Some teasers concerning conditional probabilities

A family of notorious teasers in probability is discussed. All ask for the probability that the objects of a certain pair both have some property when information exists that at least one of them does. These problems should be solved using conditional probabilities, but cause difficulties in characterizing the conditioning event appropriately. In particular, they highlight the importance of determining the way information is being obtained. A probability space for modeling verbal problems should allow for the representation of the given outcome and the statistical experiment which yielded it. The paper gives some psychological reasons for the tricky nature of these problems, and some practical tips for handling them.     

On the significance of conditional probabilities

The orthodoxy that conditional probabilities reflect what are for a subject evidential bearings is seconded. This significance suggests that there should be principles equating rationally revised probabilities on new information with probabilities reached by conditionalizing on this information. Several principles, two of which are endorsed, are considered. A book is made against a violator of these, and it is argued that there must be something wrong with a person against whom such books can be made. Appendices comment on Popper-functions, elaborate on bets and odds, and relate dutch books and strategies to conditions of inconsistency (Ramsey's idea) and imperfection.This paper corrects and elaborates two-fold Conditional Probabilities, Conditionalization, and Dutch Books, PSA 1990: Proceedings of the Biennial Meeting of the Philosophy of Science Association, Volume 1, edited by A. Fine, M. Forbes, and L. Wessels, East Lansing, Michigan, 1990.     

Understanding conditional promises and threats

Conditional promises and threats are speech acts that are used to manipulate other people's behaviour. Studies on human reasoning typically use propositional logic to analyse what people infer from such inducements. While this approach is sufficient to uncover conceptual features of inducements, it fails to explain them. To overcome this limitation, we propose a multilevel analysis integrating motivational, linguistic, deontic, behavioural, and emotional aspects. Commonalities and differences between conditional promises and threats on various levels were examined in two experiments. The first shows that both types of inducements are understood as being complementary on the linguistic level, but not reversible, due to the specific temporal order of their actions. In addition, it gives a first assessment of emotional reactions. The second experiment investigated the novel question of whether complementary promises and threats, despite semantic differences, both imply an obligation to cooperate on the deontic level. The data corroborate this hypothesis, and they support various appraisal-theoretical assumptions on the elicitation of emotions. They also reveal that content affects not only the attribution of emotions, but also the deontic interpretation.     

On the logic of nonmonotonic conditionals and conditional probabilities

I will describe the logics of a range of conditionals that behave like conditional probabilities at various levels of probabilistic support. Families of these conditionals will be characterized in terms of the rules that their members obey. I will show that for each conditional, , in a given family, there is a probabilistic support level r and a conditional probability function P such that, for all sentences C and B, CB holds just in case P[B|C]r. Thus, each conditional in a given family behaves like conditional probability above some specific support level.Chris Swoyer provided very helpful comments on drafts of this paper.     

Nonmonotonic conditionals that behave like conditional probabilities above a threshold

I'll describe a range of systems for nonmonotonic conditionals that behave like conditional probabilities above a threshold. The rules that govern each system are probabilistically sound in that each rule holds when the conditionals are interpreted as conditional probabilities above a threshold level specific to that system. The well-known preferential and rational consequence relations turn out to be special cases in which the threshold level is 1. I'll describe systems that employ weaker rules appropriate to thresholds lower than 1, and compare them to these two standard systems.     

The use of conditional probabilities in applications of Holland's theory

The hexagon of J. L. Holland (1973, Making Vocational Choices: A Theory of Careers, Englewood Cliffs, NJ, Prentice-Hall) is a useful heuristic device for predicting congruency and consistency among the six personality/environment types specified by his theory. One limitation of the traditional hexagon is the implicit bidirectionality: the hexagon displays a single distance between two types. People in applied areas may assume a symmetry which does not exist. Examples are given of situations in which a consistent asymmetry occurs; e.g., in the illustrative data presented, C (Conventional) types are more likely to have E (Enterprising) as a secondary interest than E types are to have C as a secondary interest. Implications of such asymmetries are discussed and suggestions made concerning counseling and other applications.     

