Schurz (2019, ch. 4) argues that probabilistic accounts of induction fail. In particular, he criticises probabilistic accounts of induction that appeal to direct inference principles, including subjective Bayesian approaches (e.g., Howson 2000) and objective Bayesian approaches (see, e.g., Williamson 2017). In this paper, I argue that Schurz’ preferred direct inference principle, namely Reichenbach’s Principle of the Narrowest Reference Class, faces formidable problems in a standard probabilistic setting. Furthermore, the main alternative direct inference principle, Lewis’ Principal Principle, is also hard to (...) reconcile with standard probabilism. So, I argue, standard probabilistic approaches cannot appeal to direct inference to explicate the logic of induction. However, I go on to defend a non-standard objective Bayesian account of induction: I argue that this approach can both accommodate direct inference and provide a viable account of the logic of induction. I then defend this account against Schurz’ criticisms. (shrink)

– We offer a new motivation for imprecise probabilities. We argue that there are propositions to which precise probability cannot be assigned, but to which imprecise probability can be assigned. In such cases the alternative to imprecise probability is not precise probability, but no probability at all. And an imprecise probability is substantially better than no probability at all. Our argument is based on the mathematical phenomenon of non-measurable sets. Non-measurable propositions cannot receive precise probabilities, but there is a natural (...) way for them to receive imprecise probabilities. The mathematics of non-measurable sets is arcane, but its epistemological import is far-reaching; even apparently mundane propositions are liable to be affected by non-measurability. The phenomenon of non-measurability dramatically reshapes the dialectic between critics and proponents of imprecise credence. Non-measurability offers natural rejoinders to prominent critics of imprecise credence. Non-measurability even reverses some of the critics’ arguments—by the very lights that have been used to argue against imprecise credences, imprecise credences are better than precise credences. (shrink)

Statistical tests that detect and measure deviation from the Hardy-Weinberg equilibrium (HWE) have been devised but are limited when testing for deviation at multiallelic DNA loci is attempted. Here we present the full Bayesian significance test (FBST) for the HWE. This test depends neither on asymptotic results nor on the number of possible alleles for the particular locus being evaluated. The FBST is based on the computation of an evidence index in favor of the HWE hypothesis. A great deal of (...) forensic inference based on DNA evidence assumes that the HWE is valid for the genetic loci being used. We applied the FBST to genotypes obtained at several multiallelic short tandem repeat loci during routine parentage testing; the locus Penta E exemplifies those clearly in HWE while others such as D10S1214 and D19S253 do not appear to show this. (shrink)

The Full Bayesian Significance Test, FBST, is extensively reviewed. Its test statistic, a genuine Bayesian measure of evidence, is discussed in detail. Its behavior in some problems of statistical inference like testing for independence in contingency tables is discussed.

The conservative idea that it is a philosopher’s job to clarify common sense beliefs about ordinary concepts is being weeded out from the population and replaced by a revisionist agenda: philosophers should not merely describe but also analyze and suggest ways to improve our concepts. This project is called "conceptual engineering." The conceptual engineering literature is growing rapidly as more philosophers undertake normative conceptual work. However, many philosophers are practicing conceptual engineering untethered to an explicit methodology. Analyses addressing how we (...) should engineer a concept and what should be considered successful are scarce. The thesis of this dissertation is that Carnapian Explication solves this methodological deficiency. (shrink)

Nowadays, almost everyone seems to agree that science is facing an epistemological crisis – namely the replicability crisis – and that we need to take action. But as to precisely what to do or how...

Can sentence comprehension impairments in aphasia be explained by difficulties arising from dependency completion processes in parsing? Two distinct models of dependency completion difficulty are investigated, the Lewis and Vasishth (2005) activation-based model and the direct-access model (DA; McElree, 2000). These models' predictive performance is compared using data from individuals with aphasia (IWAs) and control participants. The data are from a self-paced listening task involving subject and object relative clauses. The relative predictive performance of the models is evaluated using k-fold (...) cross-validation. For both IWAs and controls, the activation-based model furnishes a somewhat better quantitative fit to the data than the DA. Model comparisons using Bayes factors show that, assuming an activation-based model, intermittent deficiencies may be the best explanation for the cause of impairments in IWAs, although slowed syntax and lexical delayed access may also play a role. This is the first computational evaluation of different models of dependency completion using data from impaired and unimpaired individuals. This evaluation develops a systematic approach that can be used to quantitatively compare the predictions of competing models of language processing. (shrink)

An important line of response to scepticism appeals to the best explanation. But anti-sceptics have not engaged much with work on explanation in the philosophy of science. I plan to investigate whether plausible assumptions about best explanations really do favour anti-scepticism. I will argue that there are ways of constructing sceptical hypotheses in which the assumptions do favour anti-scepticism, but the size of the support for anti-scepticism is small.

