Interpretation of Probability

Background: Microbiome biomarker discovery for patient diagnosis, prognosis, and risk evaluation is attracting broad interest. Selected groups of microbial features provide signatures that characterize host disease states such as cancer or cardio-metabolic diseases. Yet, the current predictive models stemming from machine learning still behave as black boxes and seldom generalize well. Their interpretation is challenging for physicians and biologists, which makes them difficult to trust and use routinely in the physician-patient decision-making process. Novel methods that provide interpretability and biological insight (...) are needed. Here, we introduce "predomics", an original machine learning approach inspired by microbial ecosystem interactions that is tailored for metagenomics data. It discovers accurate predictive signatures and provides unprecedented interpretability. The decision provided by the predictive model is based on a simple, yet powerful score computed by adding, subtracting, or dividing cumulative abundance of microbiome measurements. -/- Results: Tested on >100 datasets, we demonstrate that predomics models are simple and highly interpretable. Even with such simplicity, they are at least as accurate as state-of-the-art methods. The family of best models, discovered during the learning process, offers the ability to distil biological information and to decipher the predictability signatures of the studied condition. In a proof-of-concept experiment, we successfully predicted body corpulence and metabolic improvement after bariatric surgery using pre-surgery microbiome data. -/- Conclusions: Predomics is a new algorithm that helps in providing reliable and trustworthy diagnostic decisions in the microbiome field. Predomics is in accord with societal and legal requirements that plead for an explainable artificial intelligence approach in the medical field. (shrink)

Imprecise probabilities are often modelled with representors, or sets of probability functions. In the recent literature, two ways of interpreting representors have emerged as especially prominent: vagueness interpretations, according to which each probability function in the set represents how the agent's beliefs would be if any vagueness were precisified away; and comparativist interpretations, according to which the set represents those comparative confidence relations that are common to all probability functions therein. I argue that these interpretations have some important limitations. I (...) also propose an alternative—the functional interpretation—according to which representors are best interpreted by reference to the roles they play in the theories that make use of them. (shrink)

It is often claimed that credences are not reducible to ordinary beliefs about probabilities. Such a reduction appears to be decisively ruled out by certain sorts of triviality results–analogous to those often discussed in the literature on conditionals. I show why these results do not, in fact, rule out the view. They merely give us a constraint on what such a reduction could look like. In particular they show that there is no single proposition belief in which suffices for having (...) a particular credence, regardless of one’s evidence. But if we allow such propositions to vary with evidence–as we should–then the results do not rule out a reduction. So, at least on this count, credences might very well just be beliefs about probabilities. (shrink)

General epistemic polarization arises when the beliefs of a population grow further apart, in particular when all agents update on the same evidence. Epistemic factionalization arises when the beliefs grow further apart, but different beliefs also become correlated across the population. I present a model of how factionalization can emerge in a population of ideally rational agents. This kind of factionalization is driven by probabilistic relations between beliefs, with background beliefs shaping how the agents’ beliefs evolve in the light of (...) new evidence. Moreover, I show that in such a model, the only possible outcomes from updating on identical evidence are general convergence or factionalization. Beliefs cannot spread out in all directions: if the beliefs overall polarize, then it must result in factionalization. (shrink)

In this work, we focus on the philosophical aspects and technical challenges that underlie the axiomatization of the non-Kolmogorovian probability framework, in connection with the problem of quantum contextuality. This fundamental feature of quantum theory has received a lot of attention recently, given that it might be connected to the speed-up of quantum computers—a phenomenon that is not fully understood. Although this problem has been extensively studied in the physics community, there are still many philosophical questions that should be properly (...) formulated. We analyzed different problems from a conceptual standpoint using the non-Kolmogorovian probability approach as a technical tool. (shrink)

There is widespread excitement in the literature about the method of arbitrary functions: many take it to show that it is from the dynamics of systems that the objectivity of probabilities emerge. In this paper, I differentiate three ways in which a probability function might be objective, and I argue that the method of arbitrary functions cannot help us show that dynamics objectivise probabilities in any of these senses.

