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## Über dieses Buch

Inspired by the eternal beauty and truth of the laws governing the run of stars on heavens over his head, and spurred by the idea to catch, perhaps for the smallest fraction of the shortest instant, the Eternity itself, man created such masterpieces of human intellect like the Platon's world of ideas manifesting eternal truths, like the Euclidean geometry, or like the Newtonian celestial me­ chanics. However, turning his look to the sub-lunar world of our everyday efforts, troubles, sorrows and, from time to time but very, very seldom, also our successes, he saw nothing else than a world full of uncertainty and tem­ porariness. One remedy or rather consolation was that of the deep and sage resignation offered by Socrates: I know, that I know nothing. But, happy or unhappy enough, the temptation to see and to touch at least a very small por­ tion of eternal truth also under these circumstances and behind phenomena charged by uncertainty was too strong. Probability theory in its most sim­ ple elementary setting entered the scene. It happened in the same, 17th and 18th centuries, when celestial mechanics with its classical Platonist paradigma achieved its greatest triumphs. The origins of probability theory were inspired by games of chance like roulettes, lotteries, dices, urn schemata, etc. and probability values were simply defined by the ratio of successful or winning results relative to the total number of possible outcomes.

## Inhaltsverzeichnis

### 1. Introduction

Abstract
From a certain point of view, position of every individual human being, every society, and the mankind as a whole in the surrounding world can be seen as that of the subject of a continual sequence of decision making acts, terminated by one’s death in the case of an individual, and perhaps infinite in the case of a collective agent. Just a very small portion of the decision problem we have to solve are of the deterministic nature when the consequences of the accepted decision can be completely, and with the absolute degree of certainty, foreseen so that we can choose a good, appropriate, acceptable, the best, or almost the best solution, supposing that the adjectives just introduced are sensefully, and with a sufficient degree of preciseness defined. As good examples of uncertainty-free decision procedures let us mention those ones applied in an artificial environment when any influence of uncertainty is avoided a priori. Decision-making in mathematics, or in formal systems in general, or games like chess can be remembered here. One common feature of all these cases consists in the fact that if the decision making fails, e.g., if the accepted decision is a posteriori proved not to possess the expected properties, then the only source of this failure can consist in an error made during the realization of the decision procedure (a computational error or a wrong step in a mathematical proof, for example).
Ivan Kramosil

### 2. Preliminaries on Axiomatic Probability Theory

Abstract
This work has been conceived as a purely theoretical and mathematical study dealing with the subject of its interest at a highly abstract and formalized level. Probability theory will serve as one and, as a matter of fact, the most important and the most powerful formal tool used below in order to achieve this goal. Therefore, beginning with a brief survey of the most elementary notions of probability theory, just the most elementary abstract ideas and construction of the axiomatic probability theory, as settled by Kolmogorov in Kolmogorov (1974) are presented in this chapter, intentionally leaving aside all the informal discussions, motivations, and practical examples preceding the formalized explanations of probability theory in the greatest part of textbooks and monographs dealing with this theory. The reader interested in these informal parts of probability theory is kindly invited to consult an appropriate textbook or monograph, let us mention explicitly the already classical textbooks Feller (1957) and Gnedenko (1965), where just these informal parts are explained very carefully, in detail, and with a lot of various examples. On the other side, Loève (1960) treates probability theory at an exclusively abstract and formalized level.
Ivan Kramosil

### 3. Probabilistic Model of Decision Making under Uncertainty

Abstract
Like the last chapter, also this one could be conceived at a purely formalized level, speaking about sets, mappings, functions, relations and ordered n-tuples of such objects satisfying some mathematically formalized demands. The difference between the two chapters consists in the fact that the intuition, interpretation and motivation behind the axiomatic probability theory can be found in most of the textbooks and monographs dealing with this theory, on the other side, in the case of general probabilistic and statistical models of decision making under uncertainty the situation is not so simple, let us mention here Lehman (1947) or Blackwell and Girshick (1954) as good introductory texts. Therefore we begin our explanation using informal terms charged by some extra-mathematical semantics, but our intention will be to get back to a formalized mathematical language as soon as possible.
Ivan Kramosil

