A main message from the causal modelling literature in the last several decades is that under some plausible assumptions, there can be statistically consistent procedures for inferring (features of) the causal structure of a set of random variables from observational data. But whether we can control the error probabilities with a finite sample size depends on the kind of consistency the procedures can achieve. It has been shown that in general, under the standard causal Markov and Faithfulness assumptions, the (...) procedures can only be pointwise but not uniformly consistent without substantial background knowledge. This implies the impossibility of choosing a finite sample size to control the worst case error probabilities. In this paper, I consider the simpler task of inferring causal directions when the skeleton of the causal structure is known, and establish a similarly negative result concerning the possibility of controlling error probabilities. Although the result is negative in form, it has an interesting positive implication for causal discovery methods. (shrink)

The Bayesian theory is outlined and its status as a logic defended. In this it is contrasted with the development and extension of Neyman-Pearson methodology by Mayo in her recently published book (1996). It is shown by means of a simple counterexample that the rule of inference advocated by Mayo is actually unsound. An explanation of why error-probablities lead us to believe that they supply a sound rule is offered, followed by a discussion of two apparently powerful objections to (...) the Bayesian theory, one concerning old evidence and the other optional stopping. (shrink)

Suppose that an informant (test, expert, device, perceptual system, etc.) is unlikely to err when pronouncing on a particular subject matter. When this is so, it might be tempting to defer to that informant when forming beliefs about that subject matter. How is such an inferential process expected to fare in terms of truth (leading to true beliefs) and evidential fit (leading to beliefs that fit one’s total evidence)? Using a medical diagnostic test as an example, we set out a (...) formal framework to investigate this question. We establish seven results and make one conjecture. The first four results show that when the test’s error probabilities are low, the process of deferring to the test can score well in terms of (i) both truth and evidential fit, (ii) truth but not evidential fit, (iii) evidential fit but not truth, or (iv) neither truth nor evidential fit. Anything is possible. The remaining results and conjecture generalize these results in certain ways. These results are interesting in themselves—especially given that the diagnostic test is not sensitive to the target disease’s base rate—but also have broader implications for the more general process of deferring to an informant. Additionally, our framework and diagnostic example can be used to create test cases for various reliabilist theories of inferential justification. We show, for example, that they can be used to motivate evidentialist process reliabilism over process reliabilism. (shrink)

Bernoulli’s 1713 golden theorem is viewed retrospectively in the context of modern model-based frequentist inference that revolves around the concept of a prespecified statistical model Mθx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{M}}_{{{\varvec{\uptheta}}}} \left( {\mathbf{x}} \right)$$\end{document}, defining the inductive premises of inference. It is argued that several widely-accepted claims relating to the golden theorem and frequentist inference are either misleading or erroneous: (a) Bernoulli solved the problem of inference ‘from probability to frequency’, and thus (b) the golden (...) theorem cannot justify an approximate Confidence Interval (CI) for the unknown parameter θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document}, (c) Bernoulli identified the probability PA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P\left( A \right)$$\end{document} with the relative frequency 1n∑k=1nxk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{n}\sum\nolimits_{k = 1}^{n} {x_{k} }$$\end{document} of event A as a result of conflating f(x0|θ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\mathbf{x}}_{0} |\theta )$$\end{document} with f(θ|x0),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\theta |{\mathbf{x}}_{0} ),$$\end{document} where x0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{x}}_{0}$$\end{document} denotes the observed data, and (d) the same ‘swindle’ is currently perpetrated by the p value testers. In interrogating the claims (a)–(d), the paper raises several foundational issues that are particularly relevant for statistical induction as it relates to the current discussions on the replication crises and the trustworthiness of empirical evidence, arguing that: [i] The alleged Bernoulli swindle is grounded in the unwarranted claim θ^nx0≃θ∗,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\theta }_{n} \left( {{\mathbf{x}}_{0} } \right) \simeq \theta^{*},$$\end{document} for a large enough n, where θ^nX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\theta }_{n} \left( {\mathbf{X}} \right)$$\end{document} is an optimal estimator of the true value θ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta^{*}$$\end{document} of θ. [ii] Frequentist error probabilities are not conditional on hypotheses (H0 and H1) framed in terms of an unknown parameter θ since θ is neither a random variable nor an event. [iii] The direct versus inverse inference problem is a contrived and misplaced charge since neither conditional distribution f(x0|θ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\mathbf{x}}_{0} |\theta )$$\end{document} and f(θ|x0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\theta |{\mathbf{x}}_{0} )$$\end{document} exists (formally or logically) in model-based (Mθx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{M}}_{{{\varvec{\uptheta}}}} \left( {\mathbf{x}} \right)$$\end{document}) frequentist inference. (shrink)