La epistemología bayesiana tiene como concepto capital la condicionalización simple. Para comprender de buena forma cómo opera esta regla, se debe dar cuenta de la concepción subjetiva de la probabilidad. Sobre la base de lo anterior es posible esclarecer alcances y límites de la condicionalización simple. En general, cuando esta regla enfrenta una dificultad se hacen esfuerzos por resolver dicha particular cuestión, pero no es usual encontrar propuestas unificadas con la intención de resolver varias de las complicaciones subyacentes al bayesianismo (...) ortodoxo. Por lo mismo, el propósito de esta investigación es justamente proponer aquello: un marco bayesiano ampliado que tendrá por objeto solucionar más de una cuestión particular. Más específicamente, el problema de la incorregibilidad, el problema del surgimiento del cero, el problema de la evidencia antigua y el problema de las teorías nuevas. Lo anterior se logrará adicionando al bayesianismo ortodoxo dos cuestiones, a saber, la condicionalización épsilon y una reconstrucción racional. Palabras clave: Probabilidad Subjetiva, Grados de Creencia, Dogmatismo Bayesiano, Evidencia Antigua, Nuevas Teorías. (shrink)

The interplay between the virus, infected cells and immune responses to SARS-CoV-2 is still under debate. By extending the basic model of viral dynamics, we propose here a formal approach to describe neutralisation versus weak (or non-)neutralisation scenarios and compare them with the possible effects of antibody-dependent enhancement (ADE). The theoretical model is consistent with the data available in the literature; we show that both weakly neutralising antibodies and ADE can result in final viral clearance or disease progression, but that (...) the immunodynamics are different in each case. As a significant proportion of the world’s population is already naturally immune or vaccinated, we also discuss the implications for secondary infections after vaccination or in the presence of immune system dysfunctions. (shrink)

A view that emancipates free will by means of quantum indeterminism is frequently rejected based on arguments pointing out its incompatibility with what we know about quantum physics. However, if one carefully examines what classical physical causal determinism and quantum indeterminism are according to physics, it becomes clear what they really imply–and, especially, what they do not imply–for agent-causation theories. Here, we will make necessary conceptual clarifications on some aspects of physical determinism and indeterminism, review some of the major objections (...) against libertarian conjectures, and show that there is no conceptual incompatibility preventing us from taking a ‘quantum-libertarian’ approach to the problem of free will. In particular, we will illustrate the possible role of self-causation (causa sui) as a potential solution to otherwise apparently incompatible or even paradoxical statements concerning free will and quantum indeterminism. (shrink)

[Accepted for publication in Lakatos's Undone Work: The Practical Turn and the Division of Philosophy of Mathematics and Philosophy of Science, special issue of Kriterion: Journal of Philosophy. Edited by S. Nagler, H. Pilin, and D. Sarikaya.] Lakatos’s analysis of progress and degeneration in the Methodology of Scientific Research Programmes is well-known. Less known, however, are his thoughts on degeneration in Proofs and Refutations. I propose and motivate two new criteria for degeneration based on the discussion in Proofs and Refutations (...) – superfluity and authoritarianism. I show how these criteria augment the account in Methodology of Scientific Research Programmes, providing a generalized Lakatosian account of progress and degeneration. I then apply this generalized account to a key transition point in the history of entropy – the transition to an information-theoretic interpretation of entropy – by assessing Jaynes’s 1957 paper on information theory and statistical mechanics. (shrink)

How do multiple reasons combine to support a conclusion about what to do or believe? This question raises two challenges: How can we represent the strength of a reason? How do the strengths of multiple reasons combine? Analogous challenges about confirmation have been answered using probabilistic tools. Can reductive and nonreductive theories of reasons use these tools to answer their challenges? Yes, or more exactly: reductive theories can answer both challenges. Nonreductive theories, with the help of a result in confirmation (...) theory, can answer one, and there are grounds for optimism that they can answer the other. (shrink)