### 4. Basic Elements of Dempster-Shafer Theory

Abstract
The greatest part of works dealing with the fundamentals of Dempster-Shafer theory is conceived either on the combinatoric, or on the axiomatic, but in both the cases on a very abstract level. The first approach begins by the assumption that S is a nonempty finite set, that m is a mapping which ascribes to each A ⊂ S a real number m(A) from the unit interval [0,1] in such a way that ∑A⊂ S m(A) = 1 (m is called a basic probability assignment on S), and that the (normalized) belief function induced by m is the mapping bel m : P(S) → [0,1] defined, for each A ⊂ S, by bel m (A) = (1 - m(ø))-1ø≠ B⊂A m(B), if m(ø) < 1, bei m being undefined otherwise Shafer (1976) and elsewhere). The other (axiomatic) approach begins with the idea that belief function on a finite nonempty set S is a mapping bel: P(S) → [0,1], satisfying certain conditions (obeying certain axioms, in other terms). If these conditions (axioms) are strong and reasonable enough, it can be proved that it is possible to define uniquely a basic probability assignment m on S such that the belief function induced by m is identical with the original belief function defined by axioms, so that both the approaches meet each other and yield the same notion of belief function (Smets (1994)). The problems how to understand and obtain the probability distribution m over P(S) in the first case, or how to justify the particular choice of the demands imposed to belief functions in the other case, are put aside or are “picked before brackets” and they are not taken as a part of Dempster-Shafer theory in its formalized setting.
Ivan Kramosil

### 5. Elementary Properties of Belief Functions

Abstract
In this chapter we shall survey the most elementary properties of belief functions and some other characteristics derived from them (cf. Smets (1992) and the references mentioned in the end of the last chapter, e. g., for more detail). We shall suppose, throughout this chapter, that the probability space ‹Ω, A, P› and the measurable spaces ‹P (S), S› and ‹E, ε› are fixed, the dependence of belief functions on possible variations or modifications of these basic stones of our constructions will be investigated in some of the following chapters. We shall also suppose that if the state space S is finite, then the σ-field S is the maximal one, i. e., S = P(P(S)), so that the values m(A), bel*(A) (and bel(A), if m(φ) < 1) are defined for each AS and obey the usual combinatoric definitions. The properties of belief functions concerning their possible combinations and actualizations will be investigated in the next chapter dealing with the Dempster combination rule.
Ivan Kramosil

### 6. Probabilistic Analysis of Dempster Combination Rule

Abstract
In the real world around us, a subject’s knowledge concerning this world in general, and investigated system(s) and their (its) environments in particular, are not of static, but rather of dynamic nature. In other words, this knowledge is subjected to changes involved by the time passing. These changes can be caused either by the changes taking places either in the world itself, or by changes of the body of evidence and laws of the nature known to the subject. The changes should be applied to the knowledge of sure deterministic nature (more correctly, the knowledge taken as sure in the given context and under the given circumstances), as well as to the knowledge charged by uncertainty. In this work we focus our attention to the knowledge expressed in the terms of compatibility relations, basic probability assignments and belief functions, so that our aim will be, in this chapter, to investigate the ways in which one compatibility relation, b.p.a., or belief function can and should be modified when obtaining some more information described by another compatibility relation, b.p.a., or belief function. As a rule, in Dempster-Shafer theory such a modification (actualization) is realized applying the so called Dempster combination rule. In this chapter we shall introduce this rule using the probabilistic model and terms presented above and we shall discover and formalize explicitly the usually only tacitly assumed hidden assumptions behind this combination rule. As in the foregoing chapters, we shall begin with an informal intuition behind our explanation, leaving this intuition aside and returning to a formalized mathematical level of presentation as soon as possible.
Ivan Kramosil

### 7. Nonspecificity Degrees of Basic Probability Assignments

Abstract
As we have already defined, in the case of a finite basic space S, basic probability assignment (BPA) on S is a probability distribution on the power-set P(S) (set of all subsets of S). In this chapter we shall define and investigate the nonspecificity degree of a BPA given by the normalized expected value of the size (cardinality) of subsets of S with respect to the probability distribution defining the BPA in question. This notion enables to express formally and to prove the intuitive feelings of improving one’s basic probability assignment and belief function when combining it with another one by the Dempster combination rule. It enables also to define a basic probability assignment which can play the role, at least in certain relations, of the BPA inverse to the original one with respect to the Dempster combination rule, even if we know that such an inverse BPA cannot be defined up to the most trivial case of the vacuous BPA m s (m s (S) = 1). Analogous properties of the combination rule dual to the Dempster one will be also briefly investigated.
Ivan Kramosil

