Japanese consumers are still hesitant to purchase products from Fukushima, although 7 years have passed since the Fukushima nuclear disaster and these products are officially considered safe. In this study, we examined whether Japanese consumers have negative implicit attitudes towards agricultural and aquatic products from the Fukushima region and whether these attitudes are independent of their explicit attitudes. Japanese students completed an implicit association test and a questionnaire to assess their implicit and explicit attitudes towards products from Fukushima relative to (...) another region. The results of two experiments reliably demonstrated that the public has negative implicit attitudes towards Fukushima products, whereas their explicit attitudes are consistently positive. These observations predominantly held for participants living close to Fukushima (Tokyo) as opposed to participants living far away (Hiroshima): Experiment 1 (n = 40). Furthermore, individual differences in aversion to germs contributed to the implicit attitudes; the implicit negative attitudes were attenuated among the participants with a relatively low aversion to germs: Experiment 2 (n = 60). These results suggest that the implicit attitudes associated with the behavioral immune system, which is conceptualized as a suite of psychological mechanisms designed to proactively resist pathogenic threats, may underlie the hesitation to purchase products from Fukushima. (shrink)

A probability distribution is regular if it does not assign probability zero to any possible event. While some hold that probabilities should always be regular, three counter-arguments have been posed based on examples where, if regularity holds, then perfectly similar events must have different probabilities. Howson and Benci et al. have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson’s “isomorphic” events are not in fact isomorphic, but Howson is speaking (...) of set-theoretic representations of events in a probability model. While those sets are not isomorphic, Williamson’s physical events are, in the relevant sense. Benci et al. claim that all three arguments rest on a conflation of different models, but they do not. They are founded on the premise that similar events should have the same probability in the same model, or in one case, on the assumption that a single rotation-invariant distribution is possible. Having failed to refute the symmetry arguments on such technical grounds, one could deny their implicit premises, which is a heavy cost, or adopt varying degrees of instrumentalism or pluralism about regularity, but that would not serve the project of accurately modelling chances. (shrink)

In probability textbooks, it is widely claimed that zero probability does not mean impossibility. But what stands behind this claim? In this paper I offer an explanation to this claim based on Kolmogorov's formalism. As such, this explanation is relevant to all interpretations of Kolmogorov's probability theory. I start by clarifying that this claim refers only to nonempty events, since empty events are always considered as impossible. Then, I offer the following three reasons for the claim that nonempty events with (...) zero probability are considered as possible: The main reason is simply because they are nonempty. Hence, they are considered as possible despite their zero probability. The second reason is that sometimes the zero probability is taken to be an approximation of some infinitesimal probability value. Such a value is strictly positive and as such does not imply impossibility in a strict sense. Finally, the third reason is that there are interpretations according to which the same event can have different probabilities. Specifically, it is assumed that an event with exactly zero probability can have strictly positive probabilities. This means that such an event can be possible which implies that its zero probability does not mean impossibility. (shrink)

Objective: Clock variance is an important statistic in many clinical and developmental studies. Existing methods require a large number of trials for accurate clock variability assessment, which is problematic in studies using clinical or either young or aged participants. Furthermore, these existing methods often implicitly convolute clock and memory processes, making it difficult to disentangle whether the clock or memory system are driving the observed deviations. Here we assessed whether twenty repeated productions of a well-engrained interval (1 second), a task (...) that does not incorporate memory updating nor the processing of feedback, could provide an accurate assessment of clock variability. Method: Sixty-eight undergraduate students completed two tasks: a 1-second production task in which they were asked to produce a 1-second duration by ending a tone by a keypress, and a multi-duration reproduction task. Durations presented in the reproduction task were tones lasting 1.17, 1.4 and 1.68 seconds. No feedback was presented in either task, and the order of presentation was counterbalanced between participants. Results: The observed central tendency in the reproduction task was better explained by models including the measures of clock variability derived from the 1-second production task than by models without it. Three clock variability measures were calculated for each participant (standard deviation, root mean squared residuals (RMSR) from an estimated linear slope, and RMSR scaled by mean production duration). The model including the scaled RMSR was prefered over the alternative models, and no notable effects of the order of presentation were observed. These results suggest that (1) measures of variability should account for drift, (2) the presentation of another timing task before a 1-second production task did not influence the assessment of the clock variability, and (3) the observed variability adheres to the scalar property and predicts temporal performance, and is thus a usable index of clock variability. Conclusion: This study shows that just 20 repeated productions of 1 second provide a reliable index of clock variability. As administering this task is fast and easy, it could prove to be useful in a large variety of developmental and clinical populations. (shrink)