Wenmackers and Romeijn (2016) formalize ideas going back to Shimony (1970) and Putnam (1963) into an open-minded Bayesian inductive logic, that can dynamically incorporate statistical hypotheses proposed in the course of the learning process. In this paper, we show that Wenmackers and Romeijn’s proposal does not preserve the classical Bayesian consistency guarantee of merger with the true hypothesis. We diagnose the problem, and offer a forward-looking open-minded Bayesians that does preserve a version of this guarantee.

I will argue, pace a great many of my contemporaries, that there's something right about Boltzmann's attempt to ground the second law of thermodynamics in a suitably amended deterministic time-reversal invariant classical dynamics, and that in order to appreciate what's right about (what was at least at one time) Boltzmann's explanatory project, one has to fully apprehend the nature of microphysical causal structure, time-reversal invariance, and the relationship between Boltzmann entropy and the work of Rudolf Clausius.

Millions of children in the United States experience parental incarcera- tion, yet it is unclear how this experience might shape social cognition. We asked children of incarcerated parents (N = 24) and children whose parents were not incarcerated (N = 58) to describe their parents. Both groups of children also rated the extent to which they agree that they feel positive and, separately, negative emotions when thinking about their parent and best friend. This approach allowed us to test between two (...) alternative hypotheses. On the one hand, cultural narratives in the United States convey negative messages about incarcerated people, and these messages could prompt children to report negativity when thinking about their incarcerated parents. On the other hand, chil- dren’s positivity toward close others is robust. Thus, when thinking about their incarcerated parents, children may report a great deal of positivity. Consistent with the latter possibility, children were more likely to describe their incarcerated parents using positive rather than negative terms. Moreover, children of incarcerated parents were more likely to agree that thinking about close others made them feel posi- tive emotions than they were to agree that thinking about close others made them feel negative emotions. A similar pattern of results emerged among children whose parents were not incarcerated. These findings demonstrate the robustness of children’s positivity and can inform debates regarding contact between incarcerated par- ents and their children. (shrink)

Evidence has shown that negative distracting stimuli are most difficult to control when we are focused in a relevant task, while positive and neutral distractors might be equally overcome. Still, r...

A naturalistic impulse has taken speculative analytic metaphysics in its critical sights. Importantly, the claim that it is desirable or requisite to give metaphysics scientific moorings rests on underlying epistemological assumptions or principles. If the naturalistic impulse toward metaphysics is to be well-founded and its prescriptions to have normative force, those assumptions or principles should be spelled out and justified. In short, advocates of naturalized or scientific metaphysics require epistemic infrastructure. This paper begins to supply it. The author first sketches (...) her conception of suitably naturalized or scientific metaphysics. She then lays out a number of candidate epistemic principles centring around the notion of theoretical constraint. The author offers several arguments for the principles, based on statistical likeliness, agreement, falsity avoidance, and methodological efficiency and inefficiency. Finally, she shows how scientific metaphysics satisfies the epistemic principles and is therefore preferable to its traditional rivals. (shrink)

Background. Attending to goal-relevant information can leave us metaphorically ‘blind’ or ‘deaf’ to the next relevant information while searching among distracters. This temporal cost lasting for about a half a second on the human selective attention has been long explored using the attentional blink paradigm. Although there is evidence that certain visual stimuli relating to one’s area of expertise can be less susceptible to attentional blink effects, it remains unexplored whether the dynamics of temporal selective attention vary with expertise and (...) objects types in the auditory modality. Methods. Using the auditory version of the attentional blink paradigm, the present study investigates whether certain auditory objects relating to musical and perceptual expertise could have an impact on the transient costs of selective attention. In this study, expert cellists and novice participants were asked to first identify a target sound, and then to detect instrumental timbres of cello or organ, or human voice as a second target in a rapid auditory stream. Results. The results showed moderate evidence against the attentional blink effect for voices independent of participants’ musical expertise. Experts outperformed novices in their overall accuracy levels of target identification and detection, reflecting a clear benefit of musical expertise. Importanly, the musicianship advantage disappeared when the human voices served as the second target in the stream. Discussion. The results are discussed in terms of stimulus salience, the advantage of voice processing, as well as perceptual and musical expertise in relation to attention and working memory performances. (shrink)