This is Part 1 of a four part paper, intended to redress some of the most fundamental confusions in the subject of physical time directionality, and represent the concepts accurately. There are widespread fallacies in the subject that need to be corrected in introductory courses for physics students and philosophers. We start in Part 1 by analysing the time reversal symmetry of quantum probability laws. Time reversal symmetry is defined as the property of invariance under the time reversal transformation, T: (...) t --> -t. It is shown that quantum mechanics (classical or relativistic) is strongly time asymmetric in its probability laws. This contradicts the orthodox analysis, found throughout the conventional literature on physical time, which claims that quantum mechanics is time symmetric or reversible. This is widely claimed as settled scientific fact, and large philosophical and scientific conclusions are drawn from it. But it is an error. The fact is that while quantum mechanics is widely claimed to be reversible on the basis of two formal mathematical properties (that it does have), these properties do not represent invariance under the time reversal transformation. A recent experiment (Batalhão at alia, 2015) showing irreversibility of quantum thermodynamics is discussed as an illustration of this result. Most physicists remain unaware of the errors, decades after they were first demonstrated. Orthodox specialists in the philosophy of time who are aware of the error continue to refer to the ‘time symmetry’ or ‘reversibility’ of quantum mechanics anyway – and exploit the ambiguity to claim false implications about physical time reversal symmetry in nature. The excuse for perpetrating the confusion is that, since it is has now become customary to refer to the formal properties of quantum mechanics as ‘reversibility’ or ‘time reversal symmetry’, we should just keep referring to them by this name, even though they are not time reversal symmetry. This causes endless confusion, in attempts to explain the physical irreversibility of our universe, and in philosophical discussions of implications of physics for the nature of time. The failure of genuine time reversal symmetry in quantum mechanics changes the interpretation of modern physics in a deep way. It changes the problem of explaining the real irreversibility found throughout nature. (shrink)

A fundamental problem in understanding the nature of time is explaining its directionality. This 1990 PhD thesis re-examines the concepts of time flow, the physical directionality of time, and the semantics of tensed language. Several novel results are argued for that contradict the orthodox anti-realist views still dominant in the subject. Specifically, the concept of "metaphysical time flow" is supported as a valid scientific concept, and argued to be intrinsic to the directionality of objective probabilities in quantum mechanics; the common (...) claims that quantum probability theory is time reversible is shown to be based on an analytic error, stemming from a false choice for the criterion for reversibility of probabilistic theories (recognized by Satosi Watanabe in the 1950s but ignored in all philosophical discussions); and a consistent semantics for tensed language (adapted from the tree model of Storrs McCall) is constructed, showing that the common rejection of "time flow" as having no meaningful semantics is false. These debates are still ongoing in almost exactly the same state they were pre-1990, and there is appears to be no visible progress in the subject. Critical points made against errors in the orthodox account (which has been sustained for 70 years by the anti-realist philosophy of time, typified by the "Pittsburg School" of Grunbaum-Earman-Norton-Roberts), are still not recognized in the philosophy of time or physics. Some key technical proofs in this thesis have been published in physics proper. See p.ii-iii for full original abstract. (This pdf is uploaded from the Massey University archive.). (shrink)

Математическата величина на вероятността се определя стандартно като положително реално число в затворения интервал от нула до единица, еднозначно опредимо в съотвествие с няколко аксиоми, напр. тези на Колмпгоров. Нейната философска интерпретация е на мярка за част от цяло. В теорията на квантовата информация, изследваща явленията на сдвояване [entanglement] в квантовата механика, се въвеждат отрицателни и комплексни вероятности. Статията обсъжда проблема какво би следвало да бъде тяхното релевантно философско тълкувание.

