A generalization of comparative probability on finite sets

Let ? be a binary relation on a finite algebra $\text{A}$ of events A, B,…, where A ? B is interpreted as “A is more probable than B.” Conventional subjective probability is concerned with the existence of a probability measure P on $\text{A}$ that agrees with ? in the sense that A ? B ? P(A) > P(B). Because evidence suggests that some people's comparative probability judgments do not admit an agreeing probability measure, this paper explores a more flexible scheme for representing ? numerically. The new representation has A ? B ? p(A, B) > 0, where p is a monotonic and normalized skew-symmetric function on $\text{A}$ × $\text{A}$ that replaces P's additivity by a conditional additivity property. Conditional additivity says that $\text{p(A ? B, C) + p(?, C) = p(A, C) + p(B, C)}$ whenever A and B are disjoint. The paper examines consequences of this representation, presents examples of ? that it accommodates but which violate the conventional representation, formulates axioms for ? on $\text{A}$ that are necessary and sufficient for the representation, and discusses specializations in which p in separable in its arguments.     

Comparing the meanings of “if” and “all”

In this study, we compared the everyday meanings of conditionals (“if p then q”) and universally quantified statements (“all p are q”) when applied to sets of elements. The interpretation of conditionals was predicted to be directly related to the conditional probability, such that P(“if p then q”) = P(q|p). Quantified statements were assumed to have two interpretations. According to an instance-focused interpretation, quantified statements are equivalent to conditionals, such that P(“all p are q”) = P(q|p). According to a set-focused interpretation, “all p are q” is true if and only if every instance in set p is an instance of q, so that the statement would be accepted when P(q|p) = 1 and rejected when this probability was below 1. We predicted an instance-focused interpretation of “all” when the relation between p and q expressed a general law for an infinite set of elements. A set-focused interpretation of “all” was predicted when the relation between p and q expressed a coincidence among the elements of a finite set. Participants were given short context stories providing information about the frequency of co-occurrence of cases of p, q, not-p, and not-q in a population. They were then asked to estimate the probability that a statement (conditional or quantified) would be true for a random sample taken from that population. The probability estimates for conditionals were in accordance with an instance-focused interpretation, whereas the estimates for quantified statements showed features of a set-focused interpretation. The type of the relation between p and q had no effect on this outcome.     

Naive Probability: Model‐Based Estimates of Unique Events

We describe a dual‐process theory of how individuals estimate the probabilities of unique events, such as Hillary Clinton becoming U.S. President. It postulates that uncertainty is a guide to improbability. In its computer implementation, an intuitive system 1 simulates evidence in mental models and forms analog non‐numerical representations of the magnitude of degrees of belief. This system has minimal computational power and combines evidence using a small repertoire of primitive operations. It resolves the uncertainty of divergent evidence for single events, for conjunctions of events, and for inclusive disjunctions of events, by taking a primitive average of non‐numerical probabilities. It computes conditional probabilities in a tractable way, treating the given event as evidence that may be relevant to the probability of the dependent event. A deliberative system 2 maps the resulting representations into numerical probabilities. With access to working memory, it carries out arithmetical operations in combining numerical estimates. Experiments corroborated the theory's predictions. Participants concurred in estimates of real possibilities. They violated the complete joint probability distribution in the predicted ways, when they made estimates about conjunctions: P(A), P(B), P(A and B), disjunctions: P(A), P(B), P(A or B or both), and conditional probabilities P(A), P(B), P(B|A). They were faster to estimate the probabilities of compound propositions when they had already estimated the probabilities of each of their components. We discuss the implications of these results for theories of probabilistic reasoning.     

The conjunction fallacy: a misunderstanding about conjunction?

It is easy to construct pairs of sentences X, Y that lead many people to ascribe higher probability to the conjunction X-and-Y than to the conjuncts X, Y. Whether an error is thereby committed depends on reasoners’ interpretation of the expressions “probability” and “and.” We report two experiments designed to clarify the normative status of typical responses to conjunction problems.     

