Verbal probabilities: Ambiguous, context-dependent, or both?

To investigate the communication value of verbal probabilistic phrases, like “possibly,” “probably,” and “perhaps,” three experiments were conducted. Subjects were asked to judge the degree of probability expressed by such phrases in different contexts: in sentences reflecting opinions on current events, in a medical discussion of treatment effectiveness, and in videotaped news reports. Judgments of degree of probability were performed in the first study on a 0–100% probability scale and in the other two on 7-point rating scales. Results indicated that different contexts influence the interpretation of probability terms and in many cases, but not always, lead to higher between-subject variability than when the terms are judged in isolation, presumably because the interpretation of probability terms tends to be correlated with the judges' personal opinions on the topics. Special communication problems arise from the fact that most people are not fully aware of the ambiguity of these phrases and underestimate the variability of such ratings in the general population. Miscommunication between experts and the general public was illustrated by answers to a questionnaire given to general medical practitioners and to parents of small children. The latter preferred numerical probabilities to words, but thinking from an individual-oriented perspective, they often misunderstood the intended statistical meanings.     

Should the probabilities count?

When facing a choice between saving one person and saving many, some people have argued that fairness requires us to decide without aggregating numbers; rather we should decide by coin toss or some form of lottery, or alternatively we should straightforwardly save the greater number but justify this in a non-aggregating contractualist way. This paper expands the debate beyond well-known number cases to previously under-considered probability cases, in which not (only) the numbers of people, but (also) the probabilities of success for saving people vary. It is shown that, in these latter cases, both the coin toss and the lottery lead to what is called an awkward conclusion, which makes probabilities count in a problematic way. Attempts to avoid this conclusion are shown to lead into difficulties as well. Finally, it is shown that while the greater number method cannot be justified on contractualist grounds for probability cases, it may be replaced by another decision method which is so justified. This decision method is extensionally equivalent to maximising expected value and seems to be the least problematic way of dealing with probability cases in a non-aggregating manner.     

Grapheme—phoneme probabilities in British English

A computational analysis of a large British English database was performed and frequencies of occurrence of grapheme—phoneme correspondences were obtained. A computer program was implemented, which used these frequencies to predict the probabilities of all possible pronunciations of any given string of graphemes. These results led to a proposal for a quantitative method of measurement of the orthographic depth of different languages.     

Surprises: low probabilities or high contrasts?

Outcome expectations can be expressed prospectively in terms of probability estimates, and retrospectively in terms of surprise. Surprise ratings and probability estimates differ, however, in some important ways. Surprises are generally created by low-probability outcomes, yet, as shown by several experiments, not all low-probability outcomes are equally surprising. To account for surprise, we propose a contrast hypothesis according to which the level of surprise associated with an outcome is mainly determined by the extent to which it contrasts with the default, expected alternative. Three ways by which contrasts can be established are explored: contrasts due to relative probabilities, where the obtained outcome is less likely than a default alternative; contrasts formed by novelty and change, where a contrast exists between the obtained outcome and the individual's previous experience; and contrasts due to the perceptual or conceptual distance between the expected and the obtained. In all these cases, greater contrast was accompanied by higher ratings of surprise.     

Verbal probabilities: a question of frame?

Verbal expressions of probability and uncertainty are of two kinds: positive (‘probable’, ‘possible’) and negative (‘not certain’, ‘doubtful’). Choice of term has implications for predictions and decisions. The present studies show that positive phrases are rated to be more optimistic (when the target outcome is positive), and more correct, when the target outcome actually occurs, even in cases where positive and negative phrases are perceived to convey the same probabilities (Experiments 1 and 2). Selection of phrase can be determined by linguistic frame. Positive quantifiers (‘some’, ‘several’) support positive probability phrases, whereas negative quantifiers (‘not all’) suggest negative phrases (Experiment 3). Positive frames induced by numeric frequencies (e.g. the number of students to be admitted) imply positive probability phrases, whereas negative frames (e.g. the number of students to be rejected) call for negative probability phrases (Experiment 4). It is concluded that choice of verbal phrase is based not only on level of probability, but also on situational and linguistic cues. Copyright © 2002 John Wiley & Sons, Ltd.     