Condorcet's famous jury theorem reaches an optimistic conclusion on the correctness of majority decisions, based on two controversial premises about voters: they are competent and vote independently, in a technical sense. I carefully analyse these premises and show that: whether a premise is justi…ed depends on the notion of probability considered; none of the notions renders both premises simultaneously justi…ed. Under the perhaps most interesting notions, the independence assumption should be weakened.

Many widely divergent accounts of luck have been offered or employed in discussing an equally wide range of philosophical topics. We should, then, expect to find some unified philosophical conception of luck of which moral luck, epistemic luck, and luck egalitarianism are species. One of the attempts to provide such an account is that offered by Duncan Pritchard, which he refers to as the modal account. This view commits us to calling an event lucky when it obtains in this world, (...) but fails to obtain in a wide class of nearby possible worlds. In support of this account, Pritchard argues that a theory of luck ought to capture the fact that luck comes in degrees and that luck is closely associated with risk. I argue against this claim by suggesting that an understanding of luck grounded in considerations of probability is better able to satisfy these demands, and that the probability theory better explains exemplary cases of luck like those brought up by Pritchard. (shrink)

Brentano never published on Bertrand’s paradox but claimed to have a solution. Adrian Maître has recovered from the Franz Brentano Archive Brentano’s remarks on his solution. They do not give us a worked demonstration of his solution but only an incomplete and in places obscure justification of it. Here I attempt to identify his solution, to explain what seem to me the clearly discernible parts of his justification and to discuss the extent to which the justification succeeds in the light (...) of current work on Bertrand’s paradox. (shrink)

Although probabilistic statements are ubiquitous, probability is still poorly understood. This shows itself, for example, in the mere stipulation of policies like expected utility maximisation and in disagreements about the correct interpretation of probability. In this work, we provide an account of probabilistic predictions that explains when, how, and why they can be useful for decision-making. We demonstrate that a calibration criterion on finite sets of predictions allows one to anticipate the distribution of utilities that a given policy will yield. (...) Based on this, we specify assumptions under which expected utility maximisation is a sensible decision criterion. We also introduce the notion of prediction methods, highlighting the dependence on schemes of abstraction, and argue that all probabilities are outputs of such prediction methods. This helps explain how the calibration criterion can be satisfied and shows that also supposedly objective probabilities are model-dependent. We compare our account of probability with conventional interpretations and show that it recovers key intuitions behind the latter. We, thus, provide a general account of what probabilities are and how they can enable successful decision-making under uncertainty. (shrink)

Events between which we have no epistemic reason to discriminate have equal epistemic probabilities. Bertrand’s chord paradox, however, appears to show this to be false, and thereby poses a general threat to probabilities for continuum sized state spaces. Articulating the nature of such spaces involves some deep mathematics and that is perhaps why the recent literature on Bertrand’s Paradox has been almost entirely from mathematicians and physicists, who have often deployed elegant mathematics of considerable sophistication. At the same time, the (...) philosophy of probability has been left out. In particular, left out entirely are the philosophical ground of the principle of indifference, the nature of the principle itself, the stringent constraint this places on the mathematical representation of the principle needed for its application to continuum sized event spaces, and what these entail for rigour in developing the paradox itself. This book puts the philosophy and its entailments back in and in so doing casts a new light on the paradox, giving original analyses of the paradox, its possible solutions, the source of the paradox, the philosophical errors we make in attempting to solve it and what the paradox proves for the philosophy of probability. The book finishes with the author’s proposed solution—a solution in the spirit of Bertrand’s, indeed—in which an epistemic principle more general than the principle of indifference offers a principled restriction of the domain of the principle of indifference. (shrink)

