Partially Undetermined Many-Valued Events and Their Conditional Probability

A logic for classical conditional events was investigated by Dubois and Prade. In their approach, the truth value of a conditional event may be undetermined. In this paper we extend the treatment to many-valued events. Then we support the thesis that probability over partially undetermined events is a conditional probability, and we interpret it in terms of bets in the style of de Finetti. Finally, we show that the whole investigation can be carried out in a logical and algebraic setting, and we find a logical characterization of coherence for assessments of partially undetermined events.     

假言推理中的概率效应

近来Oaksford等人的研究表明,概率信息对假言推理的认知过程有着显著的影响,在一定程度上可以用它来解释和预测个体在推理中的行为反应。该文针对这种新的观点,就假言推理中所涉及的前后件概率信息、条件概率信息、连接概率信息、因果概率信息以及额外前提条件等对假言推理的影响分别做了简要的阐述,并介绍了一些相关的实验研究,最后作了总结和展望。     

Conditional Probability in the Light of Qualitative Belief Change

We explore ways in which purely qualitative belief change in the AGM tradition throws light on options in the treatment of conditional probability. First, by helping see why it can be useful to go beyond the ratio rule defining conditional from one-place probability. Second, by clarifying what is at stake in different ways of doing that. Third, by suggesting novel forms of conditional probability corresponding to familiar variants of qualitative belief change, and conversely. Likewise, we explain how recent work on the qualitative part of probabilistic inference leads to a very broad class of ‘proto-probability’ functions.     

有关条件推理认知机制的概率理论

条件推理主要有两种研究范式：演绎形式(MP、DA、AC、MT)和变通形式(四卡问题)。已往研究主要探讨了内容因素、情境因素以及元认知因素等对条件推理的影响,形成了若干理论模型和观点。近来、Oaksford等人研究了概率因素对条件推理的影响．提出了条件概率模型。本文对这种新的研究取向作了详细的介绍和评价,并对未来研究进行了展望。     

条件推理的条件概率模型述评

该文从理论模型和实验范式两个方面对Oaksford等人提出的条件推理概率模型作了综合评述,并充分肯定该模型将演绎推理与归纳推理结合起来进行研究对推理心理学的发展所作出的重要贡献。     

Conditional Probabilities, Potential Surprise, and the Conjunction Fallacy

Tversky and Kahneman (1983) found that a relationship of positive conditional dependence between the components of a conjunction of two events increases the prevalence of the conjunction fallacy. Consistent with this finding, the results of two experiments reveal that dependence leads to higher estimates for the conjunctive probability and a higher incidence of the fallacy. However, contrary to the theoretical account proposed by Tversky and Kahneman, the actual magnitude of the conditional relationship does not directly affect the extent of the fallacy; all that is necessary is for a positive conditional relationship to exist. The pattern of results obtained can be accounted for in terms of Shackle's (1969) 'potential surprise' theory of subjective probability. Surprise theory predicts that the impact of the conditional event will be at its maximum where the relationship is a negative one. Tversky and Kahneman's model, on the other hand, predicts the maximum effect when the relationship is positive. In all 12 scenarios tested, multiple regression analysis revealed that the standardized beta weight associated with the conditional event was greater when the relationship was a negative one. Thus the outcome was supportive of the surprise model rather than Tversky and Kahneman's account.     

Conditional Probability and Defeasible Inference

We offer a probabilistic model of rational consequence relations (Lehmann and Magidor, 1990) by appealing to the extension of the classical Ramsey–Adams test proposed by Vann McGee in (McGee, 1994). Previous and influential models of non-monotonic consequence relations have been produced in terms of the dynamics of expectations (Gärdenfors and Makinson, 1994; Gärdenfors, 1993).Expectation is a term of art in these models, which should not be confused with the notion of expected utility. The expectations of an agent are some form of belief weaker than absolute certainty. Our model offers a modified and extended version of an account of qualitative belief in terms of conditional probability, first presented in (van Fraassen, 1995). We use this model to relate probabilistic and qualitative models of non-monotonic relations in terms of expectations. In doing so we propose a probabilistic model of the notion of expectation. We provide characterization results both for logically finite languages and for logically infinite, but countable, languages. The latter case shows the relevance of the axiom of countable additivity for our probability functions. We show that a rational logic defined over a logically infinite language can only be fully characterized in terms of finitely additive conditional probability. The research of both authors was supported in part by a grant from NSF, and, for Parikh, also by support from the research foundation of CUNY.     

有关条件推理中概率效应的实验研究

通过预备实验选取了四种不同条件概率的条件规则和四种不同前后件概率组合的条件规则作为实验材料,以大学生为被试,考察了两种概率因素(条件概率和前后件概率)对条件推理的演绎形式(MP、DA、AC、MT)以及变通形式(四卡问题)的影响。结果表明,两种概率因素对四种条件推理的影响都非常显著,研究进一步证实了人们对四种推理的认可程度主要与范畴前提的概率成正比的结论;但概率因素对四卡问题的解决影响不明显。     