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)

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)

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)

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)

The idea that simplicity matters in science is as old as science itself, with the much cited example of Ockham's Razor, 'entia non sunt multiplicanda praeter necessitatem': entities are not to be multiplied beyond necessity. A problem with Ockham's razor is that nearly everybody seems to accept it, but few are able to define its exact meaning and to make it operational in a non-arbitrary way. Using a multidisciplinary perspective including philosophers, mathematicians, econometricians and economists, this 2002 monograph examines simplicity (...) by asking six questions: what is meant by simplicity? How is simplicity measured? Is there an optimum trade-off between simplicity and goodness-of-fit? What is the relation between simplicity and empirical modelling? What is the relation between simplicity and prediction? What is the connection between simplicity and convenience? The book concludes with reflections on simplicity by Nobel Laureates in Economics. (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)

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)

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)

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)

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)

Two of the most influential theories about scientific inference are inference to the best explanation and Bayesianism. How are they related? Bas van Fraassen has claimed that IBE and Bayesianism are incompatible rival theories, as any probabilistic version of IBE would violate Bayesian conditionalization. In response, several authors have defended the view that IBE is compatible with Bayesian updating. They claim that the explanatory considerations in IBE are taken into account by the Bayesian because the Bayesian either does or should (...) make use of them in assigning probabilities to hypotheses. I argue that van Fraassen has not succeeded in establishing that IBE and Bayesianism are incompatible, but that the existing compatibilist response is also not satisfactory. I suggest that a more promising approach to the problem is to investigate whether explanatory considerations are taken into account by a Bayesian who assigns priors and likelihoods on his or her own terms. In this case, IBE would emerge from the Bayesian account, rather than being used to constrain priors and likelihoods. I provide a detailed discussion of the case of how the Copernican and Ptolemaic theories explain retrograde motion, and suggest that one of the key explanatory considerations is the extent to which the explanation a theory provides depends on its core elements rather than on auxiliary hypotheses. I then suggest that this type of consideration is reflected in the Bayesian likelihood, given priors that a Bayesian might be inclined to adopt even without explicit guidance by IBE. The aim is to show that IBE and Bayesianism may be compatible, not because they can be amalgamated, but rather because they capture substantially similar epistemic considerations. 1 Introduction2 Preliminaries3 Inference to the Best Explanation4 Bayesianism5 The Incompatibilist View : Inference to the Best Explanation Contradicts Bayesianism5. 1 Criticism of the incompatibilist view6 Constraint - Based Compatibilism6. 1 Criticism of constraint - based compatibilism7 Emergent Compatibilism7. 1 Analysis of inference to the best explanation7. 1. 1 Inference to the best explanation on specific hypotheses7. 1. 2 Inference to the best explanation on general theories7. 1. 3 Copernicus versus Ptolemy7. 1. 4 Explanatory virtues7. 1. 5 Summary7. 2 Bayesian account8 Conclusion. (shrink)

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)

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)

Proponents of Bayesian confirmation theory believe that they have the solution to a significant, recalcitrant problem in philosophy of science. It is the identification of the logic that governs evidence and its inductive bearing in science. That is the logic that lets us say that our catalog of planetary observations strongly confirms Copernicus’ heliocentric hypothesis; or that the fossil record is good evidence for the theory of evolution; or that the 3oK cosmic background radiation supports big bang cosmology. The definitive (...) solution to this problem would be a significant achievement. The problem is of central importance to philosophy of science, for, in the end, what distinguishes science from myth making is that we have good evidence for the content of science, or at least of mature sciences, whereas myths are evidentially ungrounded fictions. The core ideas shared by all versions of Bayesian confirmation theory are, at a good first approximation, that a scientist’s beliefs are or should conform to a probability measure; and that the incorporation of new evidence is through conditionalization using Bayes’ theorem. While the burden of this chapter will be to inventory why critics believe this theory may not be the solution after all, it is worthwhile first to summarize here the most appealing virtues of this simple account. There are three. First, the theory reduces the often nebulous notion of a logic of.. (shrink)