This paper generalises an argument for probabilism due to Lindley [9]. I extend the argument to a number of non-classical logical settings whose truth-values, seen here as ideal aims for belief, are in the set $\{0,1\}$, and where logical consequence $\models $ is given the “no-drop” characterization. First I will show that, in each of these settings, an agent’s credence can only avoid accuracy-domination if its canonical transform is a (possibly non-classical) probability function. In other words, if an agent values (...) accuracy as the fundamental epistemic virtue, it is a necessary requirement for rationality that her credence have some probabilistic structure. Then I show that for a certain class of reasonable measures of inaccuracy, having such a probabilistic structure is sufficient to avoid accuracy-domination in these non-classical settings. (shrink)

MỘT SỐ QUÁ TRÌNH NGẪU NHIÊN CÓ BƯỚC NHẢY -/- Hoàng Thị Phương Thảo -/- Luận án Tiến sỹ -/- TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN ĐẠI HỌC QUỐC GIA HÀ NỘI -/- Hà Nội - 2015 .

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or "toy" model of quantum mechanics over sets (QM/sets). There are two parts. The notion of an "event" is reinterpreted from being an epistemological state of indefiniteness to being an objective state of indefiniteness. And the mathematical framework of finite probability theory is recast as the quantum probability calculus for QM/sets. The point is not to (...) clarify finite probability theory but to elucidate quantum mechanics itself by seeing some of its quantum features in a classical setting. (shrink)

Frequency-based arguments against rational belief in a miracle occurring have been present for centuries, the most notable being from David Hume. In this essay, I will show Hume's argument rests on an equivocation of probability, with him using the term interchangeably to refer to two different and incompatible perspectives: Bayesianism and Frequentism. Additionally, I will show that any frequentist arguments against miracles relies on a view of probability that is only dubiously linked to rationality. In other words, the frequentist cannot (...) have it both ways: if probability is indeed frequency, then miracles are indeed improbable but not necessarily irrational to believe. On the other hand, if probability is an agent's confidence in a given belief, than under certain assumptions miracles are indeed highly probable and thus rational to believe despite their rarity. As a result, regardless of which view of probability one takes, it does not follow that believing in a miracle is irrational simply because miracles are rare. (shrink)

It is argued that Maximum Likelihood Estimation (MLE) is wrong, both conceptually and in terms of results it produces (except in two very special cases, which are discussed). While the use of MLE can still be justified on the basis of its practical performance, we argue there are better estimation methods that overcome MLE's empirical and philosophical shortcomings while retaining all of MLE's benefits.

One justification for preregistering research hypotheses, methods, and analyses is that it improves the transparent evaluation of the severity of hypothesis tests. In this article, I consider two cases in which preregistration does not improve this evaluation. First, I argue that, although preregistration can facilitate the transparent evaluation of severity in Mayo’s error statistical philosophy of science, it does not facilitate this evaluation in Popper’s theory-centric approach. To illustrate, I show that associated concerns about Type I error rate inflation are (...) only relevant in the error statistical approach and not in a theory-centric approach. Second, I argue that a preregistered test procedure that allows deviations in its implementation does not provide a more transparent evaluation of Mayoian severity than a non-preregistered procedure. In particular, I argue that sample-based validity-enhancing deviations cause an unknown inflation of the test procedure’s Type I (familywise) error rate and, consequently, an unknown reduction in its capability to license inferences severely. I conclude that preregistration does not improve the transparent evaluation of severity in Popper’s philosophy of science or when deviations are allowed. (shrink)

Although probabilistic statements are ubiquitous, probability is still poorly understood. This shows itself, for example, in the mere stipulation of policies like expected utility maximisation and in disagreements about the correct interpretation of probability. In this work, we provide an account of probabilistic predictions that explains when, how, and why they can be useful for decision-making. We demonstrate that a calibration criterion on finite sets of predictions allows one to anticipate the distribution of utilities that a given policy will yield. (...) Based on this, we specify assumptions under which expected utility maximisation is a sensible decision criterion. We also introduce the notion of prediction methods, highlighting the dependence on schemes of abstraction, and argue that all probabilities are outputs of such prediction methods. This helps explain how the calibration criterion can be satisfied and shows that also supposedly objective probabilities are model-dependent. We compare our account of probability with conventional interpretations and show that it recovers key intuitions behind the latter. We, thus, provide a general account of what probabilities are and how they can enable successful decision-making under uncertainty. (shrink)