Elicitation methods in decision-making under risk allow us to infer the utilities of outcomes as well as the probability weights from the observed preferences of an individual. An optimally efficient elicitation method is proposed, which takes the inevitable distortion of preferences by random errors into account and minimizes the effect of such errors on the inferred utility and probability weighting functions. Under mild assumptions, the optimally efficient method for eliciting utilities and probability weights is the following three-stage (...) procedure. First, a probability is elicited whose subjective weight is one half. Second, the utility function is elicited through the midpoint chaining certainty equivalent method using the probability elicited at the first stage. Finally, the probability weighting function is elicited through the probability equivalent method. (shrink)

Probabilistic reasoning is heavily investigated in decision research. Violations of probability theory have been demonstrated numerously, for instance, the tendency to overestimate the joint probab...

The error statistical account of testing uses statistical considerations, not to provide a measure of probability of hypotheses, but to model patterns of irregularity that are useful for controlling, distinguishing, and learning from errors. The aim of this paper is (1) to explain the main points of contrast between the error statistical and the subjective Bayesian approach and (2) to elucidate the key errors that underlie the central objection raised by Colin Howson at our PSA 96 Symposium.

For Ivor Grattan-Guinness on the occasion of his retirement. The work of Augustus De Morgan on symbolic logic in the mid-nineteenth century is familiar to historians of logic and mathematics alike. What is less well known is his work on probability and, more specifically, the use of probabilistic ideas and methods in his logic. The majority of De Morgan's work on probability was undertaken around 1837???1838, with his earliest publications on logic appearing from 1839, a period which culminated (...) with the publication of his Formal Logic in 1847. This article examines the overlap between his work on probability theory and logic during the earliest period of his interest in both. (shrink)

ABSTRACT This essay examines error and both external freedom and internal freedom. There is no external freedom (the latitude to choose) without internal freedom (the capacity to choose). Concerning external freedom, it suggests that errors serve as a derivative basis for natural rights. Concerning internal freedom, it overviews four groundbreaking papers from the 1990s by Stephen Boydstun, who suggested that there is no external freedom without internal indeterminism—specifically that associated with quantum probabilities related to neuronal control processes. Also reviewed (...) is work by Elio Conte and by Andrei Khrennikov. Experiments involving a quantum-like model of cognition are discussed. (shrink)

Maxwell's deduction of the probability distribution over the velocity of gas molecules—one of the most important passages in physics (Truesdell)—presents a riddle: a physical discovery of the first importance was made in a single inferential leap without any apparent recourse to empirical evidence. -/- Tychomancy proposes that Maxwell's derivation was not made a priori; rather, he inferred his distribution from non-probabilistic facts about the dynamics of intermolecular collisions. Further, the inference is of the same sort as everyday reasoning about (...) the physical probabilities attached to such canonical chance setups as tossed coins or rolled dice. The structure of this reasoning is investigated and some simple rules for inferring physical probabilities from symmetries and other causally relevant properties of physical systems are proposed. -/- Not only physics but evolutionary biology and population ecology, the science of measurement error, and climate modeling have benefited enormously from the same kind of reasoning, the book goes on to argue. Inferences from dynamics to probability are so obvious to us, however, that their methodological importance has been largely overlooked. (shrink)

The replication crisis has caused researchers to distinguish between exact replications, which duplicate all aspects of a study that could potentially affect the results, and direct replications, which duplicate only those aspects of the study that are thought to be theoretically essential to reproduce the original effect. The replication crisis has also prompted researchers to think more carefully about the possibility of making Type I errors when rejecting null hypotheses. In this context, the present article considers the utility of two (...) types of Type I error probability: the Neyman–Pearson long run Type I error rate and the Fisherian sample-specific Type I error probability. It is argued that the Neyman–Pearson Type I error rate is inapplicable in social science because it refers to a long run of exact replications, and social science deals with irreversible units that make exact replications impossible. Instead, the Fisherian sample-specific Type I error probability is recommended as a more meaningful way to conceptualize false positive results in social science because it can be applied to each sample-specific decision about rejecting the same substantive null hypothesis in a series of direct replications. It is concluded that the replication crisis may be partly due to researchers’ unrealistic expectations about replicability based on their consideration of the Neyman–Pearson Type I error rate across a long run of exact replications. (shrink)

In seeking general accounts of evidence, confirmation, or inference, philosophers have looked to logical relationships between evidence and hypotheses. Such logics of evidential relationship, whether hypothetico-deductive, Bayesian, or instantiationist fail to capture or be relevant to scientific practice. They require information that scientists do not generally have (e.g., an exhaustive set of hypotheses), while lacking slots within which to include considerations to which scientists regularly appeal (e.g., error probabilities). Building on my co-symposiasts contributions, I suggest some directions in which (...) a new and more adequate philosophy of evidence can move. (shrink)