Objects, events and "to be" verbs in Spanish--an ERP study of the syntax-semantics interface

In Spanish, objects and events at subject position constrain the selection of different forms of the auxiliary verb “to be”: locative predicates about objects require “estar en”, while those relating to events require “ser en”, both translatable as “to be in”. Subjective ratings showed that while the “object + ser + en” is considered as incorrect, the “event + estar + en” combination is also perceived as unacceptable but to a lesser degree. In an ERP study, we evaluated the impact of a purely semantic distinction (object versus events) on the subsequent processing of these auxiliary verbs followed by locatives in Spanish. For the “ser en” predicate, the P600 component was larger when the subject was an object than when it was an event. This P600 effect is consistent with an online repair of the defining predicate when it does not fit with the adequate semantic properties of the subject. On the other hand, for the “estar en” predicate, event subjects when compared to object subjects showed more positive ongoing amplitudes between 280 and 380 ms after the presentation of the “en” preposition, followed by a longer positive wave starting around 400 ms and lasting until 700 ms after the presentation of the following determiner, with central and frontal scalp distributions respectively. Thus, the different subject-predicate combinations, depending on the semantic features of the subjects, triggered syntactic reparatory processes at a structural level. These findings are consistent with an incremental interpretation of sentence meaning based on the interaction between syntactic and semantic information.     

Sources and influence of perceptual variance: Comment on Dzhafarov's Regular Minimality Principle

Dzhafarov [(2002). Multidimensional Fechnerian scaling: Pairwise comparisons, regular minimality, and nonconstant self-similarity. Journal of Mathematical Psychology, 46, 583-608] claims that Regular Minimality (RM) is a fundamental property of “same-different” discrimination probabilities and supports his claim with some empirical evidence. The key feature of RM is that the mapping, h, between two observation areas based on minimum discrimination probability is invertible. Dzhafarov [(2003a). Thurstonian-type representations for “same-different” discriminations: Deterministic decisions and independent images. Journal of Mathematical Psychology, 47, 184-204; (2003b). Thurstonian-type representations for “same-different” discriminations: Probabilistic decisions and interdependent images. Journal of Mathematical Psychology, 47, 229-243] also demonstrates that well-behaved Thurstonian models of “same-different” judgments are incompatible with RM and Nonconstant Self-Similarity (NCSS). There is extensive empirical support for the latter. Stimulus and neural sources of perceptual noise are discussed and two points are made:

Point 1: Models that require discrimination probabilities for noisy stimuli to possess the property that h is invertible would be too restrictive.

Point 2: In the absence of stimulus noise, violations of RM may be so subtle that their detection would be unlikely.

     

Subjunctive Conditional Probability

There seem to be two ways of supposing a proposition: supposing “indicatively” that Shakespeare didn’t write Hamlet, it is likely that someone else did; supposing “subjunctively” that Shakespeare hadn’t written Hamlet, it is likely that nobody would have written the play. Let P(B//A) be the probability of B on the subjunctive supposition that A. Is P(B//A) equal to the probability of the corresponding counterfactual, A □→B? I review recent triviality arguments against this hypothesis and argue that they do not succeed. On the other hand, I argue that even if we can equate P(B//A) with P(A □→B), we still need an account of how subjunctive conditional probabilities are related to unconditional probabilities. The triviality arguments reveal that the connection is not as straightforward as one might have hoped.     

Conditional Random Quantities and Compounds of Conditionals

In this paper we consider conditional random quantities (c.r.q.’s) in the setting of coherence. Based on betting scheme, a c.r.q. X|H is not looked at as a restriction but, in a more extended way, as \({XH + \mathbb{P}(X|H)H^c}\) ; in particular (the indicator of) a conditional event E|H is looked at as EH +  P(E|H)H ^c . This extended notion of c.r.q. allows algebraic developments among c.r.q.’s even if the conditioning events are different; then, for instance, we can give a meaning to the sum X|H + Y|K and we can define the iterated c.r.q. (X|H)|K. We analyze the conjunction of two conditional events, introduced by the authors in a recent work, in the setting of coherence. We show that the conjoined conditional is a conditional random quantity, which may be a conditional event when there are logical dependencies. Moreover, we introduce the negation of the conjunction and by applying De Morgan’s Law we obtain the disjoined conditional. Finally, we give the lower and upper bounds for the conjunction and disjunction of two conditional events, by showing that the usual probabilistic properties continue to hold.     