### 8. Belief Functions Induced by Partial Compatibility Relations

Abstract
A common feature of the following three chapters consists in their aim to go beyond the framework of the already classical mathematical model for Dempster—Shafer theory, as explained and analyzed till now, in at least the three following directions: (i) to weaken the demands imposed to the notion of compatibility relation as the basic relation binding the empirical data being at the user’s (observer’s) disposal with the hypothetical internal states of the system under investigation (this chapter); (ii) to abandon the assumption that the state space S is finite and to extend the definition of degrees of beliefs to at least some subsets of an infinite space S (the next chapter); (iii) to replace the probabilistic measures used in our definitions of basic probability assignments and belief functions by more general set functions, e. g., by measures or signed measures, in order to generalize the notion of basic probability assignment and belief function so that an operation inverse to the Dempster combination rule were definable if not totally, so at least for a large class of generalized basic probability assignments (Chapter 10).
Ivan Kramosil

### 9. Belief Functions over Infinite State Spaces

Abstract
In order to make the following considerations more transparent, let us recall the basic idea of our definition of belief function (Def. 4.2.1) in the terms of setvalued (generalized) random variables and their probabilistic numerical characteristics (generalized quantiles). Let S be a nonempty set, let SP (P(S)) be a nonempty σ-field of systems of subsets of S, let 〈Ω, A, P〉 be a fixed abstract probability space. Let 〈E, ε〉 be a measurable space over the nonempty space E of possible empirical values, let X: 〈Ω, A, P〉 → 〈E,ε〉 be a random variable, let ρ: S × E →{0,1} be a compatibility relation, let U ρ,X(x) = {s∈S: ρ(s, x) = 1} for each xE. Then the value bel*ρ,X(A) is defined by
$$be{l^{*}}_{{p,X}}(A) = P(\omega \in \Omega :\O \ne {U_{{p,X}}}\left( {X(w)} \right) \subset A\} )$$
(9.1.1)
for each AS for which this probability is defined. In other terms we can say: let Uρ,X (X(·)) be a set-valued (generalized) random variable, i.e. measurable mapping, which takes the probability space 〈Ω, A, P〉 into a measurable space 〈P(S), S〉. Then the (non-normalized) degree of belief bel*ρ,X(A) is defined by (9.1.1) for each A ⊂ S such that the probability in (9.1.1) is defined, hence, the inverse image of A is in A, in other terms, P(A) = {B: B ⊂ A} G SP(P(S)) holds. If, moreover, {θ} e S and P({ω ∈Ω: U ρ,X (ω) = θ}) < 1 hold, the (normalized) degree of belief belρ,X (A) is defined by the conditional probability
$$be{l_{{p,X}}}(A) = P(\omega \in \Omega :{U_{{p,X}}}(\omega ) \subset A\} /\{ \omega \in \Omega :{U_{{p,X}}}(\omega ) \ne \O \} )$$
(9.1.2)
Ivan Kramosil

### 10. Boolean Combinations of Set-Valued Random Variables

Abstract
As we remember, the role of one of the basic building stones in our definition of belief and plausibility functions over infinite sets S was played by a set-valued random variable U, defined on the abstract probability space (Ω,A, P) and taking its values in a measurable space (P(S), S) over the power-set P(S) of all subsets of S. Having at hand two or more such set-valued random variables, an immediate idea arises to define new set-valued random variables, applying boolean set-theoretical operations to the values of the original variables. Namely, let U be a nonempty set of random variables defined on (Cl, A, P) and taking their values in (P(S), S), let UU. We may define set-valued mappings ∩U, ∪U and S — U setting, for each ω∈Ω,
$$\begin{gathered} \left( { \cap ^u } \right)\left( \omega \right) = \cap \{ U\left( \omega \right):u \in u\} , \hfill \\ \left( { \cup ^u } \right)\left( \omega \right) = \cup \{ U\left( \omega \right):u \in u\} , \hfill \\ \left( {S - U} \right)\left( \omega \right) = S - U\left( \omega \right). \hfill \\ \end{gathered}$$
(10.1.1)
.
Ivan Kramosil