We show that if among the tested hypotheses the number of true hypotheses is not equal to the number of false hypotheses, then Neyman-Pearson theory of testing hypotheses does not warrant minimal epistemic reliability (the feature of driving to true conclusions more often than to false ones). We also argue that N-P does not protect from the possible negative effects of the pragmatic value-laden unequal setting of error probabilities on N-P’s epistemic reliability. Most importantly, we argue that in the (...) case of a negative impact no methodological adjustment is available to neutralize it, so in such cases the discussed pragmatic-value-ladenness of N-P inevitably compromises the goal of attaining truth. (shrink)

This comprehensive study of probability considers the approaches of Pascal, Laplace, Poisson, and others. It also discusses Laws of Large Numbers, the theory of errors, and other relevant topics.

“What data will show the truth?” is a fundamental question emerging early in any empirical investigation. From a statistical perspective, experimental design is the appropriate tool to address this question by ensuring control of the error rates of planned data analyses and of the ensuing decisions. From an epistemological standpoint, planned data analyses describe in mathematical and algorithmic terms a pre-specified mapping of observations into decisions. The value of exploratory data analyses is often less clear, resulting in confusion about (...) what characteristics of design and analysis are necessary for decision making and what may be useful to inspire new questions. This point is addressed here by illustrating the Popper-Miller theorem in plain terms and using a graphical support. Popper and Miller proved that probability estimates cannot generate hypotheses on behalf of investigators. Consistently with Popper-Miller, we show that probability estimation can only reduce uncertainty about the truth of a merely possible hypothesis. This fact clearly identifies exploratory analysis as one of the tools supporting a dynamic process of hypothesis generation and refinement which cannot be purely analytic. A clear understanding of these facts will enable stakeholders, mathematical modellers and data analysts to better engage on a level playing field when designing experiments and when interpreting the results of planned and exploratory data analyses. (shrink)

My aim in this essay is largely defensive. I aim to discuss some problems for moral error theory and to offer plausible solutions. A full positive defense of moral error theory would require substantial investigations of rival metaethical views, but that is beyond the scope of this essay. I will, however, try to motivate moral error theory and to clarify its commitments. Moral error theorists typically accept two claims – one conceptual and one ontological – about (...) moral facts. The conceptual claim is that moral facts are or entail facts about categorical reasons (and correspondingly that moral claims are or entail claims about categorical reasons); the ontological claim is that there are no categorical reasons – and consequently no moral facts – in reality. I accept this version of moral error theory and I try to unpack what it amounts to in Section 2.1 In the course of doing so I consider two preliminary objections: that moral error theory is (probably) false because its implications are intuitively unacceptable (what I call the Moorean objection) and that the general motivation for moral error theory is self-undermining in that it rests on a hidden appeal to norms. The above characterization seems to entail the standard formulation of moral error theory, according to which first-order moral claims are uniformly false. Critics have argued that the standard formulation is incoherent since – by the law of excluded middle – the negation of a false claim is true. Hence if ‘Torture is wrong’ is false, ‘Torture is not wrong’ is true. Contrary to what moral error theorists contend, then, moral error theory seems to carry first-order moral implications that by the theory’s own lights are uniformly false. In Section 3 I suggest a formulation that is consistent with the standard formulation of moral error theory, free of first-order moral implications, and subject to no logical difficulties. In Section 4 I consider and rebut Stephen Finlay’s recent attack on moral error theory. According to Finlay the conceptual claim is false because all moral claims – and indeed all normative claims – are, or should be understood as, relativized to some moral standard or system of ends. Moral error theorists thus attribute to ordinary speakers an error that simply isn’t there. I argue that Finlay’s view has some very implausible implications and that it does not avoid commitment to various forms of error theory. This becomes especially clear when we focus on fundamental moral claims. In Section 5 I consider the worry that error theorists’ rejection of categorical reasons proves too much; in particular, the worry that error theorists’ qualms about categorical reasons apply equally to claims about hypothetical reasons, that is, claims to the effect that there is reason to take the means to one’s ends. In my view error theorists such as Mackie and Joyce have failed to pay due consideration to this problem. What the challenge establishes, I submit, is that error theorists cannot just take for granted that hypothetical reasons are metaphysically unproblematic; they must offer an account of hypothetical reasons that shows that they are. I argue that the only plausible account available to error theorists is one according to which claims about hypothetical reasons reduce to non-normative claims about relations between means and ends. (shrink)