Hájek (Erkenntnis 70(2):211–235, 2009) argues that probabilities cannot be the limits of relative frequencies in counterfactual infinite sequences. I argue for a different understanding of these limits, drawing on Norton’s (Philos Sci 79(2):207–232, 2012) distinction between approximations (inexact descriptions of a target) and idealizations (separate models that bear analogies to the target). Then, I adapt Hájek’s arguments to this new context. These arguments provide excellent reasons not to use hypothetical frequencies as idealizations, but no reason not to use them as (...) approximations. (shrink)

The inflation of Type I error rates is thought to be one of the causes of the replication crisis. Questionable research practices such as p-hacking are thought to inflate Type I error rates above their nominal level, leading to unexpectedly high levels of false positives in the literature and, consequently, unexpectedly low replication rates. In this article, I offer an alternative view. I argue that questionable and other research practices do not usually inflate relevant Type I error rates. I begin (...) by introducing the concept of Type I error rates and distinguishing between statistical errors and theoretical errors. I then illustrate my argument with respect to model misspecification, multiple testing, selective inference, forking paths, exploratory analyses, p-hacking, optional stopping, double dipping, and HARKing. In each case, I demonstrate that relevant Type I error rates are not usually inflated above their nominal level, and in the rare cases that they are, the inflation is easily identified and resolved. I conclude that the replication crisis may be explained, at least in part, by researchers’ misinterpretation of statistical errors and their underestimation of theoretical errors. (shrink)

In this paper, we illustrate some serious difficulties involved in conveying information about uncertain risks and securing informed consent for risky interventions in a clinical setting. We argue that in order to secure informed consent for a medical intervention, physicians often need to do more than report a bare, numerical probability value. When probabilities are given, securing informed consent generally requires communicating how probability expressions are to be interpreted and communicating something about the quality and quantity of the evidence for (...) the probabilities reported. Patients may also require guidance on how probability claims may or may not be relevant to their decisions, and physicians should be ready to help patients understand these issues. (shrink)

Like Lewis, many philosophers hold reductionist accounts of chance (on which claims about chance are to be understood as claims that certain patterns of events are instantiated) and maintain that rationality requires that credence should defer to chance (in the sense that under certain circumstances one's credence in an event must coincide with the chance of that event). It is a shortcoming of an account of chance if it implies that this norm of rationality is unsatisfiable by computable agents. This (...) shortcoming is more common than one might have hoped. (shrink)

Laura Papish’s Kant on Evil, Self-Deception, and Moral Reform is an ambitious attempt to breath new life into old debates and a welcome contribution to a recent renaissance of interest in Kant’s theory of evil. ​The book has eight chapters, and these chapters fall into three main divisions. Chapters 1 and 2 focus on the psychology of nonmoral and immoral action. Chapters 3, 4, and 5 focus on self-deception, evil, and dissimulation. And chapters 6, 7, and 8 focus on self-cognition, (...) moral reform, and moral progress. I shall begin with a brief summary of these chapters. Then I shall turn to commentary. (shrink)

Book Review of Kenneth Westphal's Kant’s Critical Epistemology: Why Epistemology Must Consider Judgment First.

