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 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)

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 introduce a distinction, unnoticed in the literature, between four varieties of objective Bayesianism. What we call ' strong objective Bayesianism' is characterized by two claims, that all scientific inference is 'logical' and that, given the same background information two agents will ascribe a unique probability to their priors. We think that neither of these claims can be sustained; in this sense, they are 'dogmatic'. The first fails to recognize that some scientific inference, in particular that concerning evidential relations, is (...) not (in the appropriate sense) logical, the second fails to provide a non-question-begging account of 'same background information'. We urge that a suitably objective Bayesian account of scientific inference does not require either of the claims. Finally, we argue that Bayesianism needs to be fine-grained in the same way that Bayesians fine-grain their beliefs. (shrink)

Medical research makes intensive use of statistics in order to support its claims. In this paper we make explicit an epistemological tension between the conduct of clinical trials and their interpretation: statistical evidence is sometimes discarded on the basis of an underlined Bayesian reasoning. We suggest that acknowledging the potentiality of Bayesian statistics might contribute to clarify and improve comprehension of medical research. Nevertheless, despite Bayesianism may provide a better account for scientific inference with respect to the standard frequentist approach, (...) Bayesian statistics is rarely adopted in clinical research. The main reason lies in the supposed subjective elements characterizing this perspective. Hence, we discuss this objection presenting the so-called Reference analysis, a formal method which has been developed in the context of objective Bayesian statistics in order to define priors which have a minimal or null impact on posterior probabilities. Furthermore, according to this method only available data are relevant sources of information, so that it resists the most common criticisms against Bayesianism. (shrink)

Until now, antirealists have offered sketches of a theory of truth, at best. In this paper, we present a probabilist account of antirealist truth in some formal detail, and we assess its ability to deal with the problems that are standardly taken to beset antirealism.

A set of hypotheses is formulated for a connectionist approach to cognitive modeling. These hypotheses are shown to be incompatible with the hypotheses underlying traditional cognitive models. The connectionist models considered are massively parallel numerical computational systems that are a kind of continuous dynamical system. The numerical variables in the system correspond semantically to fine-grained features below the level of the concepts consciously used to describe the task domain. The level of analysis is intermediate between those of symbolic cognitive models (...) and neural models. The explanations of behavior provided are like those traditional in the physical sciences, unlike the explanations provided by symbolic models.Higher-level analyses of these connectionist models reveal subtle relations to symbolic models. Parallel connectionist memory and linguistic processes are hypothesized to give rise to processes that are describable at a higher level as sequential rule application. At the lower level, computation has the character of massively parallel satisfaction of soft numerical constraints; at the higher level, this can lead to competence characterizable by hard rules. Performance will typically deviate from this competence since behavior is achieved not by interpreting hard rules but by satisfying soft constraints. The result is a picture in which traditional and connectionist theoretical constructs collaborate intimately to provide an understanding of cognition. (shrink)

Consider the regularity thesis that each possible event has non-zero probability. Hájek challenges this in two ways: there can be nonmeasurable events that have no probability at all and on a large enough sample space, some probabilities will have to be zero. But arguments for the existence of nonmeasurable events depend on the axiom of choice. We shall show that the existence of anything like regular probabilities is by itself enough to imply a weak version of AC sufficient to prove (...) the Banach–Tarski Paradox on the decomposition of a ball into two equally sized balls, and hence to show the existence of nonmeasurable events. This provides a powerful argument against unrestricted orthodox Bayesianism that works even without AC. A corollary of our formal result is that if every partial order extends to a total preorder while maintaining strict comparisons, then the Banach–Tarski Paradox holds. This yields an argument that incommensurability cannot be avoided in ambitious versions of decision theory. (shrink)

This paper justifies the inference of probabilities from symmetries. I supply some examples of important and correct inferences of this variety. Two explanations of such inferences -- an explanation based on the Principle of Indifference and a proposal due to Poincaré and Reichenbach -- are considered and rejected. I conclude with my own account, in which the inferences in question are shown to be warranted a posteriori, provided that they are based on symmetries in the mechanisms of chance setups.