Maxwell et al. [Maxwell, J. P., Masters, R. S. W., Kerr, E., & Weedon, E. . The implicit benefit of learning without errors. The Quarterly Journal of Experimental Psychology, 54A, 1049–1068. The implicit benefit of learning without errors. The Quarterly Journal of Experimental Psychology, 54A, 1049–1068] suggested that, following unsuccessful movements, the learner forms hypotheses about the probable causes of the error and the required movement adjustments necessary for its elimination. Hypothesis testing is an explicit process that places demands (...) on cognitive resources. Demands on cognitive resources can be identified by measuring probe reaction times and movement times. Lengthened PRT and movement times reflects increased cognitive demands. Thus, PRT and movement times should be longer following errors, relative to successful, movements. This hypothesis was tested using a motor skill . Furthermore, the association between error processing and the preparation and execution phases of movement was examined. The data confirmed that cognitive demand is greater for trials following an error, relative to trials without an error. This effect was apparent throughout learning and in both the preparatory and execution phases of the movement. Cognitive effort also appeared to be higher during movement preparation, relative to movement execution. (shrink)

In this article we investigate origins of several cases of failure of Artificial Intelligence (AI) systems employing machine learning and deep learning. We focus on omission and commission errors in (a) the inputs to the AI system, (b) the processing logic, and (c) the outputs from the AI system. Our framework yields a set of 28 factors that can be used for reconstructing the path of AI failures and for determining corrective action. Our research helps identify emerging themes of inquiry (...) necessary for developing more robust AI-ML systems. We are hopeful that our work will help strengthen the use of machine-learning AI by enhancing the rates of true positive and true negative judgements from AI systems, and by lowering the probabilities of false positive and false negative judgements. (shrink)

In this paper, I show how realist moral naturalists can provide an intra-theoretic explanation of the epistemic possibility of wholesale moral error. This is a requirement on metaethical theories that has been recently defended by Akhlaghi (2021). After clarifying Akhlaghi’s argument and responding to Evers’s (2021) recent rebuttal, I argue that even under the assumption that moral facts are grounded in an appropriate subset of natural facts (N-facts), there is still a non-zero probability of wholesale moral error. (...) This is demonstrated by considering three types of epistemically possible scenarios: specifically, it could either be that N-facts do not actually exist, or that N-facts exist in a way that entails wholesale moral error, or that N-facts exist in a non-error theoretic way, but their existence is temporally restricted. (shrink)

Evidential reasoning is hard, and errors can lead to miscarriages of justice with serious consequences. Analytic methods for the correct handling of evidence come in different styles, typically focusing on one of three tools: arguments, scenarios or probabilities. Recent research used Bayesian networks for connecting arguments, scenarios, and probabilities. Well-known issues with Bayesian networks were encountered: More numbers are needed than are available, and there is a risk of misinterpretation of the graph underlying the Bayesian network, for instance as a (...) causal model. The formalism presented here models presumptive arguments about coherent hypotheses that are compared in terms of their strength. No choice is needed between qualitative or quantitative analytic styles, since the formalism can be interpreted with and without numbers. The formalism is applied to key concepts in argumentative, scenario and probabilistic analyses of evidential reasoning, and is illustrated with a fictional crime investigation example based on Alfred Hitchcock’s film ‘To Catch A Thief’. (shrink)

The main objective of the paper is to propose a frequentist interpretation of probability in the context of model-based induction, anchored on the Strong Law of Large Numbers (SLLN) and justifiable on empirical grounds. It is argued that the prevailing views in philosophy of science concerning induction and the frequentist interpretation of probability are unduly influenced by enumerative induction, and the von Mises rendering, both of which are at odds with frequentist model-based induction that dominates current practice. The (...) differences between the two perspectives are brought out with a view to defend the model-based frequentist interpretation of probability against certain well-known charges, including [i] the circularity of its definition, [ii] its inability to assign ‘single event’ probabilities, and [iii] its reliance on ‘random samples’. It is argued that charges [i]–[ii] stem from misidentifying the frequentist ‘long-run’ with the von Mises collective. In contrast, the defining characteristic of the long-run metaphor associated with model-based induction is neither its temporal nor its physical dimension, but its repeatability (in principle); an attribute that renders it operational in practice. It is also argued that the notion of a statistical model can easily accommodate non-IID samples, rendering charge [iii] simply misinformed. (shrink)

A serious error in the proof of a recent characterization of the existence of full conditional probabilities invariant under symmetries is corrected.