归类不确定情景下特征推理的综合条件概率模型

用大学生被试,通过三个实验探讨在集中呈现类别成员样本信息的归类不确定情景下的特征推理。实验结果表明,单纯的归类确定性程度和靶类别靶特征的代表性并不直接影响被试的特征推理,而是预测特征相对于目标特征的综合条件概率直接影响被试的特征推理;特征推理不是基于类别中介的间接推理,而是基于特征关联综合条件概率的直接推理。实验结果支持作者提出的预测特征综合条件概率模型。     

Probability spaces,hilbert spaces,and the axioms of test theory

A branch of probability theory that has been studied extensively in recent years, the theory of conditional expectation, provides just the concepts needed for mathematical derivation of the main results of the classical test theory with minimal assumptions and greatest economy in the proofs. The collection of all random variables with finite variance defined on a given probability space is a Hilbert space; the function that assigns to each random variable its conditional expectation is a linear operator; and the properties of the conditional expectation needed to derive the usual test-theory formulas are general properties of linear operators in Hilbert space. Accordingly, each of the test-theory formulas has a simple geometric interpretation that holds in all Hilbert spaces.     

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     

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.     

Conditionals,probability, and non-triviality

We show that the implicational fragment of intuitionism is the weakest logic with a non-trivial probabilistic semantics which satisfies the thesis that the probabilities of conditionals are conditional probabilities. We also show that several logics between intuitionism and classical logic also admit non-trivial probability functions which satisfy that thesis. On the other hand, we also prove that very weak assumptions concerning negation added to the core probability conditions with the restriction that probabilities of conditionals are conditional probabilities are sufficient to trivialize the semantics.     

Probabilistic Logic Under Coherence, Conditional Interpretations, and Default Reasoning

We study a probabilistic logic based on the coherence principle of de Finetti and a related notion of generalized coherence (g-coherence). We examine probabilistic conditional knowledge bases associated with imprecise probability assessments defined on arbitrary families of conditional events. We introduce a notion of conditional interpretation defined directly in terms of precise probability assessments. We also examine a property of strong satisfiability which is related to the notion of toleration well known in default reasoning. In our framework we give more general definitions of the notions of probabilistic consistency and probabilistic entailment of Adams. We also recall a notion of strict p-consistency and some related results. Moreover, we give new proofs of some results obtained in probabilistic default reasoning. Finally, we examine the relationships between conditional probability rankings and the notions of g-coherence and g-coherent entailment.     

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.     

If-Clauses and Probability Operators

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 quantifier theory. Some implications of the Lewis-Kratzer analysis are presented concerning the assignment of probabilities to compounds of conditionals.     

Kaufmann on the Probabilities of Conditionals

Kaufmann has recently argued that the thesis according to which the probability of an indicative conditional equals the conditional probability of the consequent given the antecedent under certain specifiable circumstances deviates from intuition. He presents a method for calculating the probability of a conditional that does seem to give the intuitively correct result under those circumstances. However, the present paper shows that Kaufmann’s method is inconsistent in that it may lead one to assign different probabilities to a single conditional at the same time.     

Iterative Probability Kinematics

Following the pioneer work of Bruno De Finetti [12], conditional probability spaces (allowing for conditioning with events of measure zero) have been studied since (at least) the 1950's. Perhaps the most salient axiomatizations are Karl Popper's in [31], and Alfred Renyi's in [33]. Nonstandard probability spaces [34] are a well know alternative to this approach. Vann McGee proposed in [30] a result relating both approaches by showing that the standard values of infinitesimal probability functions are representable as Popper functions, and that every Popper function is representable in terms of the standard real values of some infinitesimal measure.Our main goal in this article is to study the constraints on (qualitative and probabilistic) change imposed by an extended version of McGee's result. We focus on an extension capable of allowing for iterated changes of view. Such extension, we argue, seems to be needed in almost all considered applications. Since most of the available axiomatizations stipulate (definitionally) important constraints on iterated change, we propose a non-question-begging framework, Iterative Probability Systems (IPS) and we show that every Popper function can be regarded as a Bayesian IPS. A generalized version of McGee's result is then proved and several of its consequences considered. In particular we note that our proof requires the imposition of Cumulativity, i.e. the principle that a proposition that is accepted at any stage of an iterative process of acceptance will continue to be accepted at any later stage. The plausibility and range of applicability of Cumulativity is then studied. In particular we appeal to a method for defining belief from conditional probability (first proposed in [42] and then slightly modified in [6] and [3]) in order to characterize the notion of qualitative change induced by Cumulative models of probability kinematics. The resulting cumulative notion is then compared with existing axiomatizations of belief change and probabilistic supposition. We also consider applications in the probabilistic accounts of conditionals [1] and [30].     

Conditional probability and components of rt in the variable foreperiod experiment

If the passage of time during the foreperiod in a variable foreperiod experiment is marked by a series of tones, RT decreases with the conditional probability of stimulus occurrence. RTs at short foreperiods, however, are rather slower than would be expected on the basis of a simple conditional probability effect. It is suggested that this is attributable to an independent “initial slow reaction” effect, and it is shown that the degree of this effect is influenced by the duration of the prior foreperiod. The results are related to those of variable foreperiod a-reactions in which no conditional probability effect has been found, and it is argued that in a marked reaction of the kind described above, the initial slow reaction effect behaves like an a-reaction component of RT. It is suggested that the absence of a conditional probability effect in the a-reaction and its presence in the marked reaction are related to the fact that a different type of sensory process is used to identify the signal in each case.