Bets and Boundaries: Assigning Probabilities to Imprecisely Specified Events

Uncertainty and vagueness/imprecision are not the same: one can be certain about events described using vague predicates and about imprecisely specified events, just as one can be uncertain about precisely specified events. Exactly because of this, a question arises about how one ought to assign probabilities to imprecisely specified events in the case when no possible available evidence will eradicate the imprecision (because, say, of the limits of accuracy of a measuring device). Modelling imprecision by rough sets over an approximation space presents an especially tractable case to help get one’s bearings. Two solutions present themselves: the first takes as upper and lower probabilities of the event X the (exact) probabilities assigned X’s upper and lower rough-set approximations; the second, motivated both by formal considerations and by a simple betting argument, is to treat X’s rough-set approximation as a conditional event and assign to it a point-valued (conditional) probability. With rough sets over an approximation space we get a lot of good behaviour. For example, in the first construction mentioned the lower probabilities are n-monotone, for every . When we examine other models of approximation/imprecision/vagueness, and in particular, proximity spaces, we lose a lot of that good behaviour. In the literature there is not (even) agreement on the definition of upper and lower approximations for events (subsets) in the underlying domain. Betting considerations suggest one choice and, again, ways to assign upper and lower and point-valued probabilities, but nothing works well. Special Issue on Vagueness Edited by Rosanna Keefe and Libor Bêhounek     

There is a 60% probability,but I am 70% certain: communicative consequences of external and internal expressions of uncertainty

Current theories of probability recognise a distinction between external (un)certainty (frequentistic probabilities) and internal (un)certainty (degrees of belief). The present studies investigated this distinction in lay people's judgements of probability statements formulated to suggest either an internal (“I am X% certain”) or an external (“It is X% certain” or “There is an X% probability”) interpretation. These subtle differences in wording influenced participants' perceptions and endorsements of such statements, and their impressions of the speaker. External expressions were seen to signal more reliable task duration estimates, and a lower degree of external than internal certainty was deemed necessary to advise a course of action. In conversations about football, internal expressions were perceived as signalling more personal interest, and were expected to be on the average 10% higher than corresponding external probabilities. Finally, people who reported their outcome expectations for two major sports events let their degree of interest in these events influence their internal but not their external certainty. These results have implications for the communication of uncertainty and probability.     

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.     

Heuristics and biases in diagnostic reasoning: II. Congruence,information, and certainty

Six experiments were carried out to examine possible heuristics and biases in the evaluation of yes-or-no questions for the purpose of hypothesis testing. In some experiments, the prior probability of the hypotheses and the conditional probabilities of the answers given each hypothesis were elicited from the subjects; in other experiments, they were provided. We found the following biases (systematic departures from a normative model), and interviews and justifications suggested that each was the result of a corresponding heuristic: Congruence bias. Subjects overvalued questions that have a high probability of a positive result given the most likely hypothesis. This bias was apparently reduced when alternative hypotheses or probabilities of negative results are explicitly stated. Information bias. Subjects evaluated questions as worth asking even when there is no answer that can change the hypothesis that will be accepted as a basis for action. Certainty bias. Subjects overvalued questions that have the potential to establish, or rule out, one or more hypotheses with 100% probability. These heuristics are explained in terms of the idea that people fail to consider certain arguments against the use of questions that seem initially worth asking, specifically, that a question may not distinguish likely hypotheses or that no answer can change the hypothesis accepted as a basis for action.     

How probability theory explains the conjunction fallacy

The conjunction fallacy occurs when people judge a conjunctive statement B‐and‐A to be more probable than a constituent B, in contrast to the law of probability that P(B ∧ A) cannot exceed P(B) or P(A). Researchers see this fallacy as demonstrating that people do not follow probability theory when judging conjunctive probability. This paper shows that the conjunction fallacy can be explained by the standard probability theory equation for conjunction if we assume random variation in the constituent probabilities used in that equation. The mathematical structure of this equation is such that random variation will be most likely to produce the fallacy when one constituent has high probability and the other low, when there is positive conditional support between the constituents, when there are two rather than three constituents, and when people rank probabilities rather than give numerical estimates. The conjunction fallacy has been found to occur most frequently in exactly these situations. Copyright © 2008 John Wiley & Sons, Ltd.     