‘At home in the uncanny’: Freud's account of das Unheimliche in the context of his theory of religious belief

This paper serves as an exploration of Freud's comment that one of the functions of religious belief is to make humanity feel ‘at home in the uncanny’ (heimisch im Unheimlichen). The first section examines the context of Freud's comment within The Future of an Illusion. Attention is then shifted to Freud's essay on ‘The “Uncanny”’, and to his conclusion that the ‘uncanny’ is the name for everything that ought to have remained secret and hidden but has come to light. A number of interpretations of the ‘at home’ remark are discussed, and it is suggested that religion might fruitfully be viewed as an attempt to come to terms with humanity's ‘transcendental homelessness’.     

Étude développementale de l’interprétation de combinaisons conceptuelles nominales « relation » et « propriété »

The present research addresses how people interpret novel noun-noun conceptual combinations. First, we focused on two types of conceptual combinations: property and relational combinations. Secondly, we manipulated the order of the constituents. Finally, we studied if the interpretation in terms of “Property” or “Relation” changes along with age. So, four groups of 6-, 8- and 10-year-old children and adults participated in a production task. Our results indicated that the interpretations in terms of relation were more frequent for the “Relation” combinations compared to the “Property” ones. Property-transferring interpretations increased with age when Property combinations are presented. The most frequent interpretations followed the order Head noun-Modifier, which is opposite to the order observed in English.     

The role of causal models in multiple judgments under uncertainty

Two studies examined a novel prediction of the causal Bayes net approach to judgments under uncertainty, namely that causal knowledge affects the interpretation of statistical evidence obtained over multiple observations. Participants estimated the conditional probability of an uncertain event (breast cancer) given information about the base rate, hit rate (probability of a positive mammogram given cancer) and false positive rate (probability of a positive mammogram in the absence of cancer). Conditional probability estimates were made after observing one or two positive mammograms. Participants exhibited a causal stability effect: there was a smaller increase in estimates of the probability of cancer over multiple positive mammograms when a causal explanation of false positives was provided. This was the case when the judgments were made by different participants (Experiment 1) or by the same participants (Experiment 2). These results show that identical patterns of observed events can lead to different estimates of event probability depending on beliefs about the generative causes of the observations.     

Superstitious perception: Response-independent reinforcement and punishment as determinants of recurring eccentric interpretations

This study provided controlled observations of a potential mechanism for the determination of the repetitive, aberrant perceptions or interpretations of everyday events that figure prominently in a range of psychological disorders: the adventitious reinforcement of acts of cognition by the actual consequences of concurrent motor acts. Adults made a series of two-choice brightness discriminations; on 60% of trials, choosing the brighter stimulus produced a “correct” signal while errors produced an aversive sound. On 40% of trials, the choice stimuli did not in fact differ in brightness; the consequences of responding on these “identical stimuli” trials differed across blocks of trials. Thus, on these trials perceptual judgments were directly followed by events that they did not produce. When all choices on identical stimuli trials were punished with the “error” sound, subjects showed little preference for the left-side or right-side identical stimuli, but when all choices of identical stimuli were reinforced with the “correct” light, individual preferences for the left-side or the right-side stimuli substantially increased. As the consequences of responding on identical stimuli trials were independent of the stimuli chosen, these findings provide evidence for superstitious perception, the reinforcement of perceptual acts by events that do not depend upon their occurrence.     

