Preferences and Similarities

We present several relational frameworks for expressing similarities and preferences in a quantitative way. The main focus is on the occurrence of various types of transitivity in these frameworks. The first framework is that of fuzzy relations; the corresponding notion of transitivity is C-transitivity, with C a conjunctor. We discuss two approaches to the measurement of similarity of fuzzy sets: a logical approach based on biresidual operators and a cardinal approach based on fuzzy set cardinalities. The second framework is that of reciprocal relations; the corresponding notion of transitivity is cycle-transitivity. It plays a crucial role in the description of different types of transitivity arising in the comparison of random variables in terms of winning probabilities.

We start by recalling previous work leading to an interpretation of fuzzy logic in terms of metric structures defined in a set of possible worlds. These possible worlds, or possible states of a system, are characterized by the truth values of clasical logical assertions about the state of the world.

We consider next the nature of the information required to generate the underlying metrics, concluding that similarity measures are typically derived from preference relations that induce a generalized order in the space of possible worlds. Recalling the relation between similarity measures and utility functions underlying several fundamental results such as the representation theorem of Valverde, we extend our similarity-based formalism by introduction of corresponding utilitarian notions, such as marginal preference and conditional preference. On the basis of these definitions we derive a formula to infer the value of marginal preference functions.

Finally we briefly present several examples of the application of these ideas to the development of intelligent planners and controllers for robots and teams of robots. These examples motivate basic requirements for a comprehensive logic-based approach to a calculus of similarities and preferences. We note that logic-based preference formalisms permit to explicitly specify context-dependent formulas that tradeoff the relative utilities of attaining potential system states from the perspective of different preference criteria, allow specification of the interaction of individual agents, and control the behavior of groups of interacting agents.

Making a good decision is often a matter of listing and comparing positive and negative arguments, as studies in cognitive psychology have shown. In such cases, the evaluation scale should be considered bipolar, that is, negative and positive values are explicitly distinguished. Generally, positive and negative features are evaluated separately, as done in Cumulative Prospect Theory. However, contrary to the latter framework that presupposes genuine numerical assessments, decisions are often made on the basis of an ordinal ranking of the pros and the cons, and focusing on the most salient features, i.e., the decision process is qualitative. In this paper, we report on a project aiming at characterizing several decision rules, based on possibilistic order of magnitude reasoning, and tailored for the joint handling of positive and negative affects, and at testing their empirical validity. The simplest rules can be viewed as extensions of the maximin and maximax criteria to the bipolar case and, like them, suffer from a lack of discrimination power. More decisive rules that refine them are also proposed. They account for both the principle of Pareto-efficiency and the notion of order of magnitude reasoning. The most decisive one uses a lexicographic ranking of the pros and cons. It comes down to a special case of Cumulative Prospect Theory, and subsumes the “Take the best” heuristic.

The aim of this paper is to survey a class of logical formalizations of similarity-based reasoning models where similarity is understood as a graded notion of truthlikeness. We basically identify two different kinds of logical approaches that have been used to formalize fuzzy similarity reasoning: syntacticelly-oriented approaches based on a notion of approximate proof, and semantically-oriented approaches based on several notions of approximate entailments. In particular, for these approximate entailments we provide four different formalisations in terms of suitable systems of modal and conditional logics, including for each class a system of graded operators with classical semantics, as well as a system with many-valued operators. Finally, we also explore some nonmonotonic issues of similarity-based reasoning.

We present several classes of logics for reasoning with information stored in information systems. The logics enable us to cope with the phenomena of incompleteness of information and uncertainty of knowledge derived from such an information. Relational inference systems for these logics are developed in the style of dual tableaux.

We review similarity and distance measures used in Statistics for clustering and classification. We are motivated by the lack of most measures to adequately utilize a non uniform distribution defined on the data or sample space.

Such measures are mappings from O x O → R ₊ where O is either a finite set of objects or vector space like R ^p and R ₊ is the set of non-negative real numbers. In most cases those mappings fulfil conditions like symmetry and reflexivity. Moreover, further characteristics like transitivity or the triangle equation in case of distance measures are of concern.

We start with Hartigan’s list of proximity measures which he compiled in 1967. It is good practice to pay special attention to the type of scales of the variables involved, i.e. to nominal (often binary), ordinal and metric (interval and ratio) types of scales. We are interested in the algebraic structure of proximities as suggested by (1967) and (1971), information-theoretic measures as discussed by (1971), and the probabilistic W-distance measure as proposed by (1970). The last measure combines distances of objects or vectors with their corresponding probabilities to improve overall discrimination power. The idea is that rare events, i.e. set of values with a very low probability of observing, related to a pair of objects may be a strong hint to strong similarity of this pair.

