scholarly journals Probability functions, belief functions and infinite regresses

Synthese ◽  
2020 ◽  
Author(s):  
David Atkinson ◽  
Jeanne Peijnenburg
Keyword(s):  
Rational Belief ◽  
Belief Functions ◽  
Probability Functions ◽  
Degree Of Belief

Abstract In a recent paper Ronald Meester and Timber Kerkvliet argue by example that infinite epistemic regresses have different solutions depending on whether they are analyzed with probability functions or with belief functions. Meester and Kerkvliet give two examples, each of which aims to show that an analysis based on belief functions yields a different numerical outcome for the agent’s degree of rational belief than one based on probability functions. In the present paper we however show that the outcomes are the same. The only way in which probability functions and belief functions can yield different solutions for the agent’s degree of belief is if they are applied to different examples, i.e. to different situations in which the agent finds himself.

scholarly journals Plato's Cave in the Dempster-Shafer land--the Link between Pignistic and Plausibility Transformations

2017 ◽  
Author(s):  
Chunlai Zhou ◽  
Biao Qin ◽  
Xiaoyong Du
Keyword(s):  
Statistical Inference ◽  
Belief Function ◽  
Belief Functions ◽  
Reasoning Under Uncertainty ◽  
Function Model ◽  
Probability Functions ◽  
Degree Of Belief ◽  
Dempster Shafer Theory ◽  
Shafer Theory ◽  
Theory Of Evidence

In reasoning under uncertainty in AI, there are (at least) two useful and different ways of understanding beliefs: the first is as absolute belief or degree of belief in propositions and the second is as belief update or measure of change in belief. Pignistic and plausibility transformations are two well-known probability transformations that map belief functions to probability functions in the Dempster-Shafer theory of evidence. In this paper, we establish the link between pignistic and plausibility transformations by devising a belief-update framework for belief functions where plausibility transformation works on belief update while pignistic transformation operates on absolute belief. In this framework, we define a new belief-update operator connecting the two transformations, and interpret the framework in a belief-function model of parametric statistical inference. As a metaphor, these two transformations projecting the belief-update framework for belief functions to that for probabilities are likened to the fire projecting reality into shadows on the wall in Plato's cave.

A Modified Supervaluationist Framework for Decision-Making

2021 ◽  
Vol 12 (2) ◽  
pp. 175-191
Author(s):  
Jonas Karge ◽  
Keyword(s):  
Decision Making ◽  
Probability Function ◽  
Imprecise Probabilities ◽  
Adequate Account ◽  
Alternative Account ◽  
Semantic Approach ◽  
Probability Functions ◽  
Degree Of Belief ◽  
The Way

How strongly an agent beliefs in a proposition can be represented by her degree of belief in that proposition. According to the orthodox Bayesian picture, an agent's degree of belief is best represented by a single probability function. On an alternative account, an agent’s beliefs are modeled based on a set of probability functions, called imprecise probabilities. Recently, however, imprecise probabilities have come under attack. Adam Elga claims that there is no adequate account of the way they can be manifested in decision-making. In response to Elga, more elaborate accounts of the imprecise framework have been developed. One of them is based on supervaluationism, originally, a semantic approach to vague predicates. Still, Seamus Bradley shows that some of those accounts that solve Elga’s problem, have a more severe defect: they undermine a central motivation for introducing imprecise probabilities in the first place. In this paper, I modify the supervaluationist approach in such a way that it accounts for both Elga’s and Bradley’s challenges to the imprecise framework.

scholarly journals IMAGING ALL THE PEOPLE

Episteme ◽  
2016 ◽  
Vol 14 (4) ◽  
pp. 463-479 ◽  
Author(s):  
Hannes Leitgeb
Keyword(s):  
Decision Theory ◽  
Belief Function ◽  
Belief Functions ◽  
Degree Of Belief ◽  
Causal Decision ◽  
The People ◽  
Formal Result ◽  
Linear Pooling ◽  
The Individual ◽  
Commutativity Property

AbstractIt is well known that aggregating the degree-of-belief functions of different subjects by linear pooling or averaging is subject to a commutativity dilemma: other than in trivial cases, conditionalizing the individual degree-of-belief functions on a piece of evidence E followed by linearly aggregating them does not yield the same result as first aggregating them linearly and then conditionalizing the resulting social degree-of-belief function on E. In the present paper we suggest a novel way out of this dilemma: adapting the method of update or learning such that linear pooling commutes with it. As it turns out, the resulting update scheme – (general) imaging on the evidence – is well-known from areas such as the study of conditionals and causal decision theory, and a formal result from which the required commutativity property is derivable was supplied already by Gärdenfors (1982) in a different context. We end up determining under which conditions imaging would seem to be right method of update, and under which conditions, therefore, group update would not be affected by the commutativity dilemma.