Intuitive prediction: ecological validity versus representativeness

Insufficiently regressive intuitive predictions have been attributed to mistaken reliance on the representativeness heuristic. In contrast, we suggest that intuitive predictions stem from a conceptualization of ‘goodness of prediction’ that differs from the accepted statistical definition in terms of error minimization, namely, ecological validity—that is, representation of the substantive characteristics of the predicted variable Y and its distribution as well as of the relationship between Y and the predictor X—rather than minimization of prediction errors. Simultaneous satisfaction of the above representation requirements is achieved by multivalued prediction: The prediction of different Y′ values for the same X value, resulting in conditional distributions Y′|X for at least some X values. Empirical results supporting this hypothesis are presented and discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Correlations between rats' spatial location and intracranial stimulation administration affects rate of acquisition and asymptotic level of time allocation preference in the open field

The present experiment explores the effects of the response (1-sec occupancy of a target area in an open field)-reinforcer (intracranial stimulation) contingency on time allocation in the open field in rats. The probability of reinforcement given response (X) and the probability of reinforcement given nonresponse (Y) were varied randomly across sessions within a subject. The 21 contingency treatments explored included all possible combinations of values (0, .1, .2, .3, .4, .5) of X and Y such that X ≥ Y. The results indicate that rate of acquisition and asymptotic level of time allocation preference to the target area are negatively related to the value of Y (for any given value of X). Variations in X (for any given value of Y) were less effective. Evaluation of proposed contingency metrics revealed that the Weber fraction (X − Y)/X most closely approximates performance, and that the value of the difference detection threshold derived from the Weber fraction is a constant.     

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.     

The inverse fallacy: An account of deviations from Bayes’s theorem and the additivity principle

In judging posterior probabilities, people often answer with the inverse conditional probability--a tendency named the inverse fallacy. Participants (N = 45) were given a series of probability problems that entailed estimating both p(H / D) and p(approximately H / D). The findings revealed that deviations of participants' estimates from Bayesian calculations and from the additivity principle could be predicted by the corresponding deviations of the inverse probabilities from these relevant normative benchmarks. Methodological and theoretical implications of the distinction between inverse fallacy and base-rate neglect and the generalization of the study of additivity to conditional probabilities are discussed.     

Verbal probabilities: An alternative approach

Previous studies of verbal probabilities have tried to place expressions like a chance, possible, and certain on 0–1 numerical probability scales. We ask instead, out of a range of outcomes, which outcome a verbal probability suggests. When, for instance, a sample of laptop batteries lasts from 1.5 to 3.5 hours, what is a certain and what is a possible duration? Experiment 1 showed that speakers associate certain with low values and possible with (unlikely) high or maximal values. In Experiment 2, this methodology was applied to several positive and negative verbal probability phrases, showing a preference for high rather than low or middle values in a distribution. Experiment 3 showed that such maxima are not universally described by large numbers. For instance, maximum speed is often described in terms of a small number of time units. What can (possibly) happen is accordingly sometimes described with very low and sometimes with very high values, depending upon focus of interest. Finally, participants in Experiment 4 were given the role of hearers rather than speakers and were asked to infer outcome ranges from verbal probabilities. Hearers appeared to be partly aware of speakers' tendencies to describe outcomes at the top of the range.     

Intuition, reason, and metacognition

Dual Process Theories (DPT) of reasoning posit that judgments are mediated by both fast, automatic processes and more deliberate, analytic ones. A critical, but unanswered question concerns the issue of monitoring and control: When do reasoners rely on the first, intuitive output and when do they engage more effortful thinking? We hypothesised that initial, intuitive answers are accompanied by a metacognitive experience, called the Feeling of Rightness (FOR), which can signal when additional analysis is needed. In separate experiments, reasoners completed one of four tasks: conditional reasoning (N = 60), a three-term variant of conditional reasoning (N = 48), problems used to measure base rate neglect (N = 128), or a syllogistic reasoning task (N = 64). For each task, participants were instructed to provide an initial, intuitive response to the problem along with an assessment of the rightness of that answer (FOR). They were then allowed as much time as needed to reconsider their initial answer and provide a final answer. In each experiment, we observed a robust relationship between the FOR and two measures of analytic thinking: low FOR was associated with longer rethinking times and an increased probability of answer change. In turn, FOR judgments were consistently predicted by the fluency with which the initial answer was produced, providing a link to the wider literature on metamemory. These data support a model in which a metacognitive judgment about a first, initial model determines the extent of analytic engagement.     

