A new approach to specificity in possibility theory: Decision-making point of view

Abstract

The paper investigates the problem of defining a specificity relation in Pyt'ev possibility theory. That branch of possibility theory developed by prof. Yu.P. Pyt'ev in the late 1990s defines possibility as a monotone measure Π which numerical values except 0 and 1 are meaningless. The following information and its consequent implications are meaningful: $\mathrm{\Pi }(A)=0$ (A is impossible), or $\mathrm{\Pi }(A)>\mathrm{\Pi }(B)$ (A is more possible than B), or $\mathrm{\Pi }(A)=\mathrm{\Pi }(B)$ (A and B are equally possible). All results obtained with Pyt'ev possibility theory are invariant to any strictly increasing lower semi-continuous transformation of all possibility values with the fixed points: 0 and 1. This fact allows anyone to assign numerical values to possibilities using an individual subjective scale of possibility values. Unlike Zadeh possibility theory based on fuzzy sets, where specificity of possibility distributions is evaluated by means of fuzzy set inclusion and distribution ${\pi }_1$ is more specific than ${\pi }_2$ iff ${\pi }_1(\cdot )⩽{\pi }_2(\cdot )$ pointwise, in Pyt'ev possibility theory such an approach can not be applied if numerical values of ${\pi }_1$ and ${\pi }_2$ are assigned through different subjective scales. A thesis we put forward in this paper states that the specificity relation definition has to be consistent with the possibilistic decision-making approaches. The following is intended to be true: the more specific an underlying possibility and a corresponding possibility distribution are, the narrower (more specific) a set of the optimal decisions is (a decision is optimal if it minimizes the possibility of error). A specificity relation satisfying such a requirement is defined and studied.

Introduction

Since the mid-twentieth century, the necessity of handling the kind of uncertainty being not probabilistic in nature has resulted in developing many mathematical theories of uncertainty different from the probability one. Among them are: Choquet theory of capacities [3], Dempster–Shafer theory of evidence [7], [35], Zadeh fuzzy sets and the possibility theory based on them [11], [13], [43], Sugeno theory of fuzzy integral [40], and De Cooman possibility theory [4], [5], [6].

One of the non-probabilistic approaches to handling uncertainty was suggested by prof. Yu.P. Pyt'ev in the late 1990s [25], [26], [27], [28], [29], [30], [32]. It was originally developed to model probabilistic uncertainty called randomness using imprecise information about its probabilistic model which can be retrieved not only from statistical data or strong theoretical considerations (like in classical or geometric probability) but also from someone's intuitive knowledge about an experiment in question. Nevertheless, Pyt'ev possibility theory can be used to model non-probabilistic uncertainty as well [33], [34]. It has a probability–possibility consistency principle [31] which is stronger than the similar principle formulated by Zadeh. Since its development, the theory has been applied in experimental data analysis [20], [38], game theory [24], image analysis [44], [45], etc.

The domain of possibility measure in Pyt'ev possibility theory is an algebra of events of some random experiment, and its range is the lattice $\mathcal{L}=([0,1],⩽,\text{✢},\text{✻})$ called the scale of possibility values. Lattice $\mathcal{L}$ is a unit interval with natural order relation of real numbers “⩽” and two algebraic operations “✢” and “✻”. Any specific forms of operations “✢” and “✻” are not postulated, rather, they are deduced from the intention to make possibility theory be a framework for obtaining invariant results and appear to be “max” and “min” respectively. This intention is formulated in the form of the principle of relativity: in Pyt'ev possibility theory any statement is meaningful iff it is invariant to any strictly increasing lower semi-continuous transformation of all possibility values with fixed points 0 and 1. This fact allows anyone to assign numerical values to possibilities using his individual subjective scale of possibility values which can be mapped to the scale of possibility values used by any other person by a strictly increasing lower semi-continuous transformation.

