EL TESTIMONIO CONDUCE A LA REFLEXIÓN

The rest of the thesis is organized as follows. In Chapter 2 we briefly in- troduce the technical concepts used throughout the thesis. We first recall some basic notions of Probability Theory, Information Theory and dis- cuss some relationships between Information Theory and Statistics. We describe the basic concepts of Bayesian Decision Theory and, introduce the coordination language KLAIMand related analysis tools. Then, we give an overview of probabilistic trust approaches by describing some of the models proposed in the literature. Finally, we show some cases of reputation systems that are successfully used in real applications.

In Chapter 3 we present our general framework, based on Bayesian decision theory, for the assessment of trust and reputation models. We introduce the framework and discuss our results on the analysis of such systems. We close the chapter by presenting an extention of the frame- work for different data models, with rating values given in different ways.

In Chapter 4 we introduce a verification approach for reputation sys- tems that is based on the use of the coordination language KLAIMand related analysis tools. We define a parametric KLAIM specification of a reputation system that can be instantiated with different reputation models. Then, we consider stochastic specifications enabling quantita- tive analysis of properties of the considered system. Finally, we present verification results on some reputation systems.

In Chapter 5 we present NEVER, a software tool for network-aware evaluation of reputation systems and their rapid prototyping. The NEVER evaluation of reputation systems is carried out through exper- iments performed according to user-specified parameters. In such ex- periments the networked execution environment is explicitly taken into

account. We close the chapter by presenting some evaluation results ob- tained with our tool.

In Chapter 6 we comment on the research results presented in the the- sis by also comparing them with more closely related work. We summa- rize the main contributions and propose possible directions for further research.

Chapter 2

Preliminaries

In this chapter we briefly introduce the technical concepts used in the rest of the thesis. In Section 2.1 we recall some basic notions of Probabil- ity Theory. In Section 2.2 we introduce Information Theory and Statistics and show their relationship. In Section 2.3 we describe the basic concepts of Bayesian Decision Theory and in Section 2.4 we introduce the coordi- nation language KLAIMand related analysis tools. Finally, in Section 2.5 we give an overview of probabilistic trust approaches and in Section 2.6 we describe the operation of some reputation systems used in real appli- cations.

2.1

Probability Theory

Probability theory is technically a branch of measure theory, it reasons about chance, uncertainty, likelihood of phenomena. In this section, we first introduce few notions of measure theory then we recall some basic notions of probability theory. We refer the reader to [Wil91; Sti99; GS01; Kal02] for an extensive presentation of these topics.

Below we provide the definitions of σ-f ield, measure space, measur- able set and measure to then introduce some basic notions of probability theory.

F of subsets of Ω satisfying the following conditions: (a) ∅ ∈ F ,

(b) A ∈ F ⇒ Ac_{∈ F , where A}c_{denotes the complement set,}

(c) A1, A2, ..., An∈ F with n ∈ N ⇒ S_nAn ∈ F

The smallest σ-f ield associated with Ω is the collection F = {∅, Ω} and the largest one is the power set of Ω, written 2Ω_{. A measurable set is a}

pair (Ω, F), where Ω is a set and F is a σ-f ield on Ω.

Definition 2.1.2 Let (Ω, F ) a measurable set, ameasure on (Ω, F) is a func- tion µ : F → [0, ∞) such that:

(a) µ(∅) = 0, (b) µ(A) = P

nµ(An), where An(n ∈ N) is a sequence of pairwise dis-

joint sets in F with union A =S

nAn.

The triple (Ω, F, µ) is then called a measure space.

A probability measure P on (Ω, F) is a measure such that P(Ω) = 1. The triple (Ω, F, P) is called probability space and the set Ω is called sample space. A point ω of Ω is called a sample point or outcome. The collection F is called family of events and an event is an element of F. If A and B are two events and P(B) > 0, then the conditional probability that A occurs given that B occurs is defined as

P(A|B) = P(A ∩ B)

P(B) (2.1)

We denote with P(A|B) the conditional probability and we read “the probability of A given (or conditioned on) B”. It is not always the case that the occurrence of an event B changes the probability that an- other event A occurs. If the conditional probability P(A|B) remains un- changed, i.e. P(A|B) = P(A), then we say that A and B are independent. More formally, events A and B are called independent if

Experimental outcomes are not always numerical, but it is often bet- ter to work with numbers than with outcomes in the original sample space. We can assign a number to any outcome ω ∈ Ω using random vari- ables. A random variable is a real-valued function X : Ω → R such that {ω ∈ Ω : X(ω) ≤ x} ∈ F for each x ∈ R. We can think of a random variable just as a function mapping Ω in R. We denote random variables with upper-case letters, such as X, Y, Z and their possible numerical val- ues with lower-case letters, such as x, y, z. A distribution function is as- sociated with every random variable. A distribution function of a random variable X is a function F : R → [0, 1] such that F (x) = P(X ≤ x), where {X ≤ x} denotes the event {ω ∈ Ω : X(ω) ≤ x}. We denote with FXthe

distribution function of the random variable X.

A random variable X is called discrete if it takes values only in some countable subset X of R. We denote with calligraphic letters X , Y, Z pos- sible subsets of R. The probability mass function of X is p : R → [0, 1] such that p(x) = P(X = x). A random variable X is called continuous if it takes values in some uncountable subset X of R and if its distribution function can be expressed as F (x) = Rx

−∞p(u)du, for some integrable function

p(x)called the probability density function of X.

Let X be a discrete random variable, its expected value, denoted by E[X], is defined as

E[X] =X

x

xP(X = x) (2.3)

Notation: Let X be a random variable taking values in X , we say that X is distributed according to a probability distribution p(·) if for each x ∈ X, P(X = x) = p(x), and we write X ∼ p(·). We will use the term probability distribution both denoting probability mass function and probability density function, the use will be clear by the context. The support of p(·) is defined as supp(p) = {x ∈ X : p(x) > 0}. We let pn_{(·)denote the n-th extension of p(·), defined as p}n_(xn_{) =} Qn

i=1p(xi),

where xn _{= (x}

1, x2, ..., xn); this is in turn a probability distribution on

the set Xn_{. For any A ⊆ X we let p(A) denote}P

x∈Ap(x). When A ⊆ X n