The epistemic state of complete ignorance is not a probability distribution. In it, we assign the same, unique, ignorance degree of belief to any contingent outcome and each of its contingent, disjunctive parts. That this is the appropriate way to represent complete ignorance is established by two instruments, each individually strong enough to identify this state. They are the principle of indifference (PI) and the notion that ignorance is invariant under certain redescriptions of the outcome space, here developed into the (...) ‘principle of invariance of ignorance' (PII). Both instruments are so innocuous as almost to be platitudes. Yet the literature in probabilistic epistemology has misdiagnosed them as paradoxical or defective since they generate inconsistencies when conjoined with the assumption that an epistemic state must be a probability distribution. To underscore the need to drop this assumption, I express PII in its most defensible form as relating symmetric descriptions and show that paradoxes still arise if we assume the ignorance state to be a probability distribution. *Received February 2007; revised July 2007. †To contact the author, please write to: Department of History and Philosophy of Science, University of Pittsburgh, Pittsburgh, PA 15260; e-mail: [email protected]. (shrink)

Of course, the rationale of PME is so different from what has been taught in “orthodox” statistics courses for fifty years, that it causes conceptual hangups for many with conventional training. But beginning students have no difficulty with it, for it is just a mathematical model of the natural, common sense way in which anybody does conduct his inferences in problems of everyday life.The difficulties that seem so prominent in the literature today are, therefore, only transient phenomena that will disappear (...) automatically in time. Indeed, this revolution in our attitude toward inference is already an accomplished fact among those concerned with a few specialized applications; with a little familarity in its use its advantages are apparent and it no longer seems strange. It is the idea that inference was once thought to be tied to frequencies in random experiments, that will seem strange to future generations. (shrink)

In this paper we respond to the article An Objective Theory of Statistical Testing by D. G. Mayo (1983). We argue that the theory of testing developed by Mayo, NPT*, is neither novel nor objective. We also respond to the claims made by Mayo against Bayesian theory.

The focus of this study is cognitive choice: the selection of one cognitive option (a hypothesis, a theory, or an axiom, for instance) rather than another. The study proposes that cognitive choice should be based on the plausibilities of states posited by rival cognitive options and the utilities of these options' information outcomes. The proposal introduces a form of decision theory that is novel because comparative; it permits many choices among cognitive options to be based on merely comparative plausibilities and (...) utilities. This form of decision theory intersects with recommendations by advocates of decision theory for cognitive choice, on the one hand, and defenders of comparative evaluation of scientific hypotheses and theories, on the other. But it differs from prior decision-theoretic proposals because it requires no more than minimal precision in specifying plausibilities and utilities. And it differs from comparative proposals because none has shown how comparative evaluations can be carried out within a decision-theoretic framework. (shrink)

. The uniform prior distribution is often seen as a mathematical description of noninformativeness. This paper uses the well-known Three Prisoners Paradox to examine the impossibility of maintaining noninformativeness throughout hierarchization. The Paradox has been solved by Bayesian conditioning over the choice made by the Warder when asked to name a prisoner who will be shot. We generalize the paradox to situations of N prisoners, k executions and m announcements made by the Warder. We then extend the consequences of hierarchically (...) placing uniform and symmetrical priors for the probability p of the Warder naming Prisoner B, say. We prove that breaks of indifference and neutrality caused by assignment of uniform and symmetrical priors in lieu of degenerate indifference probabilities hold in general. Speaking of unknown probabilities or of probability of a probability must be forbidden as meaningless. Bruno de Finetti, 1977 I regard the use of hierarchical chains as a technique helping you to sharpen your subjective probabilities. I. J. Good, 1981. (shrink)

In this paper we consider the problem of how to measure the strength of statistical evidence from the perspective of evidence amalgamation operations. We begin with a fundamental measurement amalgamation principle : for any measurement, the inputs and outputs of an amalgamation procedure must be on the same scale, and this scale must have a meaningful interpretation vis a vis the object of measurement. Using the p value as a candidate evidence measure, we examine various commonly used approaches to amalgamation (...) of evidence across similar studies, including standard forms of meta-analysis. We show that none of these methods satisfies MAP. Thus an underlying measurement problem remains. We argue that a successful approach to evidence amalgamation necessitates a solution to the problem of evidence measurement, and we suggest some lines of reasoning that might guide further work towards this end. (shrink)