scholarly journals Factfinding Deconstructed

2019 ◽  
Author(s):  
Kevin M. Clermont
Keyword(s):  
Weighted Average ◽  
Belief Function ◽  
Black Box ◽  
Belief Functions ◽  
Standard Of Proof ◽  
Legal Evidence ◽  
Degree Of Belief ◽  
Weighted Arithmetic ◽  
A Chain ◽  
Belief Model

Academics have never agreed on a theory of proof. The darkest corner of disagreement concerns how legal factfinders logically should find facts. This Article pries open that cognitive black box. It does so by employing multivalent logic, which enables it to overcome the traditional probability theory that impeded all prior attempts. The result is the first-ever exposure of the proper logic for finding a fact or a case’s facts.The focus is the evidential processing phase, rather than the application of the standard of proof as tracked in my prior work. Processing evidence involves (1) reasoning inferentially from a piece of evidence to a degree of belief and of disbelief in the element to be proved, (2) aggregating pieces of evidence that all bear to some degree on one element in order to form a composite degree of belief and of disbelief in the element, and (3) considering the series of elemental beliefs to reach a decision. Zeroing in, the factfinder in step #1 should connect each item of evidence to an element to be proved by constructing a chain of inferences, employing multivalent logic’s rules for conjunction and disjunction to form a belief function that reflects the belief and the disbelief in the element and also the uncommitted belief reflecting uncertainty. The factfinder in step #2 should aggregate, by weighted arithmetic averaging, the belief functions of all the items of evidence that bear on any one element, creating a composite belief function for the element. The factfinder in step #3 does not need to combine elements, but instead should directly move to testing whether the degree of belief from each element’s composite belief function sufficiently exceeds the corresponding degree of disbelief. In sum, the factfinder should construct a chain of inferences to produce a belief function for each item of evidence bearing on an element, and then by weighted average produce for each element a composite belief function ready for the element-by-element standard of proof.This Article performs the task of mapping normatively how to reason from legal evidence to a decision on facts. More significantly, it constitutes a further demonstration of how embedded the multivalent-belief model is in our law.

Probability theory and epistemology

2018 ◽  
Author(s):  
Barry Loewer
Keyword(s):  
Probability Theory ◽  
Rational Belief ◽  
Bayesian Epistemology ◽  
Dutch Book ◽  
Degrees Of Belief ◽  
Rational Person ◽  
Degree Of Belief ◽  
New Evidence ◽  
Probabilistic Epistemology ◽  
Instrumental Goal

The primary uses of probability in epistemology are to measure degrees of belief and to formulate conditions for rational belief and rational change of belief. The degree of belief a person has in a proposition A is a measure of their willingness to act on A to obtain satisfaction of their preferences. According to probabilistic epistemology, sometimes called ‘Bayesian epistemology’, an ideally rational person’s degrees of belief satisfy the axioms of probability. For example, their degrees of belief in A and -A must sum to 1. The most important condition on changing degrees of belief given new evidence is called ‘conditionalization’. According to this, upon acquiring evidence E a rational person will change their degree of belief assigned to A to the conditional probability of A given E. Roughly, this rule says that the change should be minimal while accommodating the new evidence. There are arguments, ‘Dutch book arguments’, that are claimed to demonstrate that failure to satisfy these conditions makes a person who acts on their degrees of belief liable to perform actions that necessarily frustrate their preferences. Radical Bayesian epistemologists claim that rationality is completely characterized by these conditions. A more moderate view is that Bayesian conditions should be supplemented by other conditions specifying rational degrees of belief. Support for Bayesian epistemology comes from the fact that various aspects of scientific method can be grounded in satisfaction of Bayesian conditions. Further, it can be shown that there is a close connection between having true belief as an instrumental goal and satisfaction of the Bayesian conditions. Some critics of Bayesian epistemology reject the probabilistic conditions on rationality as unrealistic. They say that people do not have precise degrees of belief and even if they did it would not be possible in general to satisfy the conditions. Some go further and reject the conditions themselves. Others claim that the conditions are much too weak to capture rationality and that in fact almost any reasoning can be characterized so as to satisfy them. The extent to which Bayesian epistemology contributes to traditional epistemological concerns of characterizing knowledge and methods for obtaining knowledge is controversial.

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