Indicative conditionals,conditional probabilities,and the “defective truth-table”: A request for more experiments

While there is now considerable experimental evidence that, on the one hand, participants assign to the indicative conditional as probability the conditional probability of consequent given antecedent and, on the other, they assign to the indicative conditional the “defective truth-table” in which a conditional with false antecedent is deemed neither true nor false, these findings do not in themselves establish which multi-premise inferences involving conditionals participants endorse. A natural extension of the truth-table semantics pronounces as valid numerous inference patterns that do seem to be part of ordinary usage. However, coupled with something the probability account gives us—namely that when conditional-free ? entails conditional-free ψ, “if ? then ψ” is a trivial, uninformative truth—we have enough logic to derive the paradoxes of material implication. It thus becomes a matter of some urgency to determine which inference patterns involving indicative conditionals participants do endorse. Only thus will we be able to arrive at a realistic, systematic semantics for the indicative conditional.     

Response probabilities and “same-different” reaction times

A “same-different” reaction-time (RT) task with multidimensional stimuli was used with a 50:50 and a 75:25 proportion of “different” to “same” response. The “different” RTs to each dimension were similar for the two proportions. However, the “same” RT in the 50:50 condition was faster than the “same” for the 75:25 condition and faster than the slowest “different” RT. Interpretations of the results in terms of response bias and processing criteria are offered.     

Measuring local context as context–word probabilities

Context enables readers to quickly recognize a related word but disturbs recognition of unrelated words. The relatedness of a final word to a sentence context has been estimated as the probability (cloze probability) that a participant will complete a sentence with a word. In four studies, I show that it is possible to estimate local context-word relatedness based on common language usage. Conditional probabilities were calculated for sentences with published cloze probabilities. Four-word contexts produced conditional probabilities significantly correlated with cloze probabilities, but usage statistics were unavailable for some sentence contexts. The present studies demonstrate that a composite context measure based on conditional probabilities for one- to four-word contexts and the presence of a final period represents all of the sentences and maintains significant correlations (.25, .52, .53) with cloze probabilities. Finally, the article provides evidence for the effectiveness of this measure by showing that local context varies in ways that are similar to the N400 effect and that are consistent with a role for local context in reading. The Supplemental materials include local context measures for three cloze probability data sets.     

Can membership-functions capture the directionality of verbal probabilities?

Verbal probabilities are a common mean for communicating risk and uncertainties in many decision-making settings (e.g., finance, medicine, military). They are considered directional because they elicit a focus on either the outcome occurrence (e.g., there is a chance) or on its non-occurrence (e.g., it is unlikely). According to a quantitative perspective, directionality is dependent on the vague probabilistic meaning conveyed by verbal probabilities—e.g., p(outcome) > .50 = > focus on outcome occurrence. In contrast a more qualitative perspective suggests that directionality depends on contextual factors. The present study tested whether the directionality of verbal probabilities was determined by their vague probabilistic meaning, by contextually manipulated variables (i.e., representativeness and base rate), or by a combination of both. Participants provided their own expressions to describe the guilt of a suspect and then assessed the vague probabilistic meaning and directionality associated with those expressions. Results showed that directionality was mainly determined by the vague probabilistic meaning but also by the base rate of guilt. Although attention focus on the occurrence or the non-occurrence of the target outcome is dependent on vague probabilistic meaning, it cannot be fully accounted for by it.     