After a brief review of ontic and epistemic descriptions, and of subjective, logical and statistical interpretations of probability, we summarize the traditional axiomatization of calculus of probability in terms of Boolean algebras and its set-theoretical realization in terms of Kolmogorov probability spaces. Since the axioms of mathematical probability theory say nothing about the conceptual meaning of “randomness” one considers probability as property of the generating conditions of a process so that one can relate randomness with predictability (or retrodictability). In the (...) measure-theoretical codification of stochastic processes genuine chance processes can be defined rigorously as so-called regular processes which do not allow a long-term prediction. We stress that stochastic processes are equivalence classes of individual point functions so that they do not refer to individual processes but only to an ensemble of statistically equivalent individual processes. Less popular but conceptually more important than statistical descriptions are individual descriptions which refer to individual chaotic processes. First, we review the individual description based on the generalized harmonic analysis by Norbert Wiener. It allows the definition of individual purely chaotic processes which can be interpreted as trajectories of regular statistical stochastic processes. Another individual description refers to algorithmic procedures which connect the intrinsic randomness of a finite sequence with the complexity of the shortest program necessary to produce the sequence. Finally, we ask why there can be laws of chance. We argue that random events fulfill the laws of chance if and only if they can be reduced to (possibly hidden) deterministic events. This mathematical result may elucidate the fact that not all nonpredictable events can be grasped by the methods of mathematical probability theory. (shrink)

Despite the widespread use of key concepts of the Neyman–Pearson (N–P) statistical paradigm—type I and II errors, significance levels, power, confidence levels—they have been the subject of philosophical controversy and debate for over 60 years. Both current and long-standing problems of N–P tests stem from unclarity and confusion, even among N–P adherents, as to how a test's (pre-data) error probabilities are to be used for (post-data) inductive inference as opposed to inductive behavior. We argue that the relevance of error probabilities (...) is to ensure that only statistical hypotheses that have passed severe or probative tests are inferred from the data. The severity criterion supplies a meta-statistical principle for evaluating proposed statistical inferences, avoiding classic fallacies from tests that are overly sensitive, as well as those not sensitive enough to particular errors and discrepancies. Introduction and overview 1.1 Behavioristic and inferential rationales for Neyman–Pearson (N–P) tests 1.2 Severity rationale: induction as severe testing 1.3 Severity as a meta-statistical concept: three required restrictions on the N–P paradigm Error statistical tests from the severity perspective 2.1 N–P test T(): type I, II error probabilities and power 2.2 Specifying test T() using p-values Neyman's post-data use of power 3.1 Neyman: does failure to reject H warrant confirming H? Severe testing as a basic concept for an adequate post-data inference 4.1 The severity interpretation of acceptance (SIA) for test T() 4.2 The fallacy of acceptance (i.e., an insignificant difference): Ms Rosy 4.3 Severity and power Fallacy of rejection: statistical vs. substantive significance 5.1 Taking a rejection of H0 as evidence for a substantive claim or theory 5.2 A statistically significant difference from H0 may fail to indicate a substantively important magnitude 5.3 Principle for the severity interpretation of a rejection (SIR) 5.4 Comparing significant results with different sample sizes in T(): large n problem 5.5 General testing rules for T(), using the severe testing concept The severe testing concept and confidence intervals 6.1 Dualities between one and two-sided intervals and tests 6.2 Avoiding shortcomings of confidence intervals Beyond the N–P paradigm: pure significance, and misspecification tests Concluding comments: have we shown severity to be a basic concept in a N–P philosophy of induction? (shrink)

Kolmogorov''s axiomatization of probability includes the familiarratio formula for conditional probability: 0).$$ " align="middle" border="0">.

Harold Jeffreys' ideas on the interpretation of probability and epistemology are reviewed. It is argued that with regard to the interpretation of probability, Jeffreys embraces a version of logicism that shares some features of the subjectivism of Ramsey and de Finetti. Jeffreys also developed a probabilistic epistemology, characterized by a pragmatical and constructivist attitude towards notions such as ‘objectivity’, ‘reality’ and ‘causality’. 1 Introductory remarks 2 The interpretation of probability 3 Jeffreys' probabilistic epistemology.