A probability distribution is regular if no possible event is assigned probability zero. While some hold that probabilities should always be regular, three counter-arguments have been posed based on examples where, if regularity holds, then perfectly similar events must have different probabilities. Howson (2017) and Benci et al. (2016) have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson’s (2007) “isomorphic” events are not in fact isomorphic, but Howson is speaking (...) of set-theoretic representations of events in a probability model. While those sets are not isomorphic, Williamson’s physical events are, in the relevant sense. Benci et al. claim that all three arguments rest on a conflation of different models, but they do not. They are founded on the premise that similar events should have the same probability in the same model, or in one case, on the assumption that a single rotation-invariant distribution is possible. Having failed to refute the symmetry arguments on such technical grounds, one could deny their implicit premises, which is a heavy cost, or adopt varying degrees of instrumentalism or pluralism about regularity, but that would not serve the project of accurately modelling chances. (shrink)

ABSTRACTIn this article, we discuss the benefits of Bayesian statistics and how to utilize them in studies of moral education. To demonstrate concrete examples of the applications of Bayesian statistics to studies of moral education, we reanalyzed two data sets previously collected: one small data set collected from a moral educational intervention experiment, and one big data set from a large-scale Defining Issues Test-2 survey. The results suggest that Bayesian analysis of data sets collected from moral educational studies can provide (...) additional useful statistical information, particularly that associated with the strength of evidence supporting alternative hypotheses, which has not been provided by the classical frequentist approach focusing on P-values. Finally, we introduce several practical guidelines pertaining to how to utilize Bayesian statistics, including the utilization of newly developed free statistical software, Jeffrey’s Amazing Statistics Program, and thresholding based on Bayes Factors, to scholars in the field of moral education. (shrink)

Subjective Bayesianism is a major school of uncertain reasoning and statistical inference. It is often criticized for a lack of objectivity: it opens the door to the influence of values and biases, evidence judgments can vary substantially between scientists, it is not suited for informing policy decisions. My paper rebuts these concerns by connecting the debates on scientific objectivity and statistical method. First, I show that the above concerns arise equally for standard frequentist inference with null hypothesis significance tests. Second, (...) the criticisms are based on specific senses of objectivity with unclear epistemic value. Third, I show that Subjective Bayesianism promotes other, epistemically relevant senses of scientific objectivity—most notably by increasing the transparency of scientific reasoning. (shrink)

Many philosophers deny that happiness can be equated with pleasurable experiences. Nozick introduced an experience machine thought experiment to support the idea that happiness requires pleasurable experiences that are “in contact with reality.” In this thought experiment, people can choose to plug into a machine that induces exclusively pleasurable experiences. We test Nozick’s hypothesis that people will reject this offer. We also contrast Nozick’s experience machine scenario with scenarios that are less artificial, and offer options which are less invasive or (...) disruptive than being connected to a machine, specifically scenarios in which people are offered an experience pill or a pill that improves overall functioning. (shrink)

The physicist Richard Gott defends the Copernican principle, which claims that when we have no information about our position along a given dimension among a group of observers, we should consider ourselves to be randomly located among those observers in respect to that dimension. First, I apply Copernican reasoning to the distribution of evil in the universe. I then contend that evidence for intelligent extraterrestrial life strengthens four important versions of the argument from evil. I remain neutral regarding whether this (...) result is a reductio of these arguments from evil or the statement of a genuine evidential relationship. (shrink)

Many philosophers of science and methodologists have argued that the ability to repeat studies and obtain similar results is an essential component of science. A finding is elevated from single observation to scientific evidence when the procedures that were used to obtain it can be reproduced and the finding itself can be replicated. Recent replication attempts show that some high profile results – most notably in psychology, but in many other disciplines as well – cannot be replicated consistently. These replication (...) attempts have generated a considerable amount of controversy, and the issue of whether direct replications have value has, in particular, proven to be contentious. However, much of this discussion has occurred in published commentaries and social media outlets, resulting in a fragmented discourse. To address the need for an integrative summary, we review various types of replication studies and then discuss the most commonly voiced concerns about direct replication. We provide detailed responses to these concerns and consider different statistical ways to evaluate replications. We conclude there are no theoretical or statistical obstacles to making direct replication a routine aspect of psychological science. (shrink)

