We propose a class [I,S] of loss functions for modeling the imprecise preferences of the decision maker in Bayesian Decision Theory. This class is built upon two extreme loss functions I and S which reflect the limited information about the loss function. We give an approximation of the set of Bayes actions for every loss function in [I,S] and every prior in a mixture class; if the decision space is a subset of R, we obtain the exact set.

According to one view, there cannot: Bayesianism fails to do justice to essential aspects of knowledge and belief, and as such it cannot provide a genuine epistemology at all. According to another view, Bayesianism should supersede traditional epistemology: where the latter has been mired in endless debates over skepticism and Gettierology, Bayesianism offers the epistemologist a thriving research program. We will advocate a more moderate view: Bayesianism can illuminate various long­standing problems of epistemology, while not addressing all of them; and (...) while Bayesianism opens up fascinating new areas of research, it by no means closes down the staple preoccupations of traditional epistemology. The contrast between the two epistemologies can be traced back to the mid­17th century. Descartes regarded belief as an all­or­nothing matter, and he sought justifications for his claims to knowledge in the face of powerful skeptical arguments. No more than four years after his death, Pascal and Fermat inaugurated the.. (shrink)

The tension between rule and judgment is well known with respect to the meaning of substantive legal commands. The same conflict is present in fact finding. The law penetrates to virtually all aspects of human affairs; irtually any interaction can generate a legal conflict. Accurate fact finding about such disputes is a necessary condition for the appropriate application of substantive legal commands. Without accuracy in fact finding, the law is unpredictable, and thus individuals cannot efficiently accommodate their affairs to its (...) commands. The need for accuracy and predictability in legal fact finding has generated a search for formal tools to apply to the task. Among the tools that have been examined are Bayes' Theorem and expected utility theory (Bayesian or statistical decision theory). These tools do not map well onto trials, which in turn has generated an examination of alternative approaches, in particular the story model and the relative plausibility theory. This paper discusses these issues in turn. It elaborates the basic structure of trials in the American tradition; examines the uneasy relationship between trials and such formalisms as Bayes' Theorem and expected utility theory; and introduces the relative plausibility theory as an explanation of the nature of juridical proof. (shrink)

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The notion of probability occupies a central role in contemporary epistemology and cognitive science. Nevertheless, the classical notion of probability is hard to reconcile with the central notions postulated by the epistemological tradition.

We pose and resolve several vexing decision theoretic puzzles. Some are variants of existing puzzles, such as ‘Trumped’ (Arntzenius and McCarthy 1997), ‘Rouble trouble’ (Arntzenius and Barrett 1999), ‘The airtight Dutch book’ (McGee 1999), and ‘The two envelopes puzzle’ (Broome 1999). Others are new. A unified resolution of the puzzles shows that Dutch book arguments have no force in infinite cases. It thereby provides evidence that reasonable utility functions may be unbounded and that reasonable credence functions need not be countably (...) additive. The resolution also shows that when infinitely many decisions are involved, the difference between making the decisions simultaneously and making them sequentially can be the difference between riches and ruin. Finally, the resolution reveals a new way in which the ability to make binding commitments can save perfectly rational agents from sure losses. (shrink)

Reichenbach’s use of ‘posits’ to defend his frequentistic theory of probability has been criticized on the grounds that it makes unfalsifiable predictions. The justice of this criticism has blinded many to Reichenbach’s second use of a posit, one that can fruitfully be applied to current debates within epistemology. We show first that Reichenbach’s alternative type of posit creates a difficulty for epistemic foundationalists, and then that its use is equivalent to a particular kind of Jeffrey conditionalization. We conclude that, under (...) particular circumstances, Reichenbach’s approach and that of the Bayesians amount to the same thing, thereby presenting us with a new instance in which chance and credence coincide. (shrink)

The objective theory of probability of Richard von Mises has been criticized by Crovelli (2009), who defends a subjective approach. This paper attempts to clarify the different meanings of ‘objective’ and ‘subjective’ when applied to probability, and then argues for an objective Bayesian theory of probability, as exemplified in the writings [...].

We investigate the discrete (finite) case of the Popper–Renyi theory of conditional probability, introducing discrete conditional probabilistic models for knowledge and conditional belief, and comparing them with the more standard plausibility models. We also consider a related notion, that of safe belief, which is a weak (non-negatively introspective) type of “knowledge”. We develop a probabilistic version of this concept (“degree of safety”) and we analyze its role in games. We completely axiomatize the logic of conditional belief, knowledge and safe belief (...) over conditional probabilistic models. We develop a theory of probabilistic dynamic belief revision, introducing probabilistic “action models” and proposing a notion of probabilistic update product, that comes together with appropriate reduction laws. (shrink)

The received view of an ad hochypothesis is that it accounts for only the observation(s) it was designed to account for, and so non-ad hocness is generally held to be necessary or important for an introduced hypothesis or modification to a theory. Attempts by Popper and several others to convincingly explicate this view, however, prove to be unsuccessful or of doubtful value, and familiar and firmer criteria for evaluating the hypotheses or modified theories so classified are characteristically available. These points (...) are obscured largely because the received view fails to adequately separate psychology from methodology or to recognise ambiguities in the use of ''ad hoc''. (shrink)

In the curve fitting problem two conflicting desiderata, simplicity and goodness-of-fit, pull in opposite directions. To this problem, we propose a solution that strikes a balance between simplicity and goodness-of-fit. Using Bayes' theorem we argue that the notion of prior probability represents a measurement of simplicity of a theory, whereas the notion of likelihood represents the theory's goodness-of-fit. We justify the use of prior probability and show how to calculate the likelihood of a family of curves. We diagnose the relationship (...) between simplicity of a theory and its predictive accuracy. (shrink)