Tyler Andrew Wunder, in his article, “Alvin Plantinga on Paul Draper’s evolutionary atheology: implications of theism’s non-contingency,” argues that Plantinga makes a serious error regarding probabilities in his critique of Draper. Properly modified, Wunder believes the argument “works,” but only in a trivial sense. This paper argues that Wunder’s objection, based on an assumed probability calculus, is merely asserted; whereas, there are other competing axiomatic systems consistent with Plantinga’s treatment of probability. As to the modified argument, it (...) is demonstrated that Wunder mistakenly concludes that two key propositions are contradictory. The consequence of this is not that Plantinga’s argument “works” in a trivial sense, but rather that the argument becomes incoherent. Lastly, this paper will explore the consequences of both Wunder’s and Plantinga’s assumptions concerning conditional probability for Draper’s evidentiary argument and Plantiga’s Evolutionary Argument Against Naturalism. (shrink)

In this paper we analyze the existence of joint probabilities for the Bell-type and GHZ entangled states. We then propose the usage of nonmonotonic upper probabilities as a tool to derive consistent joint upper probabilities for the contextual hidden variables. Finally, we show that for the extreme example of no error, the GHZ state allows for the definition of a joint upper probability that is consistent with the strong correlations.

Many scientists, businessmen, and government regulators believe that the criteria for acceptable societal risk are too stringent. Those who subscribe to this belief often accept the view which I call the probability-threshold position. Proponents of this stance maintain that society ought to ignore very small risks, i.e., those causing an average annual probability of fatality of less than 10–6.After examining the three major views in the risk-evaluation debate, viz., the probability-threshold position, the zero-risk position, and the weighted-risk (...) position, I focus on the arguments for the first of these views, since it is the position which currently undergirds most public policy (especially in the U.S.) regarding acceptable risk. After analyzing Arrow's argument from decision theory, Comar's and Gibson's argument from ontology, and Starr's and Whipple's argument from epistemology, I conclude that these defenses of the probability-threshold position err in a variety of ways. Most commonly, they fail because they tacitly accept the assumption that magnitude of probability, alone, provides a sufficient condition for judging the acceptability of a given risk. In the light of these errors, I suggest that it might be more desirable for risk assessors, decision theorists, and policymakers to weight various risk-cost-benefit parameters according to alternative ethical criteria, rather than to evaluate risks solely in terms of mathematical considerations. (shrink)

In October 2009 I decided to stop doing philosophy. This meant, in particular, stopping work on the book that I was writing on the nature of probability. At that time, I had no intention of making my unfinished draft available to others. However, I recently noticed how many people are reading the lecture notes and articles on my web site. Since this draft book contains some important improvements on those materials, I decided to make it available to anyone who (...) wants to read it. That is what you have in front of you. The account of Laplace’s theory of probability in Chapter 4 is very different to what I said in my seminar lectures, and also very different to any other account I have seen; it is based on a reading of important texts by Laplace that appear not to have been read by other commentators. The discussion of von Mises’ theory in Chapter 7 is also new, though perhaps less revolutionary. And the final chapter is a new attempt to come to grips with the popular, but amorphous, subjective theory of probability. The material in the other chapters has mostly appeared in previous articles of mine but things are sometimes expressed differently here. I would like to say again that this is an incomplete draft of a book, not the book I would have written if I had decided to finish it. It no doubt contains poor expressions, it may contain some errors or inconsistencies, and it doesn’t cover all the theories that I originally intended to discuss. Apart from this preface, I have done no work on the book since October 2009. (shrink)

The representational epistemic approach to the design of visual displays and notation systems advocates encoding the fundamental conceptual structure of a knowledge domain directly in the structure of a representational system. It is claimed that representations so designed will benefit from greater semantic transparency, which enhances comprehension and ease of learning, and plastic generativity, which makes the meaningful manipulation of the representation easier and less error prone. Epistemic principles for encoding fundamental conceptual structures directly in representational schemes are described. (...) The diagrammatic recodification of probability theory is undertaken to demonstrate how the fundamental conceptual structure of a knowledge domain can be analyzed, how the identified conceptual structure may be encoded in a representational system, and the cognitive benefits that follow. An experiment shows the new probability space diagrams are superior to the conventional approach for learning this conceptually challenging topic. (shrink)

An often-cited convention for discovery-oriented behavioral science research states that the general relative seriousness of the antecedently accepted false positive error rate of α = 0.05 be mirrored by a false negative error rate of β = 0.20. In 1965, Jacob Cohen proposed this convention to decrease a β-error rate typically in vast excess of 0.20. Thereby, we argue, Cohen (unintentionally) contributed to the wide acceptance of strongly uneven error rates in behavioral science. Although Cohen’s convention (...) can appear epistemically reasonable for an individual researcher, the comparatively low probability that published effect size estimates are replicable renders his convention unreasonable for an entire scientific field. Appreciating Cohen’s convention helps to understand why even error rates (α = β) are “non-conventional” in behavioral science today, and why Cohen’s explanatory reason for β = 0.20—that resource restrictions keep from collecting larger samples—can easily be mistaken for the justificatory reason it is not. (shrink)