Communicative functions of directional verbal probabilities: Speaker’s choice,listener’s inference,and reference points

Verbal probabilities have directional communicative functions, and most can be categorized as positive (e.g., “it is likely”) or negative (e.g., “it is doubtful”). We examined the communicative functions of verbal probabilities based on the reference point hypothesis According to this hypothesis, listeners are sensitive to and can infer a speaker’s reference points based on the speaker’s selected directionality. In four experiments (two of which examined speakers’ choice of directionality and two of which examined listeners’ inferences about a speaker’s reference point), we found that listeners could make inferences about speakers’ reference points based on the stated directionality of verbal probability. Thus, the directionality of verbal probabilities serves the communicative function of conveying information about a speaker’s reference point.     

Effects of partial reinforcement on conditioning,conditional probabilities,asymptotic performance,and extinction of the rabbit’s nictitating membrane response

Classical conditioning and extinction of the rabbit nictitating membrane response was investigated under 100 per cent, and two 50 per cent reinforcement conditions (50% equated total trials, and 50% equated reinforced trials) at average intertriai intervals (ITIs) of 30 and 60 sec. It was found that: (1) partial reinforcement reduced the rate of conditioning, but all groups eventually attained asymptomatic performance levels near 100 per cent; (2) first-order and second-order conditional probabilities following reinforced and nonreinforced trials were virtually identical; and (3) no partial reinforcement extinction effect (PREE) was observed.     

A role for backward transitional probabilities in word segmentation?

A number of studies have shown that people exploit transitional probabilities between successive syllables to segment a stream of artificial continuous speech into words. It is often assumed that what is actually exploited are the forward transitional probabilities (given XY, the probability that X will be followed by Y), even though the backward transitional probabilities (the probability that Y has been preceded by X) were equally informative about word structure in the languages involved in those studies. In two experiments, we showed that participants were able to learn the words from an artificial speech stream when the only available cues were the backward transitional probabilities. Learning is as good under those conditions as when the only available cues are the forward transitional probabilities. Implications for some current models of word segmentation, particularly the simple recurrent networks and PARSER models, are discussed.     

Ambiguous probabilities: when does p=0.3 reflect a possibility,and when does it express a doubt?

Verbal phrases denoting uncertainty are usually held to be more vague than numerical probability statements. They are, however, directionally more precise, in the sense that they are either positive, suggesting the occurrence of a target outcome, or negative, drawing attention to its non‐occurrence. A numerical probability will, in contrast, sometimes be perceived as positive and sometimes as negative. When asked to complete sentences such as ‘The operation has a 30% chance of success, because’ some people will give reasons for success (‘the doctors are expert surgeons’), whereas others will give reasons for failure (‘it is a difficult operation’). It is shown in two experiments that positive reasons are given more often than negative ones, even for p values below 0.5, especially when the probability is higher than expected, and the target outcome is non‐normal, undesirable, and phrased as a negation. We conclude that the directionality of numerical probabilities (as opposed to verbal phrases) is context‐dependent, but biased towards a positive interpretation. Copyright © 2000 John Wiley & Sons, Ltd.     

Single-case probabilities and the case of Monty Hall: Levy’s view

In Baumann (American Philosophical Quarterly 42: 71–79, 2005) I argued that reflections on a variation of the Monty Hall problem throws a very general skeptical light on the idea of single-case probabilities. Levy (Synthese, forthcoming, 2007) puts forward some interesting objections which I answer here.     

What is learned about fragments in artificial grammar learning? A transitional probabilities approach

Learning local regularities in sequentially structured materials is typically assumed to be based on encoding of the frequencies of these regularities. We explore the view that transitional probabilities between elements of chunks, rather than frequencies of chunks, may be the primary factor in artificial grammar learning (AGL). The transitional probability model (TPM) that we propose is argued to provide an adaptive and parsimonious strategy for encoding local regularities in order to induce sequential structure from an input set of exemplars of the grammar. In a variant of the AGL procedure, in which participants estimated the frequencies of bigrams occurring in a set of exemplars they had been exposed to previously, participants were shown to be more sensitive to local transitional probability information than to mere pattern frequencies.     

Are choices based on conditional or conjunctive probabilities in a sequential risk-taking task?

