Conditional Versus Joint Probability Assessments (original) (raw)

Conditional events in probability assessment and revision

IEEE Transactions on Systems, Man, and Cybernetics, 1994

By resorting to the general interpretation of an event as a proposition, we deal with its natural generalization, the concept of conditional event. No algebraic structure is put on the given family of conditional events. An important feature of our approach is that conditional probability makes sense also when the conditioning event is null. Our attention is centered on de Finetti's approach to probability, whose distinguishing features make it particularly flexible for application to both artificial intelligence and inferential statistics. In particular, the problem of comparing null events by means of conditional events is considered: so the need for the introduction of iterated conditioning naturally arises when we try to compare conditional events of zero conditional probability.

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.

The probability of conditionals: A review

Psychonomic Bulletin & Review, 2021

A major hypothesis about conditionals is the Equation in which the probability of a conditional equals the corresponding conditional probability: p(if A then C) = p(C|A). Probabilistic theories often treat it as axiomatic, whereas it follows from the meanings of conditionals in the theory of mental models. In this theory, intuitive models (system 1) do not represent what is false, and so produce errors in estimates of p(if A then C), yielding instead p(A & C). Deliberative models (system 2) are normative, and yield the proportion of cases of A in which C holds, i.e., the Equation. Intuitive estimates of the probability of a conditional about unique events: If covid-19 disappears in the USA, then Biden will run for a second term, together with those of each of its clauses, are liable to yield joint probability distributions that sum to over 100%. The error, which is inconsistent with the probability calculus, is massive when participants estimate the joint probabilities of conditionals with each of the different possibilities to which they refer. This result and others under review corroborate the model theory.

Local coherence of conditional probability assessments: definition and application

In this paper we face the problem of checking the coherence of a conditional probability partial assessment (i.e. defined on a domain without structure). A procedure known in literature has the drawback to introduce, in the worst cases, a so hight number of unknowns to become impossible to apply. Hence we introduce properties, bases on the new notion of local coherence, that, when satisfied by the assessment, allow to reduce the cardinality of domain under investigation. Once the domain is reduced the already known procedure can be applied. Since these reduction rules are based on the local coherence, the correctness is guaranteed and we prove also the efficiency of a computational strategy based on them.

Talking About Probabilities: A Logical Problem for Or/MS

Decision Sciences, 1984

The correct interpretation of a natural language statement is determined as much by convention and shared meanings as by logical content. Therefore, when the word "probability" is used in the statement of a decision problem, the intended interpretation is not always clear. If such a statement also contains explicit probability formulae, confusion and even paradox may result. In this article, a problem involving cascaded inference is interpreted in four ways, three of which are regarded by the authors as legitimate or reasonable. The problem was originally suggested by Einhorn 131 and further discussed by Libby [lo]. It is suggested here that a formal, mathematicist interpretation of the word "probability" might lead to inappropriate analyses of some decisions.

Prequential Probability: Principles and Properties

Bernoulli, 1999

Forecaster has to predict, sequentially, a string of uncertain quantities (X 1 ; X 2 ; : : :), whose values are determined and revealed, one by one, by Nature. Various criteria may be proposed to assess Forecaster's empirical performance. The Weak Prequential Principle requires that such a criterion should depend on the forecaster's behaviour or strategy only through the actual forecasts issued. A wide variety of appealling criteria is shown to respect this Principle. We further show that many such criteria also obey the Strong Prequential Principle, which requires that, when both Nature and Forecaster make their choices in accordance with a common joint distribution P for (X 1 ; X 2 ; : : :), certain stochastic properties, underlying and justifying the criterion and inferences based on it, hold regardless of the detailed speci cation of P.

A general model to handle uncertainty, based on coherent conditional probabilities

Any time we draw a partial or definite conclusion or we take a decision, in a frame of partial and revisable knowledge, we put into action a complex reasoning process. In this process we take into account our (qualitative and quantitative)information, inference rules by means we extend the information to some object in which direct information is not available. This kind of reasoning, which is clearly non monotonic is in fact at the basis of the processing of scientific theories, but also in many decision process, in which the decision maker has a partial knowledge of the involved objects and moreover it is expressed in different (some time) informal languages. During the last two decades, for handling the reasoning above described, Romano Scozzafava and I, with the important contribute of some our pupils (in particular Barbara Vantaggi), developed a model of generalized inference, based on the theory of coherent conditional probabilities (essentially introduced by de Finetti) and the relevant extension problems. This innovatory paradigm relies on many theoretical tools that are fit to manage the uncertainty caused by different circumstances but reducible in fact to a lack of complete information. Moreover it is able to handle both domains without particular Boolean structure with partial qualitative or comparative assessments, and the general knowledge acquisition process, which is in fact a dynamic enlargement both of the domain and of the qualitative or quantitative assessment of the degree of belief. It provides, directly and indirectly a frame suitable for a unitary handling of many theories present in the literature, both directly and indirectly:-directly: some of uncertainty or information measures (possibilities, belief functions, generalized measure of information, fuzzy set theory) can be re-read as particular coherent conditional probabilities and some inferential rules for non-monotonic logic (such as default logic etc.) can be regarded as an extension process of particular assessments of coherent conditional probabilities;-indirectly: for general (non additive) measures of uncertainty or of information , the method of a direct approach to the conditioning and the general point of view regarding inference can be used to solve many problems related to conditioning itself, and to the related concept of independence and relevant graphical structures. The theory is based on the most general concept of event and conditional event, in a direct introduction of conditional probability as a function defined on conditional events satisying a set of axioms. An event is any fact singled-out by a (nonambiguous) statement E, that is a (Boolean) proposition that can be either true or false (corresponding to the two values 1 or 0 of the indicator I E of E). A conditional event is any ordered pair of events (E|H), with H = ∅ which can be both true or false and are different only in the played role (H plays the role of hypothesis). It is in fact essential to regard also the conditioning

Correction of incoherent conditional probability assessments

International Journal of Approximate Reasoning, 2010

In this paper we deep in the formal properties of an already stated discrepancy measure between a conditional assessment and the class of unconditional probability distributions compatible with the assessment domain.

If-clauses and probability operators-preprint (2010, with Paul Egré)

Topoi, 2011

Adams' thesis is generally agreed to be linguistically compelling for simple conditionals with factual antecedent and consequent. We propose a derivation of Adams' thesis from the Lewis-Kratzer analysis of if-clauses as domain restrictors, applied to probability operators. We argue that Lewis's triviality result may be seen as a result of inexpressibility of the kind familiar in generalized quantifi er theory. Some implications of the Lewis-Kratzer analysis are presented concerning the assignment of probabilities to compounds of conditionals.

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.