In the curve fitting problem two conflicting desiderata, simplicity and goodness-of-fit pull in opposite directions. To solve this problem, two proposals, the first one based on Bayes's theorem criterion (BTC) and the second one advocated by Forster and Sober based on Akaike's Information Criterion (AIC) are discussed. We show that AIC, which is frequentist in spirit, is logically equivalent to BTC, provided that a suitable choice of priors is made. We evaluate the charges against Bayesianism and contend that AIC approach (...) has shortcomings. We also discuss the relationship between Schwarz's Bayesian Information Criterion and BTC. (shrink)

This paper aims to make explicit the methodological conditions that should be satisfied for the Bayesian model to be used as a normative model of human probability judgment. After noticing the lack of a clear definition of Bayesianism in the psychological literature and the lack of justification for using it, a classic definition of subjective Bayesianism is recalled, based on the following three criteria: an epistemic criterion, a static coherence criterion and a dynamic coherence criterion. Then it is shown that (...) the adoption of this framework has two kinds of implications. The first one regards the methodology of the experimental study of probability judgment. The Bayesian framework creates pragmatic constraints on the methodology that are linked to the interpretation of, and the belief in, the information presented, or referred to, by an experimenter in order for it to be the basis of a probability judgment by individual participants. It is shown that these constraints have not been satisfied in the past, and the question of whether they can be satisfied in principle is raised and answered negatively. The second kind of implications consists of two limitations in the scope of the Bayesian model. They regard (1) the background of revision (the Bayesian model considers only revising situations but not updating situations), and (2) the notorious case of the null priors. In both cases Lewis’ rule is an appropriate alternative to Bayes’ rule, but its use faces the same operational difficulties. (shrink)

In order for Bayesian inquiry to count as objective, one might argue that it must lead to a consensus among those who use it and share evidence, but presumably this is not enough. It has been proposed that one should also require that the consensus be reached from very different initial opinions by conditioning only on basic experimental evidence, evidence free from subjective, social, or psychological influence. I will argue here, however, that this notion of objectivity in Bayesian inquiry is (...) too narrow. (shrink)

Carter and Leslie (1996) have argued, using Bayes's theorem, that our being alive now supports the hypothesis of an early 'Doomsday'. Unlike some critics (Eckhardt 1997), we accept their argument in part: given that we exist, our existence now indeed favors 'Doom sooner' over 'Doom later'. The very fact of our existence, however, favors 'Doom later'. In simple cases, a hypothetical approach to the problem of 'old evidence' shows that these two effects cancel out: our existence now yields no information (...) about the coming of Doom. More complex cases suggest a move from countably additive to non-standard probability measures. (shrink)

How can an incomplete and uncertain model of the environment be used to perceive, infer, decide and act efficiently? This is the challenge that both living and artificial cognitive systems have to face. Symbolic logic is, by its nature, unable to deal with this question. The subjectivist approach to probability is an extension to logic that is designed specifically to face this challenge. In this paper, we review a number of frequently encountered cognitive issues and cast them into a common (...) Bayesian formalism. The concepts we review are ambiguities, fusion, multimodality, conflicts, modularity, hierarchies and loops. First, each of these concepts is introduced briefly using some examples from the neuroscience, psychophysics or robotics literature. Then, the concept is formalized using a template Bayesian model. The assumptions and common features of these models, as well as their major differences, are outlined and discussed. (shrink)

This paper shows how complex argumentation, analyzed as dialectical structures, can be evaluated within a Bayesian framework by interpreting them as coherence constraints on subjective degrees of belief. A dialectical structure is a set of arguments (premiss-conclusion structure) among which support- and attack-relations hold. This approach addresses the observation that some theses in a debate can be better justified than others and thus fixes a shortcoming of a theory of defeasible reasoning which applies the bivalence principle to argument evaluations by (...) assigning them the status of being either defeated or undefeated. Evaluation procedures which are based on the principle of bivalence can, however, be embedded as a special case within the Bayesian framework. The approach developed in this paper rests on the assumptions that arguments can be reconstructed as deductively valid and that complex argumentation can be reconstructed such that premisses of arguments with equivalent conclusions are pairwise independent. (shrink)

The standard Bayesian recipe for selecting the rational choice is presented. A familiar example in which the recipe fails to produce any definite result is introduced. It is argued that a generalization of Gärdenfors’ and Sahlin’s theory of unreliable probabilities — which itself does not guarantee a solution to the problem — offers the best available approach. But a number of challenges to this approach are also presented and discussed.

Probabilistic models have much to offer to philosophy. We continually receive information from a variety of sources: from our senses, from witnesses, from scientific instruments. When considering whether we should believe this information, we assess whether the sources are independent, how reliable they are, and how plausible and coherent the information is. Bovens and Hartmann provide a systematic Bayesian account of these features of reasoning. Simple Bayesian Networks allow us to model alternative assumptions about the nature of the information sources. (...) Measurement of the coherence of information is a controversial matter: arguably, the more coherent a set of information is, the more confident we may be that its content is true, other things being equal. The authors offer a new treatment of coherence which respects this claim and shows its relevance to scientific theory choice. Bovens and Hartmann apply this methodology to a wide range of much discussed issues regarding evidence, testimony, scientific theories, and voting. Bayesian Epistemology is an essential tool for anyone working on probabilistic methods in philosophy, and has broad implications for many other disciplines. (shrink)

Bovens and Hartmann provide a systematic guide to the use of probabilistic methods not just in epistemology, but also in philosophy of science, voting theory, ...

