approach combines answer set semantics with quantitative reasoning and relies on an algebraic structure to support computations and comparisons of amounts. The RASP framework provides some form of preference reasoning on resources usage. In this paper, we go further in this direction by introducing expressive constructs for supporting complex preferences specification on aggregate resources. We present a refinement of the semantics of RASP so as to take into account the new constructs. For all the extensions, we provide an encoding into plain ASP. We prove that the complexity of establishing the existence of an answer set, in such an enriched framework, remains NP-complete as in ASP. Finally, we report on raspberry, a prototypical implementation of RASP. This tool consists of a compiler that, given a ground RASP program, produces a pure ASP encoding suitable to be processed by commonly available ASP-solvers.

Inferential models and relevant algorithms in a possibilistic framework

We provide a general inferential procedure based on coherent conditional possibilities and we show, by some examples, its possible use in medical diagnosis. In particular, the role of the likelihood in possibilistic setting is discussed and once the coherence of prior possibility and likelihood is checked, we update prior possibilities.

Rank discrimination measures for enforcing monotonicity in decision tree induction

Monotone classification is a relatively recent topic in machine learning in which the classification function to learn is asked to guarantee a sort of monotonicity of the class with respect to attribute values. Nevertheless, real datasets are quite far from being monotone and this can sharply limit the performance of purely monotone classifiers while standard classifiers are simply insensitive to monotonicity. Here we focus on rank discrimination measures to be used in decision tree induction, i.e., functions able to measure the discrimination power of an attribute with respect to the class taking into account the monotonicity of the class with respect to the attribute. Three new measures are studied in detail and a hierarchical construction model is derived allowing the formal definition of a general rank discrimination measure. Our measures have been compared with other well-known proposals, quantifying both the accuracy and the monotonicity of the resulting binary decision tree classifiers.

Envelopes of conditional probabilities extending a strategy and a prior probability

Any assessment formed by a strategy and a prior probability is a coherent conditional probability and can be extended, generally not in a unique way, to a full conditional probability. The corresponding class of all extensions is studied and a closed form expression for its envelopes is provided. Subclasses of extensions meeting further analytical properties are considered by imposing conglomerability and a conditional version of conglomerability, respectively. Then, the envelopes of extensions satisfying these conditions are characterized. Envelopes of the class of all full conditional probabilities extending a strategy and a prior probability.Conglomerability and conditional conglomerability for full conditional probability extensions.Envelopes of the subclass of conglomerable full conditional probability extensions.Envelopes of the subclass of conditionally conglomerable full conditional probability extensions.

Conditional belief functions as lower envelopes of conditional probabilities in a finite setting

The aim is to provide a characterization of full conditional measures on a finite Boolean algebra, obtained as lower envelope of the extensions of a full conditional probability defined on another finite Boolean algebra. Such conditional measures are conditional belief functions defined by means of a generalized Bayesian conditioning rule relying on a linearly ordered class of belief functions. This notion of Bayesian conditioning for belief functions is compared with other well-known conditioning rules by looking for those conditional measures that can be seen as lower conditional probabilities.

Coherent T-conditional Possibility Envelopes and Nonmonotonic Reasoning

Envelopes of coherent T-conditional possibilities and coherent T-conditional necessities are studied and an analysis of some inference rules which play an important role in nonmonotonic reasoning is carried out.

Bayesian inference: the role of coherence to deal with a prior belief function

Starting from a likelihood function and a prior information represented by a belief function, a closed form expression is provided for the lower envelope of the set of all the possible “posterior probabilities” in finite spaces. The same problem, removing the hypothesis of finiteness for the domain of the prior, is then studied in the finitely additive probability framework by considering either the whole set of coherent extensions or the subset of disintegrable extensions.

Fuzzy memberships as likelihood functions in a possibilistic framework

Abstract Likelihood functions are studied in a probabilistic and possibilistic setting: inferential conclusions are drawn from a set of likelihood functions and prior information relying on the notion of disintegrability. The present study allows for a new interpretation of fuzzy membership functions as coherent conditional possibilities. The concept of possibility of a fuzzy event is then introduced and a comparison with the probability of a fuzzy event is provided.

Finitely maxitive conditional possibilities, Bayesian-like inference, disintegrability and conglomerability

The aim of the paper is to study Bayesian-like inference processes involving coherent finitely maxitive T-conditional possibilities assessed on infinite sets of conditional events. Coherence of an assessment consisting of an arbitrary possibilistic prior and an arbitrary possibilistic likelihood function is proved, thus a closed form expression for the envelopes of the relevant joint and posterior possibilities is given when T is the minimum or a strict t-norm. The notions of disintegrability and conglomerability are also studied and their relevance in the infinite version of the possibilistic Bayes formula is highlighted.