The following note proposes a simple solution for an insufficiently considered difficulty in the much-debated dialogue between Mary and Joseph, the seventh of the extant lyrical divisions of the Old English Advent . In what follows I am assuming that the usual assignment of speeches, first set forth by Thorpe in the editio princeps of the Exeter Book, and accepted in all major editions up to and including that of Campbell, is to be preferred to the various alternatives that have (...) been proposed, notably the radically different assignment by Cosijn that has been tentatively adopted and defended by Robert B. Burlin. (shrink)

This paper argues that an intersubjective interpretation of probability is indispensable in understanding the rationality of science from the point of view of confirmation. It shows how intersubjective probabilities are superior to their subjective counterparts in a number of key respects, in a peculiar class of circumstances relevant to scientific practice. It also shows how group probabilities can be superior to individual probabilities even if rationality constraints other than coherence, e.g. as advocated by Objective Bayesians, are required of degrees (...) of belief. Key considerations are the improved error correction that can occur, and the increased background knowledge that becomes available, through interaction and group decision-making. (shrink)

The term “know” is one of the ten most common verbs in English, and yet a central aspect of its usage remains mysterious. Our willingness to ascribe knowledge depends not just on epistemic factors such as the quality of our evidence. It also depends on seemingly non-epistemic factors. For instance, we become less inclined to ascribe knowledge when it’s important to be right, or once our attention is drawn to possible sources of error. Accounts of this phenomenon proliferate, but (...) no consensus has been achieved, decades of research notwithstanding. Alexander Dinges offers a fresh examination of this ongoing debate. After reviewing and complementing relevant data from both armchair and experimental philosophy, he assesses extant accounts of this data including semantic, metaphysical, pragmatic, doxastic as well as more recent psychological accounts. Against this background, he offers a novel psychological account based on the idea that non-epistemic factors affect estimates of probability. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:125, HUME'S PROBABILITY ARGUMENT OF?,??,? In the Treatise,?,??,?, Hume presents an follows:' argument which, in the barest of outlines, goes as 1 (Pl) Every proposition has a probability less than one. (P2) If reason were the basis of our beliefs, then we would have no beliefs. (follows from (Pl)) (P3) We in fact do have beliefs. Hence, (P4) Reason is not the basis of our beliefs. The (...) argument has not been particularly well received. D. C. Stove, for example, refers to it as being "not merely defective, but one of the worst 2 arguments ever to impose itself on a man of genius." While not everyone is as unsympathetic as Stove, it nonetheless is difficult to find commentators favorably disposed toward the argument. Various sections of the Treatise are notoriously unclear.?,??,? is such a section; as such, it is difficult to say just what Hume intended his argument to be. My contention in the present paper is that there are two reasonable ways of reconstructing Hume's argument, both consistent with what Hume writes in?,??,?. The first reconstruction is clearly unsound; the second reconstruction fares somewhat better. In particular, my contention will be that this second reconstruction, if not sound, is at least valid 4 and contains no obviously false premises. Allow me to make some preliminary comments before presenting these reconstructions. The reconstructions to follow will primarily be concerned with premise (P2) above; in particular, the 126. reconstructions will focus on how Hume can claim that (P2) follows from (Pl). Premise (Pl) is a common 5 sceptical claim, not obviously false, and certainly not unique to Hume. As such, it strikes me that the more interesting part of the argument is Hume's claim that (P2) follows from (Pl), and such will be the focus of the reconstructions (although (Pl) will be discussed toward the end of this paper). Concerning (P2), why does Hume think this follows from (Pl)? His reasoning is contained in the following passage; Having thus found in every probability, beside the original uncertainty inherent in the subject, a new uncertainty deriv'd from the weakness of that faculty, which judges, and having adjusted these two together, we are oblig'd by our reason to add a new doubt deriv'd from the possibility of error in the estimation we make of the truth and fidelity of our faculties. This is a doubt, which immediately occurs to us, and of which, if we wou'd closely pursue our reason, we cannot avoid giving a decision. But this decision, tho' it shou'd be favourable to our preceeding judgment, being founded only on probability, must weaken still further our first evidence, and must itself be weaken'd by a fourth doubt of the same kind, and so on in infinitum; till at last there remain nothing of the original probability, however great we may suppose it to have been, and however small the diminution by every new uncertainty. No finite object can subsist under a decrease repeated in infinitum; and even the vastest quantity, which can enter into human imagination, must in this manner be reduc'd to nothing. (T182) Not for nothing did I claim that this section of the Treatise is unclear. At best, I can discern only the outline of Hume's reasoning, which seems to go as follows: first, when we assign a probability to some 127, proposition, reason dictates that we re-evaluate this probability. In particular, reason dictates that we "add a new doubt deriv'd from the possibility of error in the estimation we make of the truth and fidelity of our faculties." Note that Hume never specifies the object of this 'new doubt.' He might mean a) that our doubt concerns whether the probability we assigned the proposition is correct, or b) that our doubt concerns the evidence on which we based the assignment of that probability. Whatever the case, this 'new doubt' must "weaken still further our first evidence," presumably resulting in a lowering of the probability we originally assigned to the proposition. We then repeat this doubting process, adding new doubts of the same kind in infinitum, "till at last there remain nothing of the original... (shrink)

