65 research outputs found
Transferrable Plausibility Model - A Probabilistic Interpretation of Mathematical Theory of Evidence
This paper suggests a new interpretation of the Dempster-Shafer theory in
terms of probabilistic interpretation of plausibility. A new rule of
combination of independent evidence is shown and its preservation of
interpretation is demonstrated.Comment: Pre-publication version of: M.A. K{\l}opotek: Transferable
Plausibility Model - A Probabilistic Interpretation of Mathematical Theory of
Evidence O.Hryniewicz, J. Kacprzyk, J.Koronacki, S.Wierzcho\'{n}: Issues in
Intelligent Systems Paradigms Akademicka Oficyna Wydawnicza EXIT, Warszawa
2005 ISBN 83-87674-90-7, pp.107--11
Reasoning From Data in the Mathematical Theory of Evidence
Mathematical Theory of Evidence (MTE) is known as a foundation for reasoning
when knowledge is expressed at various levels of detail. Though much research
effort has been committed to this theory since its foundation, many questions
remain open. One of the most important open questions seems to be the
relationship between frequencies and the Mathematical Theory of Evidence. The
theory is blamed to leave frequencies outside (or aside of) its framework. The
seriousness of this accusation is obvious: no experiment may be run to compare
the performance of MTE-based models of real world processes against real world
data.
In this paper we develop a frequentist model of the MTE bringing to fall the
above argument against MTE. We describe, how to interpret data in terms of MTE
belief functions, how to reason from data about conditional belief functions,
how to generate a random sample out of a MTE model, how to derive MTE model
from data and how to compare results of reasoning in MTE model and reasoning
from data.
It is claimed in this paper that MTE is suitable to model some types of
destructive processesComment: presented as poster M.A. K{\l}opotek: Reasoning from Data in the
Mathematical Theory of Evidence. [in:] Proc. Eighth International Symposium
On Methodologies For Intelligent Systems (ISMIS'94), Charlotte, North
Carolina, USA, October 16-19, 1994. arXiv admin note: text overlap with
arXiv:1707.0388
Evidence Against Evidence Theory (?!)
This paper is concerned with the apparent greatest weakness of the
Mathematical Theory of Evidence (MTE) of Shafer \cite{Shafer:76}, which has
been strongly criticized by Wasserman \cite{Wasserman:92ijar} - the
relationship to frequencies.
Weaknesses of various proposals of probabilistic interpretation of MTE belief
functions are demonstrated.
A new frequency-based interpretation is presented overcoming various
drawbacks of earlier interpretations.Comment: 30 pages. arXiv admin note: substantial text overlap with
arXiv:1704.0400
What Does a Belief Function Believe In ?
The conditioning in the Dempster-Shafer Theory of Evidence has been defined
(by Shafer \cite{Shafer:90} as combination of a belief function and of an
"event" via Dempster rule.
On the other hand Shafer \cite{Shafer:90} gives a "probabilistic"
interpretation of a belief function (hence indirectly its derivation from a
sample). Given the fact that conditional probability distribution of a
sample-derived probability distribution is a probability distribution derived
from a subsample (selected on the grounds of a conditioning event), the paper
investigates the empirical nature of the Dempster- rule of combination.
It is demonstrated that the so-called "conditional" belief function is not a
belief function given an event but rather a belief function given manipulation
of original empirical data.\\ Given this, an interpretation of belief function
different from that of Shafer is proposed. Algorithms for construction of
belief networks from data are derived for this interpretation.Comment: 13 page
Identification and Interpretation of Belief Structure in Dempster-Shafer Theory
Mathematical Theory of Evidence called also Dempster-Shafer Theory (DST) is
known as a foundation for reasoning when knowledge is expressed at various
levels of detail. Though much research effort has been committed to this theory
since its foundation, many questions remain open. One of the most important
open questions seems to be the relationship between frequencies and the
Mathematical Theory of Evidence. The theory is blamed to leave frequencies
outside (or aside of) its framework. The seriousness of this accusation is
obvious: (1) no experiment may be run to compare the performance of DST-based
models of real world processes against real world data, (2) data may not serve
as foundation for construction of an appropriate belief model.