We appeal to the theory of Bayesian Networks to model different strategies for obtaining confirmation for a hypothesis from experimental test results provided by less than fully reliable instruments. In particular, we consider (i) repeated measurements of a single test consequence of the hypothesis, (ii) measurements of multiple test consequences of the hypothesis, (iii) theoretical support for the reliability of the instrument, and (iv) calibration procedures. We evaluate these strategies on their relative merits under idealized conditions and show some surprising (...) repercussions on the variety-of-evidence thesis and the Duhem-Quine thesis. (shrink)

We construct a probabilistic coherence measure for information sets which determines a partial coherence ordering. This measure is applied in constructing a criterion for expanding our beliefs in the face of new information. A number of idealizations are being made which can be relaxed by an appeal to Bayesian Networks.

The coherentist theory of justification provides a response to the sceptical challenge: even though the independent processes by which we gather information about the world may be of dubious quality, the internal coherence of the information provides the justification for our empirical beliefs. This central canon of the coherence theory of justification is tested within the framework of Bayesian networks, which is a theory of probabilistic reasoning in artificial intelligence. We interpret the independence of the information gathering processes (IGPs) in (...) terms of conditional independences, construct a minimal sufficient condition for a coherence ranking of information sets and assess whether the confidence boost that results from receiving information through independent IGPs is indeed a positive function of the coherence of the information set. There are multiple interpretations of what constitute IGPs of dubious quality. Do we know our IGPs to be no better than randomization processes? Or, do we know them to be better than randomization processes but not quite fully reliable, and if so, what is the nature of this lack of full reliability? Or, do we not know whether they are fully reliable or not? Within the latter interpretation, does learning something about the quality of some IGPs teach us anything about the quality of the other IGPs? The Bayesian-network models demonstrate that the success of the coherentist canon is contingent on what interpretation one endorses of the claim that our IGPs are of dubious quality. (shrink)

The Puzzle of the Hats is a betting arrangement which seems to show that a Dutch book can be made against a group of rational players with common priors who act in the common interest and have full trust in the other players’ rationality. But we show that appearances are misleading—no such Dutch book can be made. There are four morals. First, what can be learned from the puzzle is that there is a class of situations in which credences and (...) betting rates diverge. Second, there is an analogy between ways of dealing with situations of this kind and different policies for sequential choice. Third, there is an analogy with strategic voting, showing that the common interest is not always served by expressing how things seem to you in social decision-making. And fourth, our analysis of the Puzzle of the Hats casts light on a recent controversy about the Dutch book argument for the Sleeping Beauty. (shrink)

Abstract Jonathan Weisberg (Analysis, 70(3), pp. 431–438, 2010 ) argues that, given that life exists, the fact that the universe is fine-tuned for life does not confirm the design hypothesis. And if the fact that life exists confirms the design hypothesis, fine-tuning is irrelevant. So either way, fine-tuning has nothing to do with it. I will defend a design argument that survives Weisberg’s critique—the fact that life exists supports the design hypothesis, but it only does so given fine-tuning. Content Type (...) Journal Article Category Original Article Pages 1-4 DOI 10.1007/s10670-011-9322-y Authors Darren Bradley, Philosophy Department, 160 Convent Ave, New York, NY 10031, USA Journal Erkenntnis Online ISSN 1572-8420 Print ISSN 0165-0106. (shrink)

The fine-tuning argument can be used to support the Many Universe hypothesis. The Inverse Gambler’s Fallacy objection seeks to undercut the support for the Many Universe hypothesis. The objection is that although the evidence that there is life somewhere confirms Many Universes, the specific evidence that there is life in this universe does not. I will argue that the Inverse Gambler’s Fallacy is not committed by the fine-tuning argument. The key issue is the procedure by which the universe with life (...) is selected for observation. Once we take account of the procedure, we find that the support for the Many Universe hypothesis remains. (shrink)

The fine-tuning argument can be used to support the Many Universe hypothesis. The Inverse Gambler’s Fallacy objection seeks to undercut the support for the Many Universe hypothesis. The objection is that although the evidence that there is life somewhere confirms Many Universes, the specific evidence that there is life in this universe does not. I will argue that the Inverse Gambler’s Fallacy is not committed by the fine-tuning argument. The key issue is the procedure by which the universe with life (...) is selected for observation. Once we take account of the procedure, we find that the support for the Many Universe hypothesis remains. (shrink)

This paper considers the Bayesian form of the fine-tuning argument as advanced by Richard Swinburne. An expository section aims to identify the precise character of the argument, and three lines of objection are then advanced. The first of these holds that there is an inconsistency in Swinburne's procedure, the second that his argument has an unacceptable dependence on an objectivist theory of value, the third that his method is powerless to single out traditional theism from a vast number of (...) competitors. In the final section of the paper the fine-tuning argument is considered, not now as self-standing, but as one of a number of theistic arguments taken together and applied in the manner of the final chapter of Swinburne's The Existence of God. It is argued that points already made also block the way for this line of thought. (shrink)

Philosophers have long sought to explain perceptual constancy—the fact that objects appear to remain the same color, size and shape despite changes in the illumination condition, perspective and the relative distance—in terms of a mechanism that actively categorizes variable stimuli under the same pre-formed conceptual categories. Contemporary representationalists, on the other hand, explain perceptual constancy in terms of a modular mechanism that automatically discounts variation in the visual field to represent the stable properties of objects. In this paper I argue (...) that while the former view is unmotivated by empirical evidence, the later fails to account for inter- and intra-personal variability, the influence of expectations on constancy, and the systematic and normal failures of color constancy. A Bayesian approach that builds on the representational tradition in psychology solves both problems. (shrink)

This paper provides new foundations for Bayesian Decision Theory based on a representation theorem for preferences defined on a set of prospects containing both factual and conditional possibilities. This use of a rich set of prospects not only provides a framework within which the main theoretical claims of Savage, Ramsey, Jeffrey and others can be stated and compared, but also allows for the postulation of an extended Bayesian model of rational belief and desire from which they can be derived as (...) special cases. The main theorem of the paper establishes the existence of a such a Bayesian representation of preferences over conditional prospects, i.e. the existence of a pair of real-valued functions respectively measuring the agent’s degrees of belief and desire and which satisfy the postulated rationality conditions on partial belief and desire. The representation of partial belief is shown to be unique and that of partial desire, unique up to a linear transformation. (shrink)