Statistical Data and Mathematical Propositions

Statistical tests of the primality of some numbers look similar to statistical tests of many nonmathematical, clearly empirical propositions. Yet interpretations of probability prima facie appear to preclude the possibility of statistical tests of mathematical propositions. For example, it is hard to understand how the statement that n is prime could have a frequentist probability other than 0 or 1. On the other hand, subjectivist approaches appear to be saddled with ‘coherence’ constraints on rational probabilities that require rational agents to assign extremal probabilities to logical and mathematical propositions. In the light of these problems, many philosophers have come to think that there must be some way to generalize a Bayesian statistical account. In this article I propose that a classical frequentist approach should be reconsidered. I conclude that we can give a conditional justification of statistical testing of at least some mathematical hypotheses: if statistical tests provide us with reasons to believe or bet on empirical hypotheses in the standard situations, then they also provide us with reasons to believe or bet on mathematical hypotheses in the structurally similar mathematical cases.     

Operating on Functions with Variable Domains

The sum, difference, product and quotient of two functions with different domains are usually defined only on their common domain. This paper extends these definitions so that the sum and other operations are essentially defined anywhere that at least one of the components is defined. This idea is applied to propositions and events, expressed as indicator functions, to define conditional propositions and conditional events as three-valued indicator functions that are undefined when their condition is false. Extended operations of and, or, not and conditioning are then defined on these conditional events with variable conditions. The probabilities of the disjunction (or) and of the conjunction (and) of two conditionals are expressed in terms of the conditional probabilities of the component conditionals. In a special case, these are shown to be weighted averages of the component conditional probabilities where the weights are the relative probabilities of the various conditions. Next, conditional random variables are defined to be random variables X whose domain has been restricted by a condition on a second random variable Y. The extended sum, difference, product and conditioning operations on functions are then applied to these conditional random variables. The expectation of a random variable and the conditional expectation of a conditional random variable are recounted. Theorem 1 generalizes the standard result that the conditional expectation of the sum of two conditional random variables with disjoint and exhaustive conditions is a weighted sum of the conditional expectations of the component conditional random variables. Because of the extended operations, the theorem is true for arbitrary conditions. Theorem 2 gives a formula for the expectation of the product of two conditional random variables. After the definition of independence of two random variables is extended to accommodate the extended operations, it is applied to the formula of Theorem 2 to simplify the expectation of a product of conditional random variables. Two examples end the paper. The first concerns a work force of n workers of different output levels and work shifts. The second example involves two radars with overlapping surveillance regions and different detection error rates. One radar's error rate is assumed to be sensitive to fog and the other radar's error rate is assumed to be sensitive to air traffic density. The combined error rate over the combined surveillance region given heavy fog and moderate air traffic is computed.     

A Logic For Inductive Probabilistic Reasoning

Inductive probabilistic reasoning is understood as the application of inference patterns that use statistical background information to assign (subjective) probabilities to single events. The simplest such inference pattern is direct inference: from “70% of As are Bs” and “a is an A” infer that a is a B with probability 0.7. Direct inference is generalized by Jeffrey’s rule and the principle of cross-entropy minimization. To adequately formalize inductive probabilistic reasoning is an interesting topic for artificial intelligence, as an autonomous system acting in a complex environment may have to base its actions on a probabilistic model of its environment, and the probabilities needed to form this model can often be obtained by combining statistical background information with particular observations made, i.e., by inductive probabilistic reasoning. In this paper a formal framework for inductive probabilistic reasoning is developed: syntactically it consists of an extension of the language of first-order predicate logic that allows to express statements about both statistical and subjective probabilities. Semantics for this representation language are developed that give rise to two distinct entailment relations: a relation ⊨ that models strict, probabilistically valid, inferences, and a relation that models inductive probabilistic inferences. The inductive entailment relation is obtained by implementing cross-entropy minimization in a preferred model semantics. A main objective of our approach is to ensure that for both entailment relations complete proof systems exist. This is achieved by allowing probability distributions in our semantic models that use non-standard probability values. A number of results are presented that show that in several important aspects the resulting logic behaves just like a logic based on real-valued probabilities alone.     