This paper addresses the definition of independence concepts in the context of similarity relations. After motivating the need for independence concepts basic ideas from similarity relations and their connections to fuzzy systems are reviewed. Three different independence notions are discussed and investigated in the framework of similarity relations. The results show that there are significant differences for independence concepts in a probabilistic setting and in the framework of similarityh relations.

Similarity measures based on feature matching have been designed for modelling subjective similarity judgements. In this paper, the taxonomic presence-absence feature representation is extended to assess the similarity of objects whose attributes are described by partial satisfaction of predicates or by fuzzy sets. The principle of minimum specificity is used to obtain possibilistic bounds on the combination of similarity assessments. A priority hierarchy and bipolarity are incorporated into similarity measurement to utilize inter-attribute relationships in modelling similarity judgements.

Defensive forecasting first identifies a betting strategy that succeeds if probabilistic forecasts are inaccurate and then makes forecasts that will defeat this strategy. Both the strategy and the forecasts are based on the similarity of the current situation to previous situations.

The theory of defensive forecasting uses the game-theoretic framework for probability, in which game theory replaces measure theory. In this framework, a classical theorem such as the law of large numbers is proven by a betting strategy that multiplies the capital it risks by a large factor if the theorem’s prediction fails. Theorems proven in this way apply not only to the classical case where only point predictions are made. Defensive forecasting is possible because the strategies are specified explicitly.

Arguments play two types of roles w.r.t. decision, namely helping to select an alternative, or to explain a choice. Until now, the various attempts at formalizing argument-based decision making have relied only on one type of arguments, in favor of or against alternatives. The paper¹ proposes a systematic typology that identifies eight types of arguments, some of them being weaker than others. First the setting emphasizes the bipolar nature of the evaluation of decision results by making an explicit distinction between prioritized goals to be pursued, and prioritized rejections that are stumbling blocks to be avoided. This is the basis for an argumentative framework for decision. Each decision is supported by arguments emphasizing its positive consequences in terms of goals certainly satisfied, goals possibly satisfied, rejections certainly avoided and rejections possibly avoided. A decision can also be attacked by arguments emphasizing its negative consequences in terms of certainly or possibly missed goals, or rejections certainly or possibly led to by that decision. The proposed typology partitions the set of alternatives into four classes, giving thus a status to decisions, which may be recommended, discommended, controversial or neutral. This typology is also helpful from an explanation point of view for being able to use the right type of arguments depending on the context. The paper also presents a preliminary investigation on decision principles that can be used for comparing decisions. Three classes of principles can be considered: unipolar, bipolar or non-polar principles depending on whether i) only arguments pro or only arguments cons, or ii) both types, or iii) an aggregation of them into a meta-argument are used.

Preference elicitation is a well-known bottleneck in decision analysis and decision automation tasks, especially in applications targeting lay users that cannot be assisted by a professional decision analyst. Focusing on the ordinal preferences of the users, here we discuss the principles that appear to underly various frameworks developed in the AI research for interpretation and formal reasoning about sets of qualitative preference statements.

The term “preference learning” refers to the application of machine learning methods for inducing preference models from empirical data. In the recent literature, corresponding problems appear in various guises. After a brief overview of the field, this work focuses on a particular learning scenario called label ranking where the problem is to learn a mapping from instances to rankings over a finite number of labels. Our approach for learning such a ranking function, called ranking by pairwise comparison (RPC), first induces a binary preference relation from suitable training data, using a natural extension of pairwise classification. A ranking is then derived from this relation by means of a ranking procedure. This paper elaborates on a key advantage of such an approach, namely the fact that our learner can be adapted to different loss functions by using different ranking procedures on the same underlying order relations. In particular, the Spearman rank correlation is minimized by using a simple weighted voting procedure. Moreover, we discuss a loss function suitable for settings where candidate labels must be tested successively until a target label is found. In this context, we propose the idea of “empirical conditioning” of class probabilities. A related ranking procedure, called “ranking through iterated choice”, is investigated experimentally.

Preferences are ubiquitous in real-life Moreover, preferences can be of many kinds: qualitative, quantitative, conditional, positive or negative, to name a few. Our ultimate goal is to define and study formalisms that can model problems with both constraints and many kind of preferences, possibly defined by several agents, and to develop tools to solve such problems efficiently. In this paper we briefly report on our recent work towards this goal.