De Finetti would claim that we can make sense of a draw in which each positive integer has equal probability of winning. This requires a uniform probability distribution over the natural numbers, violating countable additivity. Countable additivity thus appears not to be a fundamental constraint on subjective probability. It does, however, seem mandated by Dutch Book arguments similar to those that support the other axioms of the probability calculus as compulsory for subjective interpretations. These two lines of reasoning can be (...) reconciled through a slight generalization of the Dutch Book framework. Countable additivity may indeed be abandoned for de Finetti's lottery, but this poses no serious threat to its adoption in most applications of subjective probability. Introduction The de Finetti lottery Two objections to equiprobability 3.1 The ‘No random mechanism’ argument 3.2 The Dutch Book argument Equiprobability and relative betting quotients The re-labelling paradox 5.1 The paradox 5.2 Resolution: from symmetry to relative probability Beyond the de Finetti lottery. (shrink)

This essay is, primarily, a discussion of four results about the principle of maximizing entropy (MAXENT) and its connections with Bayesian theory. Result 1 provides a restricted equivalence between the two: where the Bayesian model for MAXENT inference uses an "a priori" probability that is uniform, and where all MAXENT constraints are limited to 0-1 expectations for simple indicator-variables. The other three results report on an inability to extend the equivalence beyond these specialized constraints. Result 2 established a sensitivity of (...) MAXENT inference to the choice of the algebra of possibilities even though all empirical constraints imposed on the MAXENT solution are satisfied in each measure space considered. The resulting MAXENT distribution is not invariant over the choice of measure space. Thus, old and familiar problems with the Laplacian principle of Insufficient Reason also plague MAXENT theory. Result 3 builds upon the findings of Friedman and Shimony (1971; 1973) and demonstrates the absence of an exchangeable, Bayesian model for predictive MAXENT distributions when the MAXENT constraints are interpreted according to Jaynes's (1978) prescription for his (1963) Brandeis Dice problem. Lastly, Result 4 generalizes the Friedman and Shimony objection to cross-entropy (Kullback-information) shifts subject to a constraint of a new odds-ratio for two disjoint events. (shrink)

It is argued that models of H. Jeffreys' axioms of probability (Jeffreys [1939] 1967) are not monotone even with I. J. Good's proposed modification (Good 1950). Hence the additivity axiom seems essential to a theory of probability as it is with Kolmogorov's system (Kolmogorov 1950).

While orthodox (Neyman-Pearson) statistical tests enjoy widespread use in science, the philosophical controversy over their appropriateness for obtaining scientific knowledge remains unresolved. I shall suggest an explanation and a resolution of this controversy. The source of the controversy, I argue, is that orthodox tests are typically interpreted as rules for making optimal decisions as to how to behave--where optimality is measured by the frequency of errors the test would commit in a long series of trials. Most philosophers of statistics, however, (...) view the task of statistical methods as providing appropriate measures of the evidential-strength that data affords hypotheses. Since tests appropriate for the behavioral-decision task fail to provide measures of evidential-strength, philosophers of statistics claim the use of orthodox tests in science is misleading and unjustified. What critics of orthodox tests overlook, I argue, is that the primary function of statistical tests in science is neither to decide how to behave nor to assign measures of evidential strength to hypotheses. Rather, tests provide a tool for using incomplete data to learn about the process that generated it. This they do, I show, by providing a standard for distinguishing differences (between observed and hypothesized results) due to accidental or trivial errors from those due to systematic or substantively important discrepancies. I propose a reinterpretation of a commonly used orthodox test to make this learning model of tests explicit. (shrink)

When can a Bayesian investigator select an hypothesis H and design an experiment (or a sequence of experiments) to make certain that, given the experimental outcome(s), the posterior probability of H will be lower than its prior probability? We report an elementary result which establishes sufficient conditions under which this reasoning to a foregone conclusion cannot occur. Through an example, we discuss how this result extends to the perspective of an onlooker who agrees with the investigator about the statistical model (...) for the data but who holds a different prior probability for the statistical parameters of that model. We consider, specifically, one-sided and two-sided statistical hypotheses involving i.i.d. Normal data with conjugate priors. In a concluding section, using an "improper" prior, we illustrate how the preceding results depend upon the assumption that probability is countably additive. (shrink)