Foundations of Bayesianism is an authoritative collection of papers addressing the key challenges that face the Bayesian interpretation of probability today. Some of these papers seek to clarify the relationships between Bayesian, causal and logical reasoning. Others consider the application of Bayesianism to artificial intelligence, decision theory, statistics and the philosophy of science and mathematics. The volume includes important criticisms of Bayesian reasoning and also gives an insight into some of the points of disagreement amongst advocates of the Bayesian approach. (...) The upshot is a plethora of new problems and directions for Bayesians to pursue.The book will be of interest to graduate students or researchers who wish to learn more about Bayesianism than can be provided by introductory textbooks to the subject. Those involved with the applications of Bayesian reasoning will find essential discussion on the validity of Bayesianism and its limits, while philosophers and others interested in pure reasoning will find new ideas on normativity and the logic of belief. (shrink)

This paper argues that when used judiciously Bayes's law has a role to play in the evaluation of scientific hypotheses. Several examples are presented in which a rational response to evidence requires a judgement whether to apply Bayes's law or whether, for example, to redistribute prior probabilities. The paper concludes that reflection on Bayes's law illustrates how an adequate account of the rational evaluation of hypotheses requires an account of judgement--a point which several philosophers have noted despite few attempts to (...) develop an adequate theory of judgement. (shrink)

Crispin Wright’s discussion of the notion of ‘transmission-failure’ promises to have important philosophical ramifications, both in epistemology and beyond. This paper offers a precise, formal characterisation of the concept within a Bayesian framework. The interpretation given avoids the serious shortcomings of a recent alternative proposal due to Samir Okasha.

The rational analysis method, first proposed by John R. Anderson, has been enormously influential in helping us understand high-level cognitive processes. -/- 'The Probabilistic Mind' is a follow-up to the influential and highly cited 'Rational Models of Cognition' (OUP, 1998). It brings together developments in understanding how, and how far, high-level cognitive processes can be understood in rational terms, and particularly using probabilistic Bayesian methods. It synthesizes and evaluates the progress in the past decade, taking into account developments in Bayesian (...) statistics, statistical analysis of the cognitive 'environment' and a variety of theoretical and experimental lines of research. The scope of the book is broad, covering important recent work in reasoning, decision making, categorization, and memory. Including chapters from many of the leading figures in this field, -/- 'The Probabilistic Mind' will be valuable for psychologists and philosophers interested in cognition. (shrink)

Kuhn’s theory of paradigm reveals a pattern of scientific progress, in which normal science alternates with scientific revolution. But Kuhn underrated too much the function of scientific test in his pattern, because he focuses all his attention on the hypothetico-deductive schema instead of Bayesian schema. This paper employs Bayesian schema to re-examine Kuhn’s theory of paradigm, to uncover its logical and rational components, and to illustrate the tensional structure of logic and belief, rationality and irrationality, in the process of (...) scientific revolution. (shrink)

Both Representation Theorem Arguments and Dutch Book Arguments support taking probabilistic coherence as an epistemic norm. Both depend on connecting beliefs to preferences, which are not clearly within the epistemic domain. Moreover, these connections are standardly grounded in questionable definitional/metaphysical claims. The paper argues that these definitional/metaphysical claims are insupportable. It offers a way of reconceiving Representation Theorem arguments which avoids the untenable premises. It then develops a parallel approach to Dutch Book Arguments, and compares the results. In each case (...) preferencedefects serve as a diagnostic tool, indicating purely epistemic defects. (shrink)

Much contemporary epistemology is informed by a kind of confirmational holism, and a consequent rejection of the assumption that all confirmation rests on experiential certainties. Another prominent theme is that belief comes in degrees, and that rationality requires apportioning one's degrees of belief reasonably. Bayesian confirmation models based on Jeffrey Conditionalization attempt to bring together these two appealing strands. I argue, however, that these models cannot account for a certain aspect of confirmation that would be accounted for in any adequate (...) holistic confirmation theory. I then survey the prospects for constructing a formal epistemology that better accommodates holistic insights. (shrink)

Because physicians use scientific inference for the generalizations of individual observations and the application of general knowledge to particular situations, the Bayesian probability solution to the problem of induction has been proposed and frequently utilized. Several problems with the Bayesian approach are introduced and discussed. These include: subjectivity, the favoring of a weak hypothesis, the problem of the false hypothesis, the old evidence/new theory problem and the observation that physicians are not currently Bayesians. To the complaint that the prior probability (...) is subjective, Bayesians reply that there will be ultimate convergence, but the rebuttal to this is that there will not be uniform convergence. Secondly, since the Bayesian scheme favors a weak hypothesis, theories turn out to be a gratuitous risk. The problem with the false hypothesis comes out in the denominator of the theorem, revealing that a factor which is not a theory at all is being considered in the reasoning. On the old evidence/new theory problem old evidence cannot confirm a new theory so that the posterior probability will equal the prior probability. Finally, empiric studies have shown that current physicians are not Bayesians. But on consideration of Bayesian inference as a system of inference, it can be reasoned that physicians should be Bayesians. However, the problem of physicians' and patients' own subjectivity continue to plague this system of medical decision making. (shrink)

Bayesian paradigm has been widely acknowledged as a coherent approach to learning putative probability model structures from a finite class of candidate models. Bayesian learning is based on measuring the predictive ability of a model in terms of the corresponding marginal data distribution, which equals the expectation of the likelihood with respect to a prior distribution for model parameters. The main controversy related to this learning method stems from the necessity of specifying proper prior distributions for all unknown parameters of (...) a model, which ensures a complete determination of the marginal data distribution. Even for commonly used models, subjective priors may be difficult to specify precisely, and therefore, several automated learning procedures have been suggested in the literature. Here we introduce a novel Bayesian learning method based on the predictive entropy of a probability model, that can combine both subjective and objective probabilistic assessment of uncertain quantities in putative models. It is shown that our approach can avoid some of the limitations of the earlier suggested objective Bayesian methods. (shrink)

The volume includes important criticisms of Bayesian reasoning and also gives an insight into some of the points of disagreement amongst advocates of the ...