A high-speed algorithm for computing conditional probabilities of substrings of sequentially observed data

An algorithm is described that computes relative frequencies of occurrence of all arbitrarily long substrings of sequential data, such as are obtained from experiments in learning/memory and verbal interaction. The algorithm offers high speed and provides systematization for the computation of empirical conditional probabilities. Use of this algorithm allows application of probabilistic and information theoretic disciplines to reveal dependencies between events separated arbitrarily in time.     

Counterfactuals vs. conditional probabilities: A critical analysis of the counterfactual theory of information

Cohen and Meskin 2006 Cohen, Jonathan and Meskin, Aaron. 2006. An Objective Counterfactual Theory of Information. Australasian Journal of Philosophy, 84: 333–352. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] recently offered a counterfactual theory of information to replace the standard probabilistic theory of information. They claim that the counterfactual theory fares better than the standard account on three grounds: first, it provides a better framework for explaining information flow properties; second, it requires a less expensive ontology; and third, because it does not refer to doxastic states of the information-receiving organism, it provides an objective basis. In this paper, I show that none of these is really an advantage. Moreover, the counterfactual theory fails to satisfy one of the basic properties of information flow, namely the Conjunction principle. Thus, I conclude, there is no reason to give up the standard probabilistic theory for the counterfactual theory of information.     

The meaning(s) of conditionals: conditional probabilities,mental models,and personal utilities

Five experiments were conducted to test the hypothesis that people understand conditional statements ("if p then q") as indicating a high conditional probability P(q/p). Participants estimated the probability that a given conditional is true (Experiments 1A, 1B, and 3) or judged whether a conditional was true or false (Experiments 2 and 4) given information about the frequencies of the relevant truth table cases. Judgments were strongly influenced by the ratio of pq to p not q cases, supporting the conditional probability account In Experiments 1A, 1B, and 3, judgments were also affected by the frequency of pq cases, consistent with a version of mental model theory. Experiments 3 and 4 extended the results to thematic conditionals and showed that the pragmatic utility associated with believing a statement also affected the degree of belief in conditionals but not in logically equivalent quantified statements.     

Physical probabilities

A conception of probability as an irreducible feature of the physical world is outlined. Propensity analyses of probability are examined and rejected as both formally and conceptually inadequate. It is argued that probability is a non-dispositional property of trial-types; probabilities are attributed to outcomes as event-types. Brier's Rule in an objectivist guise is used to forge a connection between physical and subjective probabilities. In the light of this connection there are grounds for supposing physical probability to obey some standard set of axioms. However, there is no a priori reason why this should be the case.     

Prior probabilities

The theoretical construction and practical use of prior probabilities, in particular for systems having many degrees of freedom, are investigated. It becomes clear that it is operationally unsound to use mutually consistent priors if one wishes to draw sensible conclusions from practical experiments. The prior cannot usefully be identified with a state of knowledge, and indeed it is not so identified in common scientific practice. Rather, it can be identified with the question one asks. Accordingly, priors are free constructions. Their informal, ill-defined and subjective characteristics must carry over into the conclusions one chooses to draw from experiments or observations.     

Unknown probabilities

From a point of view like de Finetti's, what is the judgmental reality underlying the objectivistic claim that a physical magnitude X determines the objective probability that a hypothesis H is true? When you have definite conditional judgmental probabilities for H given the various unknown values of X, a plausible answer is sufficiency, i.e., invariance of those conditional probabilities as your probability distribution over the values of X varies. A different answer, in terms of conditional exchangeability, is offered for use when such definite conditional probabilities are absent.     

Interpersonal comparison of subjective probabilities: Toward translating linguistic probabilities

Interpersonal variability in understanding linguistic probabilities can adversely affect decision making. Using the fact that everyone judges canonical probability events similarly in a manner consistent with axiom systems that yield a probability measure, we developed and tested a method for comparing the meanings of probability phrases across individuals. An experiment demonstrated that despite extreme heterogeneity in participants' linguistic probability lexicons, interpersonal similarity in phrase meaning is well predicted by phrase rank order within the lexicons. Thus, equally ranked phrases have similar meanings, and individual differences in linguistic probabilities may simply be explained by the phrases people use at each rank.