Betting on conditionals

A study is reported testing two hypotheses about a close parallel relation between indicative conditionals, if A then B, and conditional bets, I bet you that if A then B. The first is that both the indicative conditional and the conditional bet are related to the conditional probability, P(B|A). The second is that de Finetti's three-valued truth table has psychological reality for both types of conditional—true, false, or void for indicative conditionals and win, lose, or void for conditional bets. The participants were presented with an array of chips in two different colours and two different shapes, and an indicative conditional or a conditional bet about a random chip. They had to make judgements in two conditions: either about the chances of making the indicative conditional true or false or about the chances of winning or losing the conditional bet. The observed distributions of responses in the two conditions were generally related to the conditional probability, supporting the first hypothesis. In addition, a majority of participants in further conditions chose the third option, “void”, when the antecedent of the conditional was false, supporting the second hypothesis.     

Entailment with Near Surety of Scaled Assertions of High Conditional Probability

An assertion of high conditional probability or, more briefly, an HCP assertion is a statement of the type: The conditional probability of B given A is close to one. The goal of this paper is to construct logics of HCP assertions whose conclusions are highly likely to be correct rather than certain to be correct. Such logics would allow useful conclusions to be drawn when the premises are not strong enough to allow conclusions to be reached with certainty. This goal is achieved by taking Adams" (1966) logic, changing its intended application from conditionals to HCP assertions, and then weakening its criterion for entailment. According to the weakened entailment criterion, called the Criterion of Near Surety and which may be loosely interpreted as a Bayesian criterion, a conclusion is entailed if and only if nearly every model of the premises is a model of the conclusion. The resulting logic, called NSL, is nonmonotonic. Entailment in this logic, although not as strict as entailment in Adams" logic, is more strict than entailment in the propositional logic of material conditionals. Next, NSL was modified by requiring that each HCP assertion be scaled; this means that to each HCP assertion was associated a bound on the deviation from 1 of the conditional probability that is the subject of the assertion. Scaling of HCP assertions is useful for breaking entailment deadlocks. For example, it it is known that the conditional probabilities of C given A and of ¬ C given B are both close to one but the bound on the former"s deviation from 1 is much smaller than the latter"s, then it may be concluded that in all likelihood the conditional probability of C given A B is close to one. The resulting logic, called NSL-S, is also nonmonotonic. Despite great differences in their definitions of entailment, entailment in NSL is equivalent to Lehmann and Magidor"s rational closure and, disregarding minor differences concerning which premise sets are considered consistent, entailment in NSL-S is equivalent to entailment in Goldszmidt and Pearl"s System-Z ⁺. Bacchus, Grove, Halpern, and Koller proposed two methods of developing a predicate calculus based on the Criterion of Near Surety. In their random-structures method, which assumed a prior distribution similar to that of NSL, it appears possible to define an entailment relation equivalent to that of NSL. In their random-worlds method, which assumed a prior distribution dramatically different from that of NSL, it is known that the entailment relation is different from that of NSL.     

The Causal Structure of Utility Conditionals

The psychology of reasoning is increasingly considering agents' values and preferences, achieving greater integration with judgment and decision making, social cognition, and moral reasoning. Some of this research investigates utility conditionals, ‘‘if p then q’’ statements where the realization of p or q or both is valued by some agents. Various approaches to utility conditionals share the assumption that reasoners make inferences from utility conditionals based on the comparison between the utility of p and the expected utility of q. This article introduces a new parameter in this analysis, the underlying causal structure of the conditional. Four experiments showed that causal structure moderated utility‐informed conditional reasoning. These inferences were strongly invited when the underlying structure of the conditional was causal, and significantly less so when the underlying structure of the conditional was diagnostic. This asymmetry was only observed for conditionals in which the utility of q was clear, and disappeared when the utility of q was unclear. Thus, an adequate account of utility‐informed inferences conditional reasoning requires three components: utility, probability, and causal structure.