Bayesian epistemology postulates a probabilistic analysis of many sorts of ordinary and scientific reasoning. Huber ([2005]) has provided a novel criticism of Bayesianism, whose core argument involves a challenging issue: confirmation by uncertain evidence. In this paper, we argue that under a properly defined Bayesian account of confirmation by uncertain evidence, Huber's criticism fails. By contrast, our discussion will highlight what we take as some new and appealing features of Bayesian confirmation theory. Introduction Uncertain Evidence and Bayesian Confirmation Bayesian Confirmation (...) by Uncertain Evidence: Test Cases and Basic Principles CiteULike Connotea Del.icio.us What's this? (shrink)

This essay addresses the methodology of philosophy of science and illustrates how formal and empirical methods can be fruitfully combined. Special emphasis is given to the application of experimental methods to confirmation theory and to recent work on the conjunction fallacy, a key topic in the rationality debate arising from research in cognitive psychology. Several other issue can be studied in this way. In the concluding section, a brief outline is provided of three further examples.

Epistemologists and philosophers of science have often attempted to express formally the impact of a piece of evidence on the credibility of a hypothesis. In this paper we will focus on the Bayesian approach to evidential support. We will propose a new formal treatment of the notion of degree of confirmation and we will argue that it overcomes some limitations of the currently available approaches on two grounds: (i) a theoretical analysis of the confirmation relation seen as an extension of (...) logical deduction and (ii) an empirical comparison of competing measures in an experimental inquiry concerning inductive reasoning in a probabilistic setting. (shrink)

In everyday matters, as well as in law, we allow that someone’s reasons can be causes of her actions, and often are. That correct reasoning accords with Bayesian principles is now so widely held in philosophy, psychology, computer science and elsewhere that the contrary is beginning to seem obtuse, or at best quaint. And that rational agents should learn about the world from energies striking sensory inputs nerves in people—seems beyond question. Even rats seem to recognize the difference between correlation (...) and causation,1 and accordingly make different inferences from passive observation than from interventions. A few statisticians aside,” so do most of us. To square these views with the demands of computability, increasing numbers of psychologists and others have embraced a particular formalization, causal Bayes nets, as an account of human reasoning about and to causal connections.111 Such structures can be used by rational agents, including humans in so far as they are rational, to have degrees of belief in various conceptual contents, which they use to reason to expectations, which are realized or defeated by sensory inputs, which cause them to change their degrees of belief in other contents in accord with Bayes Rule, or some generalization of it. How is all of this supposed to be carried out? l. Representing Causal Structures The causal Bayes net framework adopted by a growing number of psychologists goes like this: Our representations of causal relations are captured in a graphical causal. (shrink)

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 units of each of the original conditioning variables. (shrink)

Behind this question are two fundamentally different approaches about how to reason with chance hypotheses. One approach, due to Ronald Fisher, rejects a chance hypothesis provided sample data appear in a prespecified rejection region. The other, due to Thomas Bayes, rejects a chance hypothesis provided an alternative hypothesis confers a bigger probability on the data in question than the original hypothesis. In the Fisherian approach, chance hypotheses are rejected in isolation for rendering data too improbable. In the Bayesian approach, chance (...) hypotheses are eliminated provided some other hypotheses render the data more probable. Whereas in the Fisherian approach the emphasis is on elimination, in the Bayesian approach the emphasis is on comparison. These approaches are incompatible, and the statistical community itself is deeply riven over which of these approaches to adopt as the right canon for statistical rationality. The difference reflects a deep divergence in fundamental intuitions about the nature of statistical rationality and in particular about what counts as statistical evidence. (shrink)

From the outside, Bayesian statistics may seem like a closed little corner of probability. Once a prior is specified you compute! From the inside the field is filled with problems, conceptual and otherwise. This paper surveys some of what remains to be done and gives examples of the work in progress via a Bayesian peek into Feller volume I.

If a group is modelled as a single Bayesian agent, what should its beliefs be? I propose an axiomatic model that connects group beliefs to beliefs of group members, who are themselves modelled as Bayesian agents, possibly with di¤erent priors and di¤erent information. Group beliefs are proven to take a simple multiplicative form if people’s information is independent, and a more complex form if information overlaps arbitrarily. This shows that group beliefs can incorporate all information spread over the individuals without (...) the individuals having to communicate their (possibly complex and hard-to-describe) private information; communicating prior and posterior beliefs su¢ ces. JEL classi…cation: D70, D71.. (shrink)

In this paper, we identify a new and mathematically well-defined sense in which the coherence of a set of hypotheses can be truth-conducive. Our focus is not, as usually, on the probability but on the confirmation of a coherent set and its members. We show that, if evidence confirms a hypothesis, confirmation is "transmitted" to any hypotheses that are sufficiently coherent with the former hypothesis, according to some appropriate probabilistic coherence measure such as Olsson’s or Fitelson’s measure. Our findings have (...) implications for scientific methodology, as they provide a formal rationale for the method of indirect confirmation and the method of confirming theories by confirming their parts. (shrink)

