A notion of conditional probability and some of its consequences (original) (raw)

When upper conditional probabilities are conditional possibility measures

Proceedings of the 2015 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and Technology, 2015

Conditioning for (non-additive) uncertainty measures is still an open problem. These measures can arise through probabilistic inference procedures, as in the case of possibility measures, that can be seen as the upper envelope of the extensions of a probability when the corresponding algebras are weakly logically independent. The aim of this paper is to define a conditioning rule (B-conditioning) such that the upper envelope of the extensions of a full conditional probability on an algebra A is a full Bconditional possibility on another algebra A under weak logical independence of A, A .

Revisiting the Conditional Construal of Conditional Probability

Logic and Logical Philosophy

We show how to extend any finite probability space into another finite one which satisfies the conditional construal of conditional probability for the original propositions, given some maximal allowed degree of nesting of the conditional. This mitigates the force of the well-known triviality results.

Multiple Perspectives on the Concept of Conditional Probability

Avances de Investigación en Educación Matemática

Conditional probability is a key to the subjectivist theory of probability; however, it plays a subsidiary role in the usual conception of probability where its counterpart, namely independence is of basic importance. The paper investigates these concepts from various perspectives in order to shed light on their multi-faceted character. We will include the mathematical, philosophical, and educational perspectives. Furthermore, we will inspect conditional probability from the corners of competing ideas and solving strategies. For the comprehension of conditional probability, a wider approach is urgently needed to overcome the well-known problems in learning the concepts, which seem nearly unaffected by teaching.Conditional probability is a key to the subjectivist theory of probability; however, it plays a subsidiary role in the usual conception of probability where its counterpart, namely independence is of basic importance. The paper investigates these concepts from various perspect...

Conditional Probability from an Ontological Point of View

2013

The RATIO analysis of conditional probability stipulates that the probability of A given B is calculated from the probabilities of A and B according to the following formula: P(A|B) = P(A & B) / P(B) (when P(B)> 0). This is criticised by Alan Hájek for the cases where the RATIO analysis fails to deliver a calculation but where our intuitive judgment of conditional probability is clear. Here it is argued that there are three counter-intuitive results of the RATIO analysis of conditional probability even when a calculation is delivered. With Hájek we conclude that conditional probabilities cannot be treated as a function of unconditional probabilities. The conclusion is broader than Hájek’s, though. While he takes conditional probabilities to be irreducible, we argue that it is conditionality as such that is basic and irreducible. This is an ontological account of conditionals, which marks a connection between the antecedent and consequent conditions and where the conditional has t...

Two Theories of Conditional Probability and Non-Conglomerability

Conglomerability of conditional probabilities is suggested by some (e.g., Walley, 1991) as necessary for rational degrees of belief. Here we give sufficient conditions for non-conglomerability of conditional probabilities in the de Finetti/Dubins sense. These sufficient conditions cover familiar cases where P(⋅) is a continuous, countably additive probability. In this regard, we contrast the de Finetti/Dubins sense of conditional probability with the more familiar account of regular conditional distributions, in the fashion of Kolmogorov.

Conditional Belief Functions: a Comparison among Different Definitions

By following the idea that a conditional measure is more than a sort of a rearrangement of an unconditional measure, we are interested in study- ing the controversial concept of conditional belief function as a generalized decomposable conditional measure, that is a suitable real function '(¢j¢), deflned on a family of conditional events, ruled by a set of axioms involved two composition rules ' and fl.

You Can't Always Get What You Want: Some considerations regarding conditional probability

Forthcoming in Erkenntnis, 2014

The standard treatment of conditional probability leaves conditional probability undefined when the conditioning proposition has zero probability. Nonetheless, some find the option of extending the scope of conditional probability to include zero-probability conditions attractive or even compelling. This articles reviews some of the pitfalls associated with this move, and concludes that, for the most part, probabilities conditional on zero-probability propositions are more trouble than they are worth. * But if you try, sometimes, you might find you get what you need.

Some basic theorems of qualitative probability

Studia Logica, 1975

The main aim of this paper is to study the logic of a binary sentential operator 'z=', with the intended meaning 'is at least as probable as'. The object language will be simple; to an ordinary language for truth-functional connectives we add '&' as the only intensional operator. Our choice of axioms is heavily dependent of a theorem due to Kraft et al. [8], which states necessary and sufficient conditions for an ordering of the elements of a finite subset algebra to be compatible with some probability measure. Following a construction due to Segerberg [ 121, we show that these conditions can be translated into our language.

Characterization of Coherent Conditional Probabilities as a Tool for Their Assessment and Extension

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 1996

A major purpose of this paper is to show the broad import and applicability of the theory of probability as proposed by de Finetti, which differs radically from the usual one (based on a measure-theoretic framework). In particular, with reference to a coherent conditional probability, we prove a characterization theorem, which provides also a useful algorithm for checking coherence of a given assessment. Moreover it allows to deepen and generalise in useful directions de Finetti’s extension theorem (dubbed as “the fundamental theorem of probability”), emphasising its operational aspects in many significant applications.