All the theories mentioned above including probability one can be used to handle uncertainty of various degrees. For example, Zadeh possibility theory based on fuzzy sets is appropriate to model perfect knowledge using a possibility distribution which equals 0 everywhere except the single point, complete ignorance using the possibility distribution which equals 1 everywhere, and any degree of uncertainty between those two limiting cases as well. In probability theory degree of uncertainty can be measured using Shannon entropy [36]. The intention to introduce a possibilistic variant U of uncertainty measure has given rise to a considerable amount of literature on the subject [2], [10], [12], [18], [19], [41], [42].

The first successful attempt to do this was made by Higashi and Klir [18] within the framework of Zadeh possibility theory: a measure of uncertainty called U-uncertainty was introduced. Yager in [41] suggested an opposite point of view for Dempster–Shafer theory and introduced a measure of specificity (not uncertainty). Those results were significantly extended later in [42]. The measures introduced in [18], [41] were inspected in [12]: the U-uncertainty by Higashi and Klir was generalized to Dempster–Shafer theory, a probabilistic interpretation of Yager's specificity was provided. In [10] an approach to comparing specificity of distributions via their relative peakedness (without resorting to numerical indexes) was suggested. Finally, the papers [2], [19] provided a broad survey on recent advances in this area and introduced a unified approach to measuring uncertainty and specificity within the frameworks of various uncertainty theories (probability and possibility theories, Dempster–Shafer theory, etc.) based on a system of uncertainty and specificity axioms.

Analysis of the literature shows that regardless of the specific definition, it is generally accepted that the uncertainty measure U shall meet the following requirements:

• its value is minimal (usually 0) in the case of perfect knowledge,

• its value is maximal (usually 1 or ∞) in the case of complete ignorance,

• it is monotone.

Requirement 3 can be expressed within the framework of Zadeh possibility theory as follows: if possibility distribution ${\pi }_1:\mathrm{\Omega }\to [0,1]$ is pointwise dominated by another distribution ${\pi }_2:\mathrm{\Omega }\to [0,1]$, i.e., ${\pi }_1(\omega )⩽{\pi }_2(\omega )$ for any $\omega \in \mathrm{\Omega }$, then $U({\pi }_1)⩽U({\pi }_2)$. This definition of monotonicity is based on the natural specificity relation on a set of possibility distributions accepted in Zadeh possibility theory and defined through fuzzy set inclusion.

However, the specificity relation can not be defined as above in Pyt'ev possibility theory if numerical values of possibilities and possibility distributions are assigned using distinct scales of possibility values, for example, if the possibility distributions are provided by different experts. In this case, to introduce a measure of uncertainty U satisfying 3 it is necessary to define the appropriate specificity relation first. The paper considers a problem of defining specificity relation in the context of decision making. To make a decision, i.e., to choose a decision d from a set of possible decisions D given a realization y of an observable ill-known element η with values in Y, one usually uses a decision rule which is a function $\delta :Y\to D$. Mathematically, the decision-making problem is an optimization problem of finding a decision rule called optimal and minimizing the possibility of error. However, many optimal decision rules may exist. For example, in the case of complete ignorance (which can be modeled using the possibility distribution being equal to 1 everywhere) as underlying possibilistic model, any decision rule is optimal, so choosing any decision $d\in D$ leads to the same possibility of error being equal to 1, i.e., such a possibilistic model and the corresponding possibility distribution provide no useful information to a decision maker. In contrast, in the case of perfect knowledge (a possibility distribution which equals 0 everywhere except for the single point), only one decision $d_{⁎}$ exists such that any optimal decision rule $\delta :Y\to D$ satisfies the condition $\delta (y)=d_{⁎}$ for any possible realization $y\in Y$ of an ill-known element η, i.e., such a possibilistic model provides a decision maker with the most precise information.

Let ${\mathrm{\Delta }}_{⁎}$ be a set of all optimal decision rules and $D_y=\{\delta (y)|\delta \in {\mathrm{\Delta }}_{⁎}\}$ be a set of all optimal decisions corresponding to a value $y\in Y$ taken by ill-known element η. A thesis advanced in this paper and used to define the specificity relation is that the narrower $D_y$ is for all possible $y\in Y$ (i.e., for all values $y\in Y$ such that possibility of $\eta =y$ is not zero), the more specific underlying possibility and corresponding possibility distribution are.