According to the so-called ‘deliberation crowds out prediction’ thesis, while deliberating about what you’ll do, you cannot rationally have credences for what you’ll do – you cannot rationally have option-credences. Versions of the thesis have been defended by authors such as Spohn, Levi, Gilboa, Price, Louise, and others. After registering a number of concerns about the thesis, I rehearse and rebut many of the main arguments for it, grouped according to their main themes: agency, vacuity, betting, and decision-theoretical considerations. I (...) go on to suggest many possible theoretical roles for option-credences. -/- I locate the debate about the thesis in a broader discussion: Are there rational credence gaps – propositions to which one cannot rationally assign credences? If there are, they spell trouble for various foundations of Bayesian epistemology, including the usual ratio formula for conditional probability, conditionalization, decision theory, and independence. According to the thesis, credence gaps are completely mundane; they arise every time someone rationally deliberates. But these foundations are safe from any threat here, I contend, since the thesis is false. Deliberation welcomes prediction. (shrink)

This article examines the effect of material evidence upon historiographic hypotheses. Through a series of successive Bayesian conditionalizations, I analyze the extended competition among several hypotheses that offered different accounts of the transition between the Bronze Age and the Iron Age in Palestine and in particular to the “emergence of Israel”. The model reconstructs, with low sensitivity to initial assumptions, the actual outcomes including a complete alteration of the scientific consensus. Several known issues of Bayesian confirmation, including the problem of (...) old evidence, the introduction and confirmation of novel theories and the sensitivity of convergence to uncertain and disputed evidence are discussed in relation to the model’s result and the actual historical process. The most important result is that convergence of probabilities and of scientific opinion is indeed possible when advocates of rival hypotheses hold similar judgment about the factual content of evidence, even if they differ sharply in their historiographic interpretation. This speaks against the contention that understanding of present remains is so irrevocably biased by theoretical and cultural presumptions as to make an objective assessment impossible. (shrink)

The field of psychology, including cognitive science, is vexed by a crisis of confidence. Although the causes and solutions are varied, we focus here on a common logical problem in inference. The default mode of inference is significance testing, which has a free lunch property where researchers need not make detailed assumptions about the alternative to test the null hypothesis. We present the argument that there is no free lunch; that is, valid testing requires that researchers test the null against (...) a well-specified alternative. We show how this requirement follows from the basic tenets of conventional and Bayesian probability. Moreover, we show in both the conventional and Bayesian framework that not specifying the alternative may lead to rejections of the null hypothesis with scant evidence. We review both frequentist and Bayesian approaches to specifying alternatives, and we show how such specifications improve inference. The field of cognitive science will benefit because consideration of reasonable alternatives will undoubtedly sharpen the intellectual underpinnings of research. (shrink)

Executive control of attention refers to processes that detect and resolve conflict among competing thoughts and actions. Despite the high-level nature of this faculty, the role of awareness in executive control of attention is not well understood. In this study, we used interocular suppression to mask the flankers in an arrow flanker task, in which the flankers and the target arrow were presented simultaneously in order to elicit executive control of attention. Participants were unable to detect the flanker arrows or (...) to reliably identify their direction when masked. There was a typical conflict effect (prolonged reaction time and increased error rate under flanker-target incongruent condition compared to congruent condition) when the flanker arrows were unmasked, while the conflict effect was absent when the flanker arrows were masked with interocular suppression. These results suggest that blocking awareness of competing stimuli with interocular suppression prevents the involvement of executive control of attention. (shrink)

Within the literature on emotion and behavioral action, studies on approach-avoidance take up a prominent place. Several experimental paradigms feature successful conceptual replications but many original studies have not yet been replicated directly. We present such a direct replication attempt of two seminal experiments originally conducted by Chen and Bargh (1999). In their first experiment, participants affectively evaluated attitude objects by pulling or pushing a lever. Participants who had to pull the lever with positively valenced attitude objects and push the (...) lever with negatively valenced attitude objects (i.e., congruent instruction) did so faster than participants who had to follow the reverse (i.e., incongruent) instruction. In Chen and Bargh's second experiment, the explicit evaluative instructions were absent and participants merely responded to the attitude objects by either always pushing or always pulling the lever. Similar results were obtained as in Experiment 1. Based on these findings, Chen and Bargh concluded that (1) attitude objects are evaluated automatically; and (2) attitude objects automatically trigger a behavioral tendency to approach or avoid. We attempted to replicate both experiments and failed to find the effects reported by Chen and Bargh as indicated by our pre-registered Bayesian data analyses; nevertheless, the evidence in favor of the null hypotheses was only anecdotal, and definitive conclusions await further study. (shrink)