In most decision-making situations, there is a plethora of information potentially available to people. Deciding what information to gather and what to ignore is no small feat. How do decision makers determine in what sequence to collect information and when to stop? In two experiments, we administered a version of the German cities task developed by Gigerenzer and Goldstein (1996), in which participants had to decide which of two cities had the larger population. Decision makers were not provided with the (...) names of the cities, but they were able to collect different kinds of cues for both response alternatives (e.g., “Does this city have a university?”) before making a decision. Our experiments differed in whether participants were free to determine the number of cues they examined. We demonstrate that a novel model, using hierarchical latent mixtures and Bayesian inference (Lee & Newell, 2011) provides a more complete description of the data from both experiments than simple conventional strategies, such as the take–the–best or the Weighted Additive heuristics. (shrink)

A set of hypotheses is formulated for a connectionist approach to cognitive modeling. These hypotheses are shown to be incompatible with the hypotheses underlying traditional cognitive models. The connectionist models considered are massively parallel numerical computational systems that are a kind of continuous dynamical system. The numerical variables in the system correspond semantically to fine-grained features below the level of the concepts consciously used to describe the task domain. The level of analysis is intermediate between those of symbolic cognitive models (...) and neural models. The explanations of behavior provided are like those traditional in the physical sciences, unlike the explanations provided by symbolic models. Higher-level analyses of these connectionist models reveal subtle relations to symbolic models. Parallel connectionist memory and linguistic processes are hypothesized to give rise to processes that are describable at a higher level as sequential rule application. At the lower level, computation has the character of massively parallel satisfaction of soft numerical constraints; at the higher level, this can lead to competence characterizable by hard rules. Performance will typically deviate from this competence since behavior is achieved not by interpreting hard rules but by satisfying soft constraints. The result is a picture in which traditional and connectionist theoretical constructs collaborate intimately to provide an understanding of cognition. (shrink)

It is a familiar argument that advocates accommodating the so-called paradoxes of decision theory by abandoning the “independence” postulate. After all, if we grant that choice reveals preference, the anomalous choice patterns of the Allais and Ellsberg problems violate postulate P2 of Savage's system. The strategy of making room for new preference patterns by relaxing independence is adopted in each of the following works: Samuelson, Kahneman and Tversky's “Prospect Theory”, Allais and Hagen, Fishburn, Chew and MacCrimmon, McClennen, and in closely (...) argued essays by Machina. (shrink)

The thesis of this article is that the nature of probability is centered on its formal properties, not on any of its standard interpretations. Section 2 is a survey of Bayesian applications. Section 3 focuses on two examples from physics that seem as completely objective as other physical concepts. Section 4 compares the conflict between subjective Bayesians and objectivists about probability to the earlier strident conflict in physics about the nature of force. Section 5 outlines a pragmatic approach to the (...) various interpretations of probability. Finally, Sect. 6 argues that the essential formal nature of probability is expressed in the standard axioms, but more explicit attention should be given to the concept of randomness. (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)

William of Ockham wrote, ‘It is futile to do with more things that which can be done with fewer .’ But what if each option uses less than its predecessor but no option uses the least? A scale perfectly balanced between a pair of kilogram weights can be tipped by adding half a kilogram to one side, or a quarter of a kilogram, or an eighth of a kilogram, or … For any choice, there is an option that gets the (...) job done with less. Relative futility does not entail absolute futility. The job can get done – but only if we use superfluous means. Is this a refutation of Ockham’s Razor or just a close shave? (shrink)

A set of hypotheses is formulated for a connectionist approach to cognitive modeling. These hypotheses are shown to be incompatible with the hypotheses underlying traditional cognitive models. The connectionist models considered are massively parallel numerical computational systems that are a kind of continuous dynamical system. The numerical variables in the system correspond semantically to fine-grained features below the level of the concepts consciously used to describe the task domain. The level of analysis is intermediate between those of symbolic cognitive models (...) and neural models. The explanations of behavior provided are like those traditional in the physical sciences, unlike the explanations provided by symbolic models. (shrink)