Various scientific theories stand in a reductive relation to each other. In a recent article, we have argued that a generalized version of the Nagel-Schaffner model (GNS) is the right account of this relation. In this article, we present a Bayesian analysis of how GNS impacts on confirmation. We formalize the relation between the reducing and the reduced theory before and after the reduction using Bayesian networks, and thereby show that, post-reduction, the two theories are confirmatory of each other. We (...) then ask when a purported reduction should be accepted on epistemic grounds. To do so, we compare the prior and posterior probabilities of the conjunction of both theories before and after the reduction and ask how well each is confirmed by the available evidence. (shrink)

Bayesian psychology, in what is perhaps its most familiar version, is incomplete: Jeffrey conditionalization is not a complete account of rational belief change. Jeffrey conditionalization is sensitive to the order in which the evidence arrives. This order effect can be so pronounced as to call for a belief adjustment that cannot be understood as an assimilation of incoming evidence by Jeffrey's rule. Hartry Field's reparameterization of Jeffrey's rule avoids the order effect but fails as an account of how new evidence (...) should be assimilated. (shrink)

It is widely believed that the so-called knowledge account of assertion best explains why sentences such as “It’s raining in Paris but I don’t believe it” and “It’s raining in Paris but I don’t know it” appear odd to us. I argue that the rival rational credibility account of assertion explains that fact just as well. I do so by providing a broadly Bayesian analysis of the said type of sentences which shows that such sentences cannot express rationally held beliefs. (...) As an interesting aside, it will be seen that these sentences also harbor a lesson for Bayesian epistemology itself. (shrink)

The Lottery Paradox and the Preface Paradox both involve the thesis that high probability is sufficient for rational acceptability. The standard solution to these paradoxes denies that rational acceptability is deductively closed. This solution has a number of untoward consequences. The present paper suggests that a better solution to the paradoxes is to replace the thesis that high probability suffices for rational acceptability with a somewhat stricter thesis. This avoids the untoward consequences of the standard solution. The new solution will (...) be defended against a seemingly obvious objection. 1 The paradoxes of rational acceptability 2 The standard solution 3 A new solution to the paradoxes 4 Basic assumptions 5 The new solution defended 6 Conclusion 7 Appendix. (shrink)

This paper is concerned with formal solutions to the lottery paradox on which high probability defeasibly warrants acceptance. It considers some recently proposed solutions of this type and presents an argument showing that these solutions are trivial in that they boil down to the claim that perfect probability is sufficient for rational acceptability. The argument is then generalized, showing that a broad class of similar solutions faces the same problem. An argument against some formal solutions to the lottery paradox The (...) argument generalized Some variations Adding modalities Anticipated objections. (shrink)

“Evidence” in the form of data collected and analysis thereof is fundamental to medicine, health and science. In this paper, we discuss the “evidence-based” aspect of evidence-based medicine in terms of statistical inference, acknowledging that this latter field of statistical inference often also goes by various near-synonymous names—such as inductive inference (amongst philosophers), econometrics (amongst economists), machine learning (amongst computer scientists) and, in more recent times, data mining (in some circles). Three central issues to this discussion of “evidence-based” are (i) (...) whether or not the statistical analysis can and/or should be objective and/or whether or not (subjective) prior knowledge can and/or should be incorporated, (ii) whether or not the analysis should be invariant to the framing of the problem (e.g. does it matter whether we analyse the ratio of proportions of morbidity to non-morbidity rather than simply the proportion of morbidity?), and (iii) whether or not, as we get more and more data, our analysis should be able to converge arbitrarily closely to the process which is generating our observed data. For many problems of data analysis, it would appear that desiderata (ii) and (iii) above require us to invoke at least some form of subjective (Bayesian) prior knowledge. This sits uncomfortably with the understandable but perhaps impossible desire of many medical publications that at least all the statistical hypothesis testing has to be classical non-Bayesian—i.e. it is not permitted to use any (subjective) prior knowledge. (shrink)

The advent of formal definitions of the simplicity of a theory has important implications for model selection. But what is the best way to define simplicity? Forster and Sober ([1994]) advocate the use of Akaike's Information Criterion (AIC), a non-Bayesian formalisation of the notion of simplicity. This forms an important part of their wider attack on Bayesianism in the philosophy of science. We defend a Bayesian alternative: the simplicity of a theory is to be characterised in terms of Wallace's Minimum (...) Message Length (MML). We show that AIC is inadequate for many statistical problems where MML performs well. Whereas MML is always defined, AIC can be undefined. Whereas MML is not known ever to be statistically inconsistent, AIC can be. Even when defined and consistent, AIC performs worse than MML on small sample sizes. MML is statistically invariant under 1-to-1 re-parametrisation, thus avoiding a common criticism of Bayesian approaches. We also show that MML provides answers to many of Forster's objections to Bayesianism. Hence an important part of the attack on Bayesianism fails. (shrink)

We criticise Shepard's notions of “invariance” and “universality,” and the incorporation of Shepard's work on inference into the general framework of his paper. We then criticise Tenenbaum and Griffiths' account of Shepard (1987b), including the attributed likelihood function, and the assumption of “weak sampling.” Finally, we endorse Barlow's suggestion that minimum message length (MML) theory has useful things to say about the Bayesian inference problems discussed by Shepard and Tenenbaum and Griffiths. [Barlow; Shepard; Tenenbaum & Griffiths].