### 11. Belief Functions with Signed and Nonstandard Values

Abstract
Both signed belief functions and belief functions with nonstandard values generalize the notion of belief function in the sense that the domain of this function is the same as in the classical case, i. e., the field of all subset of a nonempty set S (as a rule, we shall limit ourselves to finite sets S), but the values are either real numbers including those beyond the scope of the unit interval [0, 1], or even some objects from a more sophisticated structure. A theoretical motivation for such generalization can be given by our attempt to define an operation inverse to the Dempster combination rule ⊕, i. e., to define an operation ⊖ such that, given basic probability assignments (b.p.a.’s) m 1 and m 2 on S, the equality ((m 1m 2) ⊖ m 2) (A) = m 1 (A) would hold for all AS. Although the problem is stated at a purely theoretical and algebraical level, it possesses an intuitive interpretation which is perhaps worth being discussed in more detail. Let us consider a subject whose degrees of belief concerning the membership of the actual state of the investigated system in particular subsets of the set S of all states are quantified by a basic probability assignment (b.p.a.) m 1 and by the corresponding belief function bel m 1. The subject combines her/his beliefs with the beliefs of her/his colleague quantified by a b.p.a. m 2 and by bel m 2, so that she/he obtains the actualized beliefs quantified by m 1m 2 and by bel m 1bel m 2, and completely forgets the original beliefs m 1 and m 2, erasing them totally from her/his memory.
Ivan Kramosil

### 12. Jordan Decomposition of Signed Belief Functions

Abstract
In this chapter we shall try to arrive at some decompositions into generalized or even into classical probabilistic belief functions inspired by, and similar to, the Jordan decomposition of signed measure. Here generalized belief function is a particular case of signed belief function supposing that the signed measure µ used when defining the belief function in question takes only non-negative values (possibly including +∞), so that µ is just what is called simply (σ-additive) measure in Halmos (1950). We shall also prove that generalizations of basic probability assignments, generated on P(S) with a finite S by signed belief functions, are also signed measures on the measurable space 〈 P (S), P (P (S))〉. The following well-known theorem will play the key role of an inspiration, but also as a technical tool for all our reasonings and constructions throughout this chapter.
Ivan Kramosil

### 13. Monte-Carlo Estimations for Belief Functions

Abstract
Let S be a finite set, let E be an empirical space, let ρ: S × E → {0, 1} be a compatibility relation, let X: (〈Ω, A, P〉) →〈E, E〉 be a random variable, let belmρ, x be the belief function defined, for each AS, by Let us suppose that there exists, for each xE, a value sS such that ρ(s, x) = 1 so that Uρ(X(ω)) ≠ ∅ holds for every ω ∈ Ω, hence, mρ,x(∅) = 0.
Ivan Kramosil

### 14. Boolean—Valued and Boolean—Like Processed Belief Functions

Abstract
The reasons for which it may seem useful to reconsider the Dempster—Shafer model of uncertainty quantification and processing from the point of view of possible non-numerical quantification of occurring uncertainty degrees can be divided into two groups: why to refuse the numerical real-valued degrees, and why to choose just this or that set of values and structure over this set as an adequate alternative to the original numerical evaluation. First, there are some general arguments in favour of the claim that structures over sets of abstract objects of non-numerical nature can be sometimes more close to the spaces of uncertain events and structures over them than the space of real numbers with all the riches of notions, relations and operations over these numbers (over-specification of the degrees of uncertainty by real numbers, these degrees need not be dichotomic, a danger of an ontological shift from structures over real numbers to structures over uncertainties, and so on). A more detailed discussion in this direction can be found in Drossos (1990) and Novák (1989), as far as fuzzy sets are concerned, in Bundy (1985) for set-valued probability measures, and in Kramosil (1989,1991) for applications of such probabilities in uncertain data processing expert (knowledge) systems; we shall not repeat this discussion here and refer to these sources.
Ivan Kramosil

Without Abstract
Ivan Kramosil

### Backmatter

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