An important assumption underlying meaningful comparisons of scores in rater-mediated assessments is that measurement is commensurate across raters. When raters differentially apply the standards established by an instrument, scores from different raters are on fundamentally different scales and no longer preserve a common meaning and basis for comparison. In this study, we developed a method to accommodate measurement noninvariance across raters when measurements are cross-classified within two distinct hierarchical units. We conceptualized random item effects cross-classified graded response models and used (...) random discrimination and threshold effects to test, calibrate, and account for measurement noninvariance among raters. By leveraging empirical estimates of rater-specific deviations in the discrimination and threshold parameters, the proposed method allows us to identify noninvariant items and empirically estimate and directly adjust for this noninvariance within a cross-classified framework. Within the context of teaching evaluations, the results of a case study suggested substantial noninvariance across raters and that establishing an approximately invariant scale through random item effects improves model fit and predictive validity. (shrink)

Nelson Goodman cast the ‘problem of induction’ as the task of articulating the principles and standards by which to distinguish valid from invalid inductive inferences. This paper explores some logical bounds on the ability of a rational reasoner to accomplish this task. By a simple argument, either an inductive inference method cannot admit its own fallibility, or there exists some non-inferable hypothesis whose non-inferability the method cannot infer (violating the principle of ‘negative introspection’). The paper discusses some implications of this (...) limited self-knowledge for the justifiability of inductive inferences, auto-epistemic logic, and the epistemic foundations of game theory. (shrink)

Over the years several non-equivalent probabilistic measures of coherence have been discussed in the philosophical literature. In this paper we examine these measures with respect to their empirical adequacy. Using test cases from the coherence literature as vignettes for psychological experiments we investigate whether the measures can predict the subjective coherence assessments of the participants. It turns out that the participants’ coherence assessments are best described by Roche’s coherence measure based on Douven and Meijs’ average mutual support approach and the (...) conditional probability. (shrink)

Why are conditional degrees of belief in an observation E, given a statistical hypothesis H, aligned with the objective probabilities expressed by H? After showing that standard replies are not satisfactory, I develop a suppositional analysis of conditional degree of belief, transferring Ramsey’s classical proposal to statistical inference. The analysis saves the alignment, explains the role of chance-credence coordination, and rebuts the charge of arbitrary assessment of evidence in Bayesian inference. Finally, I explore the implications of this analysis for Bayesian (...) reasoning with idealized models in science. (shrink)

This volume focuses on various questions concerning the interpretation of probability and probabilistic reasoning in biology and physics. It is inspired by the idea that philosophers of biology and philosophers of physics who work on the foundations of their disciplines encounter similar questions and problems concerning the role and application of probability, and that interaction between the two communities will be both interesting and fruitful. In this introduction we present the background to the main questions that the volume focuses on (...) and summarize the highlights of the individual contributions. (shrink)

We survey a variety of possible explications of the term “Individual Risk.” These in turn are based on a variety of interpretations of “Probability,” including classical, enumerative, frequency, formal, metaphysical, personal, propensity, chance and logical conceptions of probability, which we review and compare. We distinguish between “groupist” and “individualist” understandings of probability, and explore both “group to individual” and “individual to group” approaches to characterising individual risk. Although in the end that concept remains subtle and elusive, some pragmatic suggestions for (...) progress are made. (shrink)

The p value is the probability under the null hypothesis of obtaining an experimental result that is at least as extreme as the one that we have actually obtained. That probability plays a crucial role in frequentist statistical inferences. But if we take the word ‘extreme’ to mean ‘improbable’, then we can show that this type of inference can be very problematic. In this paper, I argue that it is a mistake to make such an interpretation. Under minimal assumptions about (...) the alternative hypothesis, I explain why ‘extreme’ means ‘outside the most precise predicted range of experimental outcomes for a given upper bound probability of error’. Doing so, I rebut recent formulations of recurrent criticisms against the frequentist approach in statistics and underscore the importance of random variables. (shrink)

