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According to common judicial standard, judgment in favor of plaintiff should be made if and only if it is "more probable than not" that the defendant's action was the cause for the plaintiff's damage (or death). This paper provides formal semantics, based on structural models of counterfactuals, for the probability that event x was a necessary or sufficient cause (or both) of another event y. The paper then explicates conditions under which the probability of necessary (or sufficient) causation can be learned from statistical data, and shows how data from both experimental and nonexperimental studies can be combined to yield information that neither study alone can provide. Finally,weshow that necessity and sufficiency are two independent aspects of causation, and that both should be invoked in the construction of causal explanations for specific scenarios.

In the previous chapter (Tenenbaum, Griffiths, & Niyogi, this volume), we introduced a framework for thinking about the structure, function, and acquisition of intuitive theories inspired by an analogy to the research program of generative grammar in linguistics. We argued that a principal function for intuitive theories, just as for grammars for natural

computation. In preparation. Address for correspondence:

When participants assess the relationship between two variables, each with levels of presence and absence, the two most robust phenomena are that: (a) observing the joint presence of the variables has the largest impact on judgment and observing joint absence has the smallest impact, and (b) participants’ prior beliefs about the variables ’ relationship influence judgment. Both phenomena represent departures from the traditional normative model (the phi coefficient or related measures) and have therefore been interpreted as systematic errors. However, both phenomena are consistent with a Bayesian approach to the task. From a Bayesian perspective: (a) joint presence is normatively more informative than joint absence if the presence of variables is rarer than their absence, and (b) failing to incorporate prior beliefs is a normative error. Empirical evidence is reported showing that joint absence is seen as more informative than joint presence when it is clear that absence of the variables, rather than their presence, is rare.

It is “well known ” that in linear models: (1) testable constraints on the marginal distribution of observed variables distinguish certain cases in which an unobserved cause jointly influences several observed variables; (2) the technique of “instrumental variables ” sometimes permits an estimation of the influence of one variable on another even when the association between the variables may be confounded by unobserved common causes; (3) the association (or conditional probability distribution of one variable given another) of two variables connected by a path or pair of paths with a single common vertex (a trek) can be computed directly from the parameter values associated with each edge in the trek; (4) the association of two variables produced by multiple treks can be computed from the parameters associated with each trek; and (5) the independence of two variables conditional on a third implies the corresponding independence of the sums of the variables over all units conditional on the sums over all

Introduction: Characterizing the Questions of causal reasoning This chapter describes the mechanism approach to the study of causal reasoning. We will first offer a characterization of the central issues in human causal reasoning, and will discuss how the mechanism approach addresses these issues. In the course of this presentation, we will frequently compare the mechanism approach with alternative accounts based on analyses of covariation, or what is often termed the regularity view. The aims of this chapter are the following: to explain why covariation and mechanism are different, to discuss why such a distinction is actually a useful tool for our understanding of causal reasoning, and to explicate the complementary nature of the two views. Before presenting these two approaches, it is necessary first to offer a description of the domain or problem itself : namely, what are these alternative approaches to? Although there are a number of different ways of characterizing the study of

Leading accounts of judgment under uncertainty evaluate performance within purely statistical frameworks, holding people to the standards of classical Bayesian (Tversky & Kahneman, 1974) or frequentist (Gigerenzer & Hoffrage, 1995) norms. We argue that these frameworks have limited ability to explain the success and flexibility of people's real-world judgments, and propose an alternative normative framework based on Bayesian inferences over causal models. Deviations from traditional norms of judgment, such as "base-rate neglect", may then be explained in terms of a mismatch between the statistics given to people and the causal models they intuitively construct to support probabilistic reasoning. Four experiments show that when a clear mapping can be established from given statistics to the parameters of an intuitive causal model, people are more likely to use the statistics appropriately, and that when the classical and causal Bayesian norms differ in their prescriptions, people's judgments are more consistent with causal Bayesian norms.

this paper. Correspondence should be addressed to M. J. Dennis, Madsen Center, Augustana College, 2001 S. Summit Ave., Sioux Falls, SD 57197 (e-mail: dennis@inst.augie . edu)

Are human contingency judgments based on associationistic principles such as cue competition or on normative principles as specified by rational-cognitive models? In this study, participants learned to predict an outcome from several simultaneously presented cues. They were asked to judge the cues in regard to causal power or statistical concepts such as probability or relative frequency. Uniform application of associationistic principles implies cue-interaction effects of blocking (Experiment 1) and conditioned inhibition (Experiment 2) for all judgments. A rational-cognitive framework predicts cueinteraction effects for causality judgments, but not for probability and relative frequency judgments. The results support the rational-cognitive framework on all accounts.