Philosophers now seem to agree that frequentism is an untenable strategy to explain the meaning of probabilities. Nevertheless, I want to revive frequentism, and I will do so by grounding probabilities on typicality in the same way as the thermodynamic arrow of time can be grounded on typicality within statistical mechanics. This account, which I will call typicality frequentism, will evade the major criticisms raised against previous forms of frequentism. In this theory, probabilities arise within a physical theory from statistical (...) behavior of almost all initial conditions. The main advantage of typicality frequentism is that it shows which kinds of probabilities (that also have empirical relevance) can be derived from physics. Although one cannot recover all probability talk in this account, this is rather a virtue than a vice, because it shows which types of probabilities can in fact arise from physics and which types need to be explained in different ways, thereby opening the path for a pluralistic account of probabilities. (shrink)

Fisher criticised the Neyman-Pearson approach to hypothesis testing by arguing that it relies on the assumption of “repeated sampling from the same population.” The present article considers the responses to this criticism provided by Pearson and Neyman. Pearson interpreted alpha levels in relation to imaginary replications of the original test. This interpretation is appropriate when test users are sure that their replications will be equivalent to one another. However, by definition, scientific researchers do not possess sufficient knowledge about the relevant (...) and irrelevant aspects of their tests and populations to be sure that their replications will be equivalent to one another. Pearson also interpreted the alpha level as a personal rule that guides researchers’ behavior during hypothesis testing. However, this interpretation fails to acknowledge that the same researcher may use different alpha levels in different testing situations. Addressing this problem, Neyman proposed that the average alpha level adopted by a particular researcher can be viewed as an indicator of that researcher’s typical Type I error rate. Researchers’ average alpha levels may be informative from a metascientific perspective. However, they are not useful from a scientific perspective. Scientists are more concerned with the error rates of specific tests of specific hypotheses, rather than the error rates of their colleagues. It is concluded that neither Neyman nor Pearson adequately rebutted Fisher’s “repeated sampling” criticism. Fisher’s significance testing approach is briefly considered as an alternative to the Neyman-Pearson approach. (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 the face of continuing assumptions by many scientists and journal editors that p-values provide a gold standard for inference, counter warnings are published periodically. But the core problem is not with p-values, per se. A finding that “p-value is less than α” could merely signal that a critical value has been exceeded. The question is why, when estimating a parameter, we provide a range (a confidence interval), but when testing a hypothesis about a parameter (e.g. µ = x) we (...) proceed as if “=” entails exact equality of the parameter with x. That standard is hard to meet, and is not a standard expected for power calculations, where we are satisfied to reject a null hypothesis H0 if the result is merely “detectably” different from (exact) H0. This paper explores, with resampling (simulation) methods, the impacts on p-values, and alternatives, if the null hypothesis is defined as a thick or thin range of values. It also examines, empirically, the extent to which the p-value may or may not be a good predictor of the probability that H0 is true, given the distribution of the data. (shrink)

Where E is the proposition that [If H and O were true, H would explain O], William Roche and Elliot Sober have argued that P(H|O&E) = P(H|O). In this paper I argue that not only is this equality not generally true, it is false in the very kinds of cases that Roche and Sober focus on, involving frequency data. In fact, in such cases O raises the probability of H only given that there is an explanatory connection between them.

Early work on the frequency theory of probability made extensive use of the notion of randomness, conceived of as a property possessed by disorderly collections of outcomes. Growing out of this work, a rich mathematical literature on algorithmic randomness and Kolmogorov complexity developed through the twentieth century, but largely lost contact with the philosophical literature on physical probability. The present chapter begins with a clarification of the notions of randomness and probability, conceiving of the former as a property of a (...) sequence of outcomes, and the latter as a property of the process generating those outcomes. A discussion follows of the nature and limits of the relationship between the two notions, with largely negative verdicts on the prospects for any reduction of one to the other, although the existence of an apparently random sequence of outcomes is good evidence for the involvement of a genuinely chancy process. (shrink)

At the heart of Drazen Prelec’s chapter is the distinction between outcome utility and diagnostic utility. There is a particular distinction in the literature on causal networks (Pearl 2000), namely the distinction between observing and intervening, that maps onto Prelec’s distinction between diagnostic and outcome utility. I will explore the connection between both frameworks.