Common methods for analysing response time (RT) tasks, frequently used across different disciplines of psychology, suffer from a number of limitations such as the failure to directly measure the underlying latent processes of interest and the inability to take into account the uncertainty associated with each individual's point estimate of performance. Here, we discuss a Bayesian hierarchical diffusion model and apply it to RT data. This model allows researchers to decompose performance into meaningful psychological processes and to account optimally for (...) individual differences and commonalities, even with relatively sparse data. We highlight the advantages of the Bayesian hierarchical diffusion model decomposition by applying it to performance on Approach–Avoidance Tasks, widely used in the emotion and psychopathology literature. Model fits for two experimental data-sets demonstrate that the model performs well. The Bayesian hierarchical diffusion model overcomes important limitations of current analysis procedures and provides deeper insight in latent psychological processes of interest. (shrink)

We consider work extraction from two finite reservoirs with constant heat capacity, when the thermodynamic coordinates of the process are not fully specified, i.e., are described by probabilities only. Incomplete information refers to both the specific value of the temperature as well as the label of the reservoir to which it is assigned. Based on the concept of inference, we characterize the reduced performance resulting from this lack of control. Indeed, the estimates for the average efficiency reveal that uncertainty regarding (...) the exact labels reduces the maximal expected efficiency below the Carnot value ), its minimum value reproducing the well known Curzon–Ahlborn value: \ . We also estimate the efficiency before the value of the temperature is revealed. It is found that if the labels are known with certainty, then in the near-equilibrium limit the efficiency scales as \ , while if there is maximal uncertainty in the labels, then the average estimate for efficiency drops to \ . We also suggest how the inferred properties of the incomplete model can be mapped onto a model with complete information but with an additional source of thermodynamic irreversibility. (shrink)

According to grounded cognition, words whose semantics contain sensory-motor features activate sensory-motor simulations, which, in turn, interact with spatial responses to produce grounded congruency effects. Growing evidence shows these congruency effects do not always occur, suggesting instead that the grounded features in a word's meaning do not become active automatically across contexts. Researchers sometimes use this as evidence that concepts are not grounded, further concluding that grounded information is peripheral to the amodal cores of concepts. We first review broad evidence (...) that words do not have conceptual cores, and that even the most salient features in a word's meaning are not activated automatically. Then, in three experiments, we provide further evidence that grounded congruency effects rely dynamically on context, with the central grounded features in a concept becoming active only when the current context makes them salient. Even when grounded features are central to a word's meaning, their activation depends on task conditions. (shrink)

The Jeffreys–Lindley paradox displays how the use of a \ value ) in a frequentist hypothesis test can lead to an inference that is radically different from that of a Bayesian hypothesis test in the form advocated by Harold Jeffreys in the 1930s and common today. The setting is the test of a well-specified null hypothesis versus a composite alternative. The \ value, as well as the ratio of the likelihood under the null hypothesis to the maximized likelihood under the (...) alternative, can strongly disfavor the null hypothesis, while the Bayesian posterior probability for the null hypothesis can be arbitrarily large. The academic statistics literature contains many impassioned comments on this paradox, yet there is no consensus either on its relevance to scientific communication or on its correct resolution. The paradox is quite relevant to frontier research in high energy physics. This paper is an attempt to explain the situation to both physicists and statisticians, in the hope that further progress can be made. (shrink)

In this contribution, I comment on Raffaella Campaner’s defense of explanatory pluralism in psychiatry (in this volume). In her paper, Campaner focuses primarily on explanatory pluralism in contrast to explanatory reductionism. Furthermore, she distinguishes between pluralists who consider pluralism to be a temporary state on the one hand and pluralists who consider it to be a persisting state on the other hand. I suggest that it would be helpful to distinguish more than those two versions of pluralism – different understandings (...) of explanatory pluralism both within philosophy of science and psychiatry – namely moderate/temporary pluralism, anything goes pluralism, isolationist pluralism, integrative pluralism and interactive pluralism. Next, I discuss the pros and cons of these different understandings of explanatory pluralism. Finally, I raise the question of how to implement or operationalize explanatory pluralism in scientific practice; how to structure the “genuine dialogue” or shape “the pluralistic attitude” Campaner is referring to. As tentative answers, I explore a question-based framework for explanatory pluralism as well as social-epistemological procedures for interaction among competing approaches and explanations. (shrink)