Humphrey's paradox and the interpretation of inverse conditional propensities

The aim of this paper is to distinguish between, and examine, three issues surrounding Humphreys's paradox and interpretation of conditional propensities. The first issue involves the controversy over the interpretation of inverse conditional propensities — conditional propensities in which the conditioned event occurs before the conditioning event. The second issue is the consistency of the dispositional nature of the propensity interpretation and the inversion theorems of the probability calculus, where an inversion theorem is any theorem of probability that makes explicit (or implicit) appeal to a conditional probability and its corresponding inverse conditional probability. The third issue concerns the relationship between the notion of stochastic independence which is supported by the propensity interpretation, and various notions of causal independence. In examining each of these issues, it is argued that the dispositional character of the propensity interpretation provides a consistent and useful interpretation of the probability calculus.I would like to thank William L. Harper, Paul Humphreys, John Nicholas and Kathleen Okruhlik for helpful comments and advice on earlier drafts of this paper. Research for this paper was supported by a fellowship from the Social Sciences and Humanities Research Council of Canada (award number 452-90-2513).     

Coherent Odds and Subjective Probability

A well-known Dutch book, due to de Finetti, shows how violations of the additivity law of probability theory (Pr(A∪B)=Pr(A)+Pr(B)−Pr(A∩B)) open the door to a sequence of bets leading to a sure loss. In this paper we show that in a market environment, when bookies act strategically, it may well be optimal for them to post incoherent (nonadditive) odds. This is true even when the bookie's own preferences are expected utility, provided she plays against at least two nonexpected utility bettors.     

The mental representation of causal conditional reasoning: mental models or causal models

In this paper, two experiments are reported investigating the nature of the cognitive representations underlying causal conditional reasoning performance. The predictions of causal and logical interpretations of the conditional diverge sharply when inferences involving pairs of conditionals—such as if P₁then Q and if P₂then Q—are considered. From a causal perspective, the causal direction of these conditionals is critical: are the P_icauses of Q; or symptoms caused byQ. The rich variety of inference patterns can naturally be modelled by Bayesian networks. A pair of causal conditionals where Q is an effect corresponds to a “collider” structure where the two causes (P_i) converge on a common effect. In contrast, a pair of causal conditionals where Q is a cause corresponds to a network where two effects (P_i) diverge from a common cause. Very different predictions are made by fully explicit or initial mental models interpretations. These predictions were tested in two experiments, each of which yielded data most consistent with causal model theory, rather than with mental models.     

What Conditional Probability Could Not Be

Kolmogorov's axiomatization of probability includes the familiarratio formula for conditional probability: 0).$$ " align="middle" border="0"> Call this the ratio analysis of conditional probability. It has become so entrenched that it is often referred to as the definition of conditional probability.I argue that it is not even an adequate analysis of that concept. I prove what I call the Four Horn theorem, concluding that every probability assignment has uncountably many trouble spots. Trouble spots come in four varieties: assignments of zero togenuine possibilities; assignments of infinitesimals to such possibilities; vague assignments to such possibilities; and no assignment whatsoever to such possibilities. Each sort of trouble spot can create serious problems for the ratio analysis. I marshal manyexamples from scientific and philosophical practice against the ratio analysis. I conclude more positively: we should reverse the traditional direction of analysis. Conditional probability should be taken as the primitive notion, and unconditional probability should be analyzed in terms of it. I'd probably be famous now If I wasn't such a good waitress. Jane Siberry, Waitress     

Effects of a Relative-Frequency Elicitation Question on Likelihood Judgment Accuracy: The Case of External Correspondence

In three experiments, college students judged the likelihood that they chose the correct alternative for each of 40 two-alternative, general-knowledge items. They responded either to a relative-frequency elicitation question (“Out of 100 questions for which you felt this certain of the answer, how many would you answer correctly?”) or to a probability elicitation question (“What is the probability that you chose the correct answer?”). Judgments in response to the relative-frequency elicitation question tended to be lower, exhibit less scatter, and express complete certainty less often than judgments in response to the probability elicitation question. Two types of explanation for these effects are considered. First, the effect of the relative-frequency elicitation question may be to reduce random response error in participants' likelihood judgments. Second, the relative-frequency elicitation question may encourage the use of frequency information and simpler algorithms for making likelihood judgments.