The theoretical construction and practical use of prior probabilities, in particular for systems having many degrees of freedom, are investigated. It becomes clear that it is operationally unsound to use mutually consistent priors if one wishes to draw sensible conclusions from practical experiments. The prior cannot usefully be identified with a state of knowledge, and indeed it is not so identified in common scientific practice. Rather, it can be identified with the question one asks. Accordingly, priors are free constructions. Their (...) informal, ill-defined and subjective characteristics must carry over into the conclusions one chooses to draw from experiments or observations. (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)

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)

I I set out my view that all inference is essentially deductive and pinpoint what I take to be the major shortcomings of the induction rule.II The import of data depends on the probability model of the experiment, a dependence ignored by the induction rule. Inductivists admit background knowledge must be taken into account but never spell out how this is to be done. As I see it, that is the problem of induction.III The induction rule, far from providing a (...) method of discovery, does not even serve to detect pattern. Knowing that there is uniformity in the universe is no help to discovering laws. A critique of Reichenbach's justification of the straight rule is constructed along these lines.IV The induction rule, by itself, cannot account for the varying rates at which confidence in an hypothesis mounts with data. The mathematical analysis of this salient feature of inductive reasoning requires prior probabilities. We also argue, against orthodox statisticians, that prior probabilities make a substantive contribution to the objectivity of inductive methods, viz. to the design of experiments and the selection of decision rules.V Carnap's general criticisms of various estimation rules, like the straight rule and the ‘impervious rule’, are seen to be misguided when the prior densities to which they correspond are taken into account.VI Analysis of Hempel's definition of confirmation qua formalization of the enumerative (naive) conception of instancehood. We show that from the standpoint of the quantitative measure P(H/E):P(H) for the degree to which E confirms H, Hempel's classificatory concept yields correct results only for sampling at large from a finite population with a two-way classification all of whose compositions are equally probable. We extend the analysis to Goodman's paradox, finding cases in which grue-like hypotheses do receive as much confirmation as their opposite numbers. We argue, moreover, the irrelevancy of entrenchment, and maintain that Goodman's paradox is no more than a straightforward counter-example to the enumerative conception of instancehood embodied in Hempel's definition.VII We rebutt the objection that prior probabilities, qua inputs of Bayesian analysis, can only be obtained by enumerative induction (insofar as they are objective). The divergence in the prior densities of two rational agents is less a function of subjectivity, we maintain, than of vagueness.VIII Our concluding remarks stress that, for Bayesians, there is no problem of induction in the usual sense. (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)

This article proposes a new interpretation of mutual information. We examine three extant interpretations of MI by reduction in doubt, by reduction in uncertainty, and by divergence. We argue that the first two are inconsistent with the epistemic value of information assumed in many applications of MI: the greater is the amount of information we acquire, the better is our epistemic position, other things being equal. The third interpretation is consistent with EVI, but it is faced with the problem of (...) measure sensitivity and fails to justify the use of MI in giving definitive answers to questions of information. We propose a fourth interpretation of MI by reduction in expected inaccuracy, where inaccuracy is measured by a strictly proper monotonic scoring rule. It is shown that the answers to questions of information given by MI are definitive whenever this interpretation is appropriate, and that it is appropriate in a wide range of applications with epistemic implications. _1_ Introduction _2_ Formal Analyses of the Three Interpretations _2.1_ Reduction in doubt _2.2_ Reduction in uncertainty _2.3_ Divergence _3_ Inconsistency with Epistemic Value of Information _4_ Problem of Measure Sensitivity _5_ Reduction in Expected Inaccuracy _6_ Resolution of the Problem of Measure Sensitivity _6.1_ Alternative measures of inaccuracy _6.2_ Resolution by strict propriety _6.3_ Range of applications _7_ Global Scoring Rules _8_ Conclusion. (shrink)