The aim of this article is to discuss the conditions under which functional neuroimaging can contribute to the study of higher cognition. We begin by presenting two case studies—on moral and economic decision making—which will help us identify and examine one of the main ways in which neuroimaging can help advance the study of higher cognition. We agree with critics that functional magnetic resonance imaging (fMRI) studies seldom “refine” or “confirm” particular psychological hypotheses, or even provide details of the neural (...) implementation of cognitive functions. However, we suggest that neuroimaging can support psychology in a different way—namely, by selecting among competing hypotheses of the cognitive mechanisms underlying some mental function. One of the main ways in which neuroimaging can be used for hypothesis selection is via reverse inferences, which we here examine in detail. Despite frequent claims to the contrary, we argue that successful reverse inferences do not assume any strong or objectionable form of reductionism or functional locationism. Moreover, our discussion illustrates that reverse inferences can be successful at early stages of psychological theorizing, when models of the cognitive mechanisms are only partially developed. (shrink)

This chapter contains sections titled: Highlights of Past Literature Current Work Future Work.

The idea that perceptual and cognitive systems must incorporate knowledge about the structure of the environment has become a central dogma of cognitive theory. In a Bayesian context, this idea is often realized in terms of “tuning the prior”—widely assumed to mean adjusting prior probabilities so that they match the frequencies of events in the world. This kind of “ecological” tuning has often been held up as an ideal of inference, in fact defining an “ideal observer.” But widespread as this (...) viewpoint is, it directly contradicts Bayesian philosophy of probability, which views probabilities as degrees of belief rather than relative frequencies, and explicitly denies that they are objective characteristics of the world. Moreover, tuning the prior to observed environmental frequencies is subject to overfitting, meaning in this context overtuning to the environment, which leads (ironically) to poor performance in future encounters with the same environment. Whenever there is uncertainty about the environment—which there almost always is—an agent's prior should be biased away from ecological relative frequencies and toward simpler and more entropic priors. (shrink)

The article revisits the large n problem as it relates to the Jeffreys-Lindley paradox to compare the frequentist, Bayesian, and likelihoodist approaches to inference and evidence. It is argued that what is fallacious is to interpret a rejection of as providing the same evidence for a particular alternative, irrespective of n; this is an example of the fallacy of rejection. Moreover, the Bayesian and likelihoodist approaches are shown to be susceptible to the fallacy of acceptance. The key difference is that (...) in frequentist testing the severity evaluation circumvents both fallacies but no such principled remedy exists for the other approaches. (shrink)

This paper is an extended review of the book Error and Inference, edited by Deborah Mayo and Aris Spanos, about their frequentist and philosophical perspective on testing of hypothesis and on the criticisms of alternatives like the Bayesian approach.

In this paper I argue that it is finally time to move beyond the Nagelian framework and to break new ground in thinking about epistemic reduction in biology. I will do so, not by simply repeating all the old objections that have been raised against Ernest Nagel’s classical model of theory reduction. Rather, I grant that a proponent of Nagel’s approach can handle several of these problems but that, nevertheless, Nagel’s general way of thinking about epistemic reduction in terms of (...) theories and their logical relations is entirely inadequate with respect to what is going on in actual biological research practice. (shrink)

Some philosophers, most notably Hempel and Salmon, have tried to reduce explanation to probability by proposing analyses of explanation in probabilistic terms. Hempel claims, roughly, that a hypothesis H explains a datum D if and only if the conditional probability P is close to 1. It is well known that such an account fails in cases where H is irrelevant for D. Even though it is highly likely that Tom will not become pregnant, given that he regularly takes his wife’s (...) birth control pills, the latter does not explain the former. Neither does an idea work which is in the proximity of Salmon’s, namely, that H explains D if and only if P > P. Suppose Susan swallows a pound of arsenic in order to commit suicide. Shortly after, however, she dies because she is run over by a bus. The probability of dying, given that one ingests a pound of arsenic, is usually higher than the prior probability of dying. Nonetheless, it is not the arsenic but the collision with the bus which explains Susan’s death. The aforementioned objections are directed against …. (shrink)