The concept of randomness has been unjustly neglected in recent philosophical literature, and when philosophers have thought about it, they have usually acquiesced in views about the concept that are fundamentally flawed. After indicating the ways in which these accounts are flawed, I propose that randomness is to be understood as a special case of the epistemic concept of the unpredictability of a process. This proposal arguably captures the intuitive desiderata for the concept of randomness; at least it should suggest (...) that the commonly accepted accounts cannot be the whole story and more philosophical attention needs to be paid. (shrink)

Bayesian models of human learning are becoming increasingly popular in cognitive science. We argue that their purported confirmation largely relies on a methodology that depends on premises that are inconsistent with the claim that people are Bayesian about learning and inference. Bayesian models in cognitive science derive their appeal from their normative claim that the modeled inference is in some sense rational. Standard accounts of the rationality of Bayesian inference imply predictions that an agent selects the option that maximizes the (...) posterior expected utility. Experimental confirmation of the models, however, has been claimed because of groups of agents that probability match the posterior. Probability matching only constitutes support for the Bayesian claim if additional unobvious and untested (but testable) assumptions are invoked. The alternative strategy of weakening the underlying notion of rationality no longer distinguishes the Bayesian model uniquely. A new account of rationality—either for inference or for decision-making—is required to successfully confirm Bayesian models in cognitive science. (shrink)

To clarify and illuminate the place of probability in science Ellery Eells and James H. Fetzer have brought together some of the most distinguished philosophers ...

Bayesian epistemology suggests various ways of measuring the support that a piece of evidence provides a hypothesis. Such measures are defined in terms of a subjective probability assignment, pr, over propositions entertained by an agent. The most standard measure (where “H” stands for “hypothesis” and “E” stands for “evidence”) is: the difference measure: d(H,E) = pr(H/E) - pr(H).0 This may be called a “positive (probabilistic) relevance measure” of confirmation, since, according to it, a piece of evidence E qualitatively confirms a (...) hypothesis H if and only if pr(H/E) > pr(H), where qualitative disconfirmation is characterized by replacing “>” with “ “ with “=”. Other more or less standard positive relevance measures that have been proposed are: the log-ratio measure: r(H,E) = log[pr(H/E)/pr(H)] and the log-likelihood-ratio measure: l(H,E) = log[pr(E/H)/pr(E/~H)]. (shrink)

Many philosophers (myself included) have been converted to causal decision theory by something like the following line of argument: Evidential decision theory endorses irrational courses of action in a range of examples, and endorses “an irrational policy of managing the news”. These are fatal problems for evidential decision theory. Causal decision theory delivers the right results in the troublesome examples, and does not endorse this kind of irrational news-managing. So we should give up evidential decision theory, and be causal decision (...) theorists instead. Unfortunately, causal decision theory has its own family of problematic examples for which it endorses irrational courses of action, and its own irrational policy that it is committed to endorsing. These are, I think, fatal problems for causal decision theory. I wish that I had another theory to offer in its place. (shrink)

Wesley Salmon and John Earman have presented influential Bayesian reconstructions of Thomas Kuhn’s account of theory-change. In this paper I argue that all attempts to give a Bayesian reading of Kuhn’s philosophy of science are fundamentally misguided due to the fact that Bayesian confirmation theory is in fact inconsistent with Kuhn’s account. The reasons for this inconsistency are traced to the role the concept of incommensurability plays with reference to the ‘observational vocabulary’ within Kuhn’s picture of scientific theories. The upshot (...) of the discussion is that it is impossible to integrate both Kuhn’s claims and Bayesianism within a coherent account of theory-change. (shrink)

Three experiments investigated the effect of rarity on people's selection and interpretation of data in a variant of the pseudodiagnosticity task. For familiar (Experiment 1) but not for arbitrary (Experiment 3) materials, participants were more likely to select evidence so as to complete a likelihood ratio when the initial evidence they received was a single likelihood concerning a rare feature. This rarity effect with familiar materials was replicated in Experiment 2 where it was shown that participants were relatively insensitive to (...) explicit manipulations of the likely diagnosticity of rare evidence. In contrast to the effects for data selection, there was an effect of rarity on confidence ratings after receipt of a single likelihood for arbitrary but not for familiar materials. It is suggested that selecting diagnostic evidence necessitates explicit consideration of the alternative hypothesis and that consideration of the possible consequences of the evidence for the alternative weakens the rarity effect in confidence ratings. Paradoxically, although rarity effects in evidence selection and confidence ratings are in the spirit of Bayesian reasoning, the effect on confidence ratings appears to rely on participants thinking less about the alternative hypothesis. (shrink)

Finland is internationally known as one of the leading centers of twentieth century analytic philosophy. This volume offers for the first time an overall survey of the Finnish analytic school. The rise of this trend is illustrated by original articles of Edward Westermarck, Eino Kaila, Georg Henrik von Wright, and Jaakko Hintikka. Contributions of Finnish philosophers are then systematically discussed in the fields of logic, philosophy of language, philosophy of science, history of philosophy, ethics and social philosophy. Metaphilosophical reflections on (...) the nature of philosophy are highlighted by the Finnish dialogue between analytic philosophy, phenomenology, pragmatism, and critical theory. (shrink)

According to the Bayesian view, scientific hypotheses must be appraised in terms of their posterior probabilities relative to the available experimental data. Such posterior probabilities are derived from the prior probabilities of the hypotheses by applying Bayes'theorem. One of the most important problems arising within the Bayesian approach to scientific methodology is the choice of prior probabilities. Here this problem is considered in detail w.r.t. two applications of the Bayesian approach: (1) the theory of inductive probabilities (TIP) developed by Rudolf (...) Carnap and other epistomologists and (2) the analysis of the multinational inferences provided by Bayesian statstics (BS). ... Zie: Summary. (shrink)

This is a high quality, concise collection of articles on the foundations of probability and statistics. Its editor, Richard Swinburne, has collected five papers by contemporary leaders in the field, written a pretty thorough and even-handed introductory essay, and placed a very clean and accessible version of Reverend Thomas Bayes’s famous essay (“An Essay Towards the Solving a Problem in the Doctrine of Chances”) at the end, as an Appendix (with a brief historical introduction by the noted statistician G.A. Barnard). (...) I will briefly discuss each of the five papers in the volume, with an emphasis on certain issues arising from the use of probability as a tool for thinking about evidence. (shrink)

• Several recent Bayesian discussions make use of “approximation” – Earman on the Quantitative Old Evidence Problem – Vranas on Quantitative Approaches to the Ravens Paradox – Dorling’s Quantitative Approach to Duhem–Quine – Strevens’s Quantitative Approach to Duhem–Quine – rThere are also examples not involving confirmation: E.g.