In this paper we develop a frequentist interpretation of the DST bringing to
fall the above argument against DST. An immediate consequence of it is the
possibility to develop algorithms acquiring automatically DST belief models
from data. We propose three such algorithms for various classes of belief model
structures: for tree structured belief networks, for poly-tree belief networks
and for general type belief networks.Comment: An internal report 199
Approximation by filter functions
In this exploratory article, we draw attention to the common formal ground
among various estimators such as the belief functions of evidence theory and
their relatives, approximation quality of rough set theory, and contextual
probability. The unifying concept will be a general filter function composed of
a basic probability and a weighting which varies according to the problem at
hand. To compare the various filter functions we conclude with a simulation
study with an example from the area of item response theory
Probability as a Modal Operator
This paper argues for a modal view of probability. The syntax and semantics
of one particularly strong probability logic are discussed and some examples of
the use of the logic are provided. We show that it is both natural and useful
to think of probability as a modal operator. Contrary to popular belief in AI,
a probability ranging between 0 and 1 represents a continuum between
impossibility and necessity, not between simple falsity and truth. The present
work provides a clear semantics for quantification into the scope of the
probability operator and for higher-order probabilities. Probability logic is a
language for expressing both probabilistic and logical concepts.Comment: Appears in Proceedings of the Fourth Conference on Uncertainty in
Artificial Intelligence (UAI1988
Evidential Reasoning in a Categorial Perspective: Conjunction and Disjunction of Belief Functions
The categorial approach to evidential reasoning can be seen as a combination
of the probability kinematics approach of Richard Jeffrey (1965) and the
maximum (cross-) entropy inference approach of E. T. Jaynes (1957). As a
consequence of that viewpoint, it is well known that category theory provides
natural definitions for logical connectives. In particular, disjunction and
conjunction are modelled by general categorial constructions known as products
and coproducts. In this paper, I focus mainly on Dempster-Shafer theory of
belief functions for which I introduce a category I call Dempster?s category. I
prove the existence of and give explicit formulas for conjunction and
disjunction in the subcategory of separable belief functions. In Dempster?s
category, the new defined conjunction can be seen as the most cautious
conjunction of beliefs, and thus no assumption about distinctness (of the
sources) of beliefs is needed as opposed to Dempster?s rule of combination,
which calls for distinctness (of the sources) of beliefs.Comment: Appears in Proceedings of the Seventh Conference on Uncertainty in
Artificial Intelligence (UAI1991
A New Approach to Updating Beliefs
We define a new notion of conditional belief, which plays the same role for
Dempster-Shafer belief functions as conditional probability does for
probability functions. Our definition is different from the standard definition
given by Dempster, and avoids many of the well-known problems of that
definition. Just as the conditional probability Pr (lB) is a probability
function which is the result of conditioning on B being true, so too our
conditional belief function Bel (lB) is a belief function which is the result
of conditioning on B being true. We define the conditional belief as the lower
envelope (that is, the inf) of a family of conditional probability functions,
and provide a closed form expression for it. An alternate way of understanding
our definition of conditional belief is provided by considering ideas from an
earlier paper [Fagin and Halpern, 1989], where we connect belief functions with
inner measures. In particular, we show here how to extend the definition of
conditional probability to non measurable sets, in order to get notions of
inner and outer conditional probabilities, which can be viewed as best
approximations to the true conditional probability, given our lack of
information. Our definition of conditional belief turns out to be an exact
analogue of our definition of inner conditional probability.Comment: Appears in Proceedings of the Sixth Conference on Uncertainty in
Artificial Intelligence (UAI1990
Hybrid Probabilistic Programs: Algorithms and Complexity
Hybrid Probabilistic Programs (HPPs) are logic programs that allow the
programmer to explicitly encode his knowledge of the dependencies between
events being described in the program. In this paper, we classify HPPs into
three classes called HPP_1,HPP_2 and HPP_r,r>= 3. For these classes, we provide
three types of results for HPPs. First, we develop algorithms to compute the
set of all ground consequences of an HPP. Then we provide algorithms and
complexity results for the problems of entailment ("Given an HPP P and a query
Q as input, is Q a logical consequence of P?") and consistency ("Given an HPP P
as input, is P consistent?"). Our results provide a fine characterization of
when polynomial algorithms exist for the above problems, and when these
problems become intractable.Comment: Appears in Proceedings of the Fifteenth Conference on Uncertainty in
Artificial Intelligence (UAI1999
- …