This article scrutinizes in detail much of the extant historiography on the controversy between biometricians and Mendelians, considering in particular how this controversy is related to the evolutionary synthesis. While the historiographic critique concentrates on William Provine’s standard account, it also extends to the proposal by Donald MacKenzie and Barry Barnes. What Provine and these sociologists of scientific knowledge have in common is a set of unquestioned assumptions about the nature of Darwinism, about William Bateson’s anti-Darwinism, and about the very (...) idea of an evolutionary “synthesis.” While these assumptions make for a compelling history of the synthesis, they engender an endemically asymmetrical perspective and bias historiography toward the mere confirmation of antecedent expectations, which renders the years from 1859 to 1929 an age of ignorance and misunderstanding. In contrast, a return to what was probably the original meaning of evolutionary “synthesis” allows for a symmetrical account. It yields an appreciation of the positive contributions made both by Mendelians and biometricians to the gradual development of Darwinism. The article concludes with a synopsis of this alternative account. (shrink)

The main aim of this paper is to revisit the curve fitting problem using the reliability of inductive inference as a primary criterion for the ‘fittest' curve. Viewed from this perspective, it is argued that a crucial concern with the current framework for addressing the curve fitting problem is, on the one hand, the undue influence of the mathematical approximation perspective, and on the other, the insufficient attention paid to the statistical modeling aspects of the problem. Using goodness-of-fit as the (...) primary criterion for ‘best', the mathematical approximation perspective undermines the reliability of inference objective by giving rise to selection rules which pay insufficient attention to ‘accounting for the regularities in the data'. A more appropriate framework is offered by the error-statistical approach, where (i) statistical adequacy provides the criterion for assessing when a curve captures the regularities in the data adequately, and (ii) the relevant error probabilities can be used to assess the reliability of inductive inference. Broadly speaking, the fittest curve (statistically adequate) is not determined by the smallness if its residuals, tempered by simplicity or other pragmatic criteria, but by the nonsystematic (e.g. white noise) nature of its residuals. The advocated error-statistical arguments are illustrated by comparing the Kepler and Ptolemaic models on empirical grounds. ‡I am grateful to Deborah Mayo and Clark Glymour for many valuable suggestions and comments on an earlier draft of the paper; estimating the Ptolemaic model was the result of Glymour's prompting and encouragement. †To contact the author, please write to: Department of Economics, Virginia Tech 3019 Pamplin Hall (0316), Blacksburg, VA 24061; e-mail: [email protected]. (shrink)

The bulk of this book is the second series of John Dewey Lectures, delivered by Professor Ayer in April 1970. To this, Ayer has added a criticism of Roy Harred’s purported refutation of Hume and a chapter about "non-truth-functional" conditionals that rounds out the lectures. Leaving Harred aside, this book provides an elegant, concise, and up-to-date introduction to the problem of induction and related issues concerning probability. Hume is here vindicated. Beginning by giving what may be the best, updated (...) paraphrase of Hume’s negative argument about causal judgments, Ayer explicates the subsumed atomicity, the denial of natural necessity, and the claimed vacuity or circularity of introducing the "uniformity of nature." Ayer goes on to distinguish a priori or logical probabilities, statistical or frequency judgments, and credibility or ground floor inductive judgments, and he summarily argues that the admissibility of the first two forms of judgment, properly understood, provides no real justification for credibility judgments, which, though not defined as subjective, turn out to need some support from our decision to project and therefore entrench certain predicates, or take as lawlike certain generalizations, rather than others. Ayer here includes a very clear explication and criticism of Carnap’s way out in the Logical Concept of Probability, and elsewhere. Ayer acutely summarizes and discusses the Hempel and Goodman paradoxes, taking a position substantially in agreement with Quine and Goodman, and welds this discussion into a Humean conclusion: "In a certain sense cases are what we choose them to be. We do not decide what facts habitually go together but we do decide what combinations are to be imaginatively projected. The despised savages who beat gongs at solar eclipses to summon back the sun are not making any factual error. It is a true generalization that whenever they beat the gongs the sun does shine again, and if they always keep up the ceremony, it is also a true generalization that the sun comes out again only when they beat the gongs. If we despise them, it is because they tell a fictional story about what would happen if they did not beat the gongs, which we do not accept. They see what goes on as well as we do; it is just that we have a different and, we think, a better idea of the way the world works." To say all this is just, of course, to summarize and refurbish Hume’s position on causality. But to do that has two very real values now: 1) It underlines and exposes the epistemic basis of the position taken by Hume’s heirs; 2) It makes clear what are the departures required if one is to differ in substance from Hume and Ayer.—J. L. (shrink)