In Bayes or Bust? John Earman quickly dismisses a possible resolution (or avoidance) of the problem of old evidence. In this note, I argue that his dismissal is premature, and that the proposed resolution (when charitably reconstructed) is reasonable.

The Paradox of the Ravens (a.k.a,, The Paradox of Confirmation) is indeed an old chestnut. A great many things have been written and said about this paradox and its implications for the logic of evidential support. The first part of this paper will provide a brief survey of the early history of the paradox. This will include the original formulation of the paradox and the early responses of Hempel, Goodman, and Quine. The second part of the paper will describe attempts (...) to resolve the paradox within a Bayesian framework, and show how to improve upon them. This part begins with a discussion of how probabilistic methods can help to clarify the statement of the paradox itself. And it describes some of the early responses to probabilistic explications. We then inspect the assumptions employed by traditional (canonical) Bayesian approaches to the paradox. These assumptions may appear to be overly strong. So, drawing on weaker assumptions, we formulate a new-and-improved Bayesian confirmation-theoretic resolution of the Paradox of the Ravens. (shrink)

outlined. This account is partly inspired by the work of C.S. Peirce. When we want to consider how degree of confirmation varies with changing I show that a large class of quantitative Bayesian measures of con-.

According to Bayesian confirmation theory, evidence E (incrementally) confirms (or supports) a hypothesis H (roughly) just in case E and H are positively probabilistically correlated (under an appropriate probability function Pr). There are many logically equivalent ways of saying that E and H are correlated under Pr. Surprisingly, this leads to a plethora of non-equivalent quantitative measures of the degree to which E confirms H (under Pr). In fact, many non-equivalent Bayesian measures of the degree to which E confirms (or (...) supports) H have been proposed and defended in the literature on inductive logic. I provide a thorough historical survey of the various proposals, and a detailed discussion of the philosophical ramifications of the differences between them. I argue that the set of candidate measures can be narrowed drastically by just a few intuitive and simple desiderata. In the end, I provide some novel and compelling reasons to think that the correct measure of degree of evidential support (within a Bayesian framework) is the (log) likelihood ratio. The central analyses of this research have had some useful and interesting byproducts, including: (i ) a new Bayesian account of (confirmationally) independent evidence, which has applications to several important problems in con- firmation theory, including the problem of the (confirmational) value of evidential diversity, and (ii ) novel resolutions of several problems in Bayesian confirmation theory, motivated by the use of the (log) likelihood ratio measure, including a reply to the Popper-Miller critique of probabilistic induction, and a new analysis and resolution of the problem of irrelevant conjunction (a.k.a., the tacking problem). (shrink)

Contemporary Bayesian confirmation theorists measure degree of (incremental) confirmation using a variety of non-equivalent relevance measures. As a result, a great many of the arguments surrounding quantitative Bayesian confirmation theory are implicitly sensitive to choice of measure of confirmation. Such arguments are enthymematic, since they tacitly presuppose that certain relevance measures should be used (for various purposes) rather than other relevance measures that have been proposed and defended in the philosophical literature. I present a survey of this pervasive class of (...) Bayesian confirmation-theoretic enthymemes, and a brief analysis of some recent attempts to resolve the problem of measure sensitivity. (shrink)

The Paradox of the Ravens (a.k.a,, The Paradox of Confirmation) is indeed an old chestnut. A great many things have been written and said about this paradox and its implications for the logic of evidential support. The first part of this paper will provide a brief survey of the early history of the paradox. This will include the original formulation of the paradox and the early responses of Hempel, Goodman, and Quine. The second part of the paper will describe attempts (...) to resolve the paradox within a Bayesian framework, and show how to improve upon them. This part begins with a discussion of how probabilistic methods can help to clarify the statement of the paradox itself. And it describes some of the early responses to probabilistic explications. We then inspect the assumptions employed by traditional (canonical) Bayesian approaches to the paradox. These assumptions may appear to be overly strong. So, drawing on weaker assumptions, we formulate a new-and-improved Bayesian confirmation-theoretic resolution of the Paradox of the Ravens. (shrink)

By and large, we think (Strevens's [2005]) is a useful reply to our original critique (Fitelson and Waterman [2005]) of his article on the Quine-Duhem (QD) problem (Strevens [2001]). But, we remain unsatisfied with several aspects of his reply (and his original article). Ultimately, we do not think he properly addresses our most important worries. In this brief rejoinder, we explain our remaining worries, and we issue a revised challenge for Strevens's approach to QD.

has proposed an interesting and novel Bayesian analysis of the Quine-Duhem (Q–D) problem (i.e., the problem of auxiliary hypotheses). Strevens's analysis involves the use of a simplifying idealization concerning the original Q–D problem. We will show that this idealization is far stronger than it might appear. Indeed, we argue that Strevens's idealization oversimplifies the Q–D problem, and we propose a diagnosis of the source(s) of the oversimplification. Some background on Quine–Duhem Strevens's simplifying idealization Indications that (I) oversimplifies Q–D Strevens's argument (...) for the legitimacy of (I). (shrink)

The central problem with Bayesian philosophy of science is that it cannot take account of the relevance of simplicity and unification to confirmation, induction, and scientific inference. The standard Bayesian folklore about factoring simplicity into the priors, and convergence theorems as a way of grounding their objectivity are some of the myths that Earman's book does not address adequately. 1Review of John Earman: Bayes or Bust?, Cambridge, MA. MIT Press, 1992, £33.75cloth.