In this study, we examined participants' choice behavior in a sequential risk-taking task. We were especially interested in the extent to which participants focus on the immediate next choice or consider the entire choice sequence. To do so, we inspected whether decisions were either based on conditional probabilities (e.g., being successful on the immediate next trial) or on conjunctive probabilities (of being successful several times in a row). The results of five experiments with a simplified nine-card Columbia Card Task and a CPT-model analysis show that participants' choice behavior can be described best by a mixture of the two probability types. Specifically, for their first choice, the participants relied on conditional probabilities, whereas subsequent choices were based on conjunctive probabilities. This strategy occurred across different start conditions in which more or less cards were already presented face up. Consequently, the proportion of risky choices was substantially higher when participants started from a state with some cards facing up, compared with when they arrived at that state starting from the very beginning. The results, alternative accounts, and implications are discussed.     

A general method of obtaining exact sampling probabilities of the shannon-wiener measure of informationĤ

A two-variable generating function is described which yields the sampling probabilities of the Shannon-Wiener information measure. Expansion and collection of terms in like powers of the first variable imposes the restriction that the sum of thek category frequencies equaln; collection of terms in like powers of the second variable then produces terms whose coefficients are the required probabilities. The method may be used with either equal or unequal category probabilities for any finiten andk, and thus represents a general solution to the small sample problem. Tables of sampling probabilities are presented.The substance of this paper was reported at the meetings of the Southern Society for Philosophy and Psychology, Miami, Florida, April 12, 1963.     

Testing the Δ-∑ hypothesis in the suboptimal choice task: Same delta with different probabilities of reinforcement

In a concurrent-chain procedure, pigeons choose between 2 initial-link stimuli; one is followed by terminal link stimuli that signal reliably whether food will be delivered after a delay; the other is followed by terminal link stimuli that do not signal whether food will be delivered after the delay. Pigeons prefer the former alternative even when it yields a lower overall probability of food. Recently, we proposed the Delta-Sigma (∆-∑) hypothesis to explain the effect: Preference depends on the difference (∆) between the reinforcement probabilities associated with the terminal link stimuli, and the overall probability of reinforcement (∑) associated with the alternative. The hypothesis predicts that, for constant ∑, animals should prefer alternatives with greater ∆ values regardless of the specific probabilities of reinforcement that determine ∆. In 2 experiments, we tested this prediction by comparing a ∆ = .5 against a ∆ = 0 alternative, with the former obtained with different pairs of reinforcement probabilities across conditions. The results supported the hypothesis when the 2 probabilities defining ∆ were significantly greater than 0, but not when one of them was close to 0. The results challenge our theoretical accounts of suboptimal choice and the variables considered to determine pigeons’ preference.     

When and why do people avoid unknown probabilities in decisions under uncertainty? Testing some predictions from optimal foraging theory.

When given a choice between two otherwise equivalent options - one in which the probability information is stated and another in which it is missing - most people avoid the option with missing probability information (Camerer & Weber, 1992). This robust, frequently replicated tendency is known as the ambiguity effect. It is unclear, however, why the ambiguity effect occurs. Experiments 1 and 2, which separated effects of the comparison process from those related to missing probability information, demonstrate that the ambiguity effect is elicited by missing probabilities rather than by comparison of options. Experiments 3 and 4 test predictions drawn from the literature on behavioral ecology. It is suggested that choices between two options should reflect three parameters: (1) the need of the organism, (2) the mean expected outcome of each option; and (3) the variance associated with each option's outcome. It is hypothesized that unknown probabilities are avoided because they co-occur with high outcome variability. In Experiment 3 it was found that subjects systematically avoid options with high outcome variability regardless of whether probabilities are explicitly stated or not. In Experiment 4, we reversed the ambiguity effect: when participants' need was greater than the known option's expected mean outcome, subjects preferred the ambiguous (high variance) option. From these experiments we conclude that people do not generally avoid ambiguous options. Instead, they take into account expected outcome, outcome variability, and their need in order to arrive at a decision that is most likely to satisfy this need.