On Two Notions of Independence in Evidence Theory

Workshop on the Theory of Belief …, 2010

Many extensions of classical stochastic independence have been proposed when working with probability sets to represent uncertainty. As belief functions can be seen as particular instances of such probability sets, some authors have investigated how these extensions can be reinterpreted and retrieved in the particular framework of belief functions. They have mainly focused on the so-called notions of random set independence, fuzzy non-interaction, strong independence and unknown interaction. In this paper, we pursue this effort in two ways: first by showing that the notion of epistemic irrelevance, central in Walley theory of lower previsions, can be likewise reinterpreted in terms of belief functions; second by considering the more general case where mass assignments inducing belief functions are themselves imprecise.

5th International Symposium on Imprecise …, 2007

Philosophy of Science, 2001

A Bayesian account of independent evidential support is outlined. This account is partly inspired by the work of C.S. Peirce. I show that a large class of quantitative Bayesian measures of confirmation satisfy some basic desiderata suggested by Peirce for adequate accounts of independent evidence. I argue that, by considering further natural constraints on a probabilistic account of independent evidence, all but a very small class of Bayesian measures of confirmation can be ruled out. In closing, another application of my account to the problem of evidential diversity is also discussed.

Philosophy of Science, 2013

The variety-of-evidence thesis says that the degree of warrant given to a hypothesis by a body of evidence increases with the variety of this body of evidence, ceteris paribus. Many philosophers claimed to be able to provide a Bayesian proof of this thesis until ) proposed a Bayesian model in which independence backfires under special circumstances. This article points to two limitations in the model of Bovens and Hartmann: their conceptualization of unreliable evidential sources, and the restriction to comparing full independence to full dependence. It is shown that the variety-of-evidence thesis is rehabilitated when unreliable sources are reconceptualized. It turns out however that also allowing for degrees of independence leads to qualify the variety-ofevidence thesis: as Bovens and Hartmann claimed, more independence does not always imply stronger confirmation.

2009

The goal of the paper is to recall a recently introduced concept of conditional independence in evidence theory and to discuss Markov properties based on this independence concept.

2011

The goal of the paper is to reveal the relationships between recently introduced concept of conditional independence in evidence theory and those (dependent on the choice of conditioning rule) of conditional irrelevance.

In this paper we discuss the nature of independence of sources in the theory of evidence from an algebraic point of view, starting from an analogy with projective geometries. Independence in Dempster's rule is equivalent to independence of frames as Boolean algebras. Collection of frames, in turn, can be given several algebraic interpretations in terms of semimodular lattices, matroids, and geometric lattices. Each of those structures are endowed with a particular notion of independence, which we prove to be distinct even though related to independence of frames. We show that the latter is in fact opposed to classical linear independence, giving collection of frames the structure of "anti-matroids".

Patrick Suppes: Scientific Philosopher, 1994

As the paper explains, it is crucial to epistemology in general and to the theory of causation in particular to investigate the properties of conditional independence as completely as possible. The paper summarizes the most important results concerning conditional independence with respect to two important representations of epistemic states, namely (strictly positive) probability measures and natural conditional (or disbelief or ranking) functions. It finally adds some new observations.

International Journal of Approximate Reasoning, 2000

The aim of this paper is that of studying a notion of independence for imprecise probabilities which is essentially based on the intuitive meaning of this concept. This is expressed, in the case of two events, by the reciprocal irrelevance of the knowledge of the value of each event for evaluating the other one, and has been termed epistemic independence. In order to consider more general situations in the framework of coherent imprecise probabilities, a de®nition of (epistemic) independence is introduced referring to arbitrary sets of logically independent partitions. Logical independence is viewed as a natural prerequisite for epistemic independence. It is then proved that the de®nition is always consistent, its relationship with the factorization rule is analysed, and some of its more relevant implications are discussed. Ó

Annals of Mathematics and Artificial Intelligence, 2002

This paper investigates the concept of strong conditional independence for sets of probability measures. Couso, Moral and Walley [7] have studied different possible definitions for unconditional independence in imprecise probabilities. Two of them were considered as more relevant: epistemic independence and strong independence. In this paper, we show that strong independence can have several extensions to the case in which a conditioning to the value of additional variables is considered. We will introduce simple examples in order to make clear their differences. We also give a characterization of strong independence and study the verification of semigraphoid axioms.