No evidence exists that the accidental origin of life is too improbable to have occurred naturally, but there are numerous attempts to argue so. Dizzying statistics are cited to show that a god had to be responsible. This paper identifies the Argument from Biogenesis, then explains why all these arguments so far fail, and what would actually have to be done to make such an argument succeed. Describes seven general types of error, with examples. Includes a table of forty-seven (...) statistics used by over twenty authors to date. (shrink)

The preface paradox can be motivated by appealing to a plausible inference from an author’s reasonable assertion that her book is bound to contain errors to the author’s rational belief that her book contains errors. By evaluating and undermining the validity of this inference, I offer a resolution of the paradox. Discussions of the preface paradox have surprisingly failed to note that expressions of fallibility made in prefaces typically employ terms such as surely, undoubtedly, and bound to be. After considering (...) what these terms mean, I show that the motivating inference is invalid. Moreover, I argue that a closer consideration of our expressions of fallibility suggest that epistemically responsible authors would not be rational to believe that their books contain errors. I conclude by considering alternative expressions of fallibility that employ terms such as possible and probable, and discuss the role that expressions of fallibility play in conversation. (shrink)

Judgments of character traits tend to be overcorrelated, a bias known as the halo effect. We conducted two studies to test an explanation of the effect based on shared lexical context and connotation. Study 1 tested whether the context similarity of trait names could explain 39 participants’ ratings of the probability that two traits would co-occur. Over 126 trait pairs, cosine similarity between the word2vec vectors of the two words was a reliable predictor of the human judgments of trait (...) co-occurrence probability (cross-validated r2 =.19, p <.001). Two measures related to word similarity increased the variation accounted for in the human judgments to 45%, cross-validated (p <.001). In Experiment 2, 40 different participants judged similarity of word meaning within the pairs, confirming that the word pairs were not simply synonymous (Average [SD] = 40.8/100 [13.1/100]). Shared lexical context and word connotation play a role in shaping the halo effect. (shrink)

Randomized controlled clinical trials play an important role in the development of new medical therapies. There is, however, an ethical issue surrounding the use of randomized treatment allocation when the patient is suffering from a life threatening condition and requires immediate treatment. Such patients can only benefit from the treatment they actually receive and not from the alternative therapy, even if it ultimately proves to be superior. We discuss a novel new way to analyse data from such clinical trials based (...) on the use of the recently developed theory of imprecise probabilities. This work draws an explicit distinction between the related but nevertheless distinct questions of inference and decision in clinical trials. The traditional question of scientific interest asks 'Which treatment offers the greater chance of success?' and is the primary reason for conducting the clinical trial. The question of decision concerns the welfare of the patients in the clinical trial, asking whether the accumulated evidence favours one treatment over the other to such an extent that the next patient should decline randomization and instead express a preference for one treatment. Consideration of the decision question within the framework of imprecise probabilities leads to a mathematical definition of equipoise and a method for governing the randomization protocol of a clinical trial. This paper describes in detail the protocol for the conduct of clinical trials based on this new method of analysis, which is illustrated in a retrospective analysis of data from a clinical trial comparing the anti-emetic drugs ondansetron and droperidol in the treatment of postoperative nausea and vomiting. The proposed methodology is compared quantitatively using computer simulation studies with conventional clinical trial designs and is shown to maintain high statistical power with reduced sample sizes, at the expense of a high type I error rate that we argue is irrelevant in some specific circumstances. Particular emphasis is placed on describing the type of medical conditions and treatment comparisons where the new methodology is expected to provide the greatest benefit. (shrink)

Inductive reasoning requires exploiting links between evidence and hypotheses. This can be done focusing either on the posterior probability of the hypothesis when updated on the new evidence or on the impact of the new evidence on the credibility of the hypothesis. But are these two cognitive representations equally reliable? This study investigates this question by comparing probability and impact judgments on the same experimental materials. The results indicate that impact judgments are more consistent in time and more (...) accurate than probability judgments. Impact judgments also predict the direction of errors in probability judgments. These findings suggest that human inductive reasoning relies more on estimating evidential impact than on posterior probability. (shrink)