The economic paradigms of Ludwig von Mises on the one hand and of John Maynard Keynes on the other have been correctly recognized as antithetical at the theoretical level, and as antagonistic with respect to their practical and public policy implications. Characteristically they have also been vindicated by opposing sides of the political spectrum. Nevertheless the respective views of these authors with respect to the meaning and interpretation of probability exhibit a closer conceptual affinity than has been acknowledged in the (...) literature. In particular it is argued that in some relevant respects Ludwig von Mises´ interpretation of the concept of probability exhibits a closer affinity with the interpretation of probability developed by his opponent John Maynard Keynes than with the views on probability espoused by his brother Richard von Mises. Nevertheless there also exist significant differences between the views of Ludwig von Mises and those of John Maynard Keynes with respect to probability. One of these is highlighted more particularly: where John Maynard Keynes advocated a monist view of probability, Ludwig von Mises embraced a dualist view of probability, according to which the concept of probability has two different meanings each of which is valid in a particular area or context. It is concluded that both John Maynard Keynes and Ludwig von Mises presented highly nuanced views with respect to the meaning and interpretation of probability. (shrink)

In the following we will investigate whether von Mises’ frequency interpretation of probability can be modified to make it philosophically acceptable. We will reject certain elements of von Mises’ theory, but retain others. In the interpretation we propose we do not use von Mises’ often criticized ‘infinite collectives’ but we retain two essential claims of his interpretation, stating that probability can only be defined for events that can be repeated in similar conditions, and that exhibit frequency stabilization. The central idea (...) of the present article is that the mentioned ‘conditions’ should be well-defined and ‘partitioned’. More precisely, we will divide probabilistic systems into object, initializing, and probing subsystem, and show that such partitioning allows to solve problems. Moreover we will argue that a key idea of the Copenhagen interpretation of quantum mechanics (the determinant role of the observing system) can be seen as deriving from an analytic definition of probability as frequency. Thus a secondary aim of the article is to illustrate the virtues of analytic definition of concepts, consisting of making explicit what is implicit. (shrink)

It is argued that Maximum Likelihood Estimation (MLE) is wrong, both conceptually and in terms of results it produces (except in two very special cases, which are discussed). While the use of MLE can still be justified on the basis of its practical performance, we argue there are better estimation methods that overcome MLE's empirical and philosophical shortcomings while retaining all of MLE's benefits.

This dialogue builds on a previous critique that highlights a methodological paradox in the empirical application of probability theory. In the earlier work, it was suggested that while probability is mathematically sound, conceptual challenges arise when it is used to model real-world uncertainty. In this dialogue, Laplace, a proponent of classical probability, and Incredulus, a philosophical skeptic, engage in a thoughtful examination of these issues. Their inquiry explores the interplay between logic, probability, and empirical observation, raising questions about how past (...) experiences shape our understanding of possible outcomes and the limits this may impose on probabilistic reasoning. (shrink)

Reasoning from inconclusive evidence, or ‘induction’, is central to science and any applications we make of it. For that reason alone it demands the attention of philosophers of science. This Element explores the prospects of using probability theory to provide an inductive logic, a framework for representing evidential support. Constraints on the ideal evaluation of hypotheses suggest that overall support for a hypothesis is represented by its probability in light of the total evidence, and incremental support, or confirmation, indicated by (...) an increased probability given the evidence than otherwise. This proposal is shown to have the capacity to reconstruct many canons of the scientific method and inductive inference. Along the way, significant objections are discussed, such as the challenge from inductive scepticism and the objection that the probabilistic approach makes evidential support arbitrary. (shrink)

I argue that when we use ‘probability’ language in epistemic contexts—e.g., when we ask how probable some hypothesis is, given the evidence available to us—we are talking about degrees of support, rather than degrees of belief. The epistemic probability of A given B is the mind-independent degree to which B supports A, not the degree to which someone with B as their evidence believes A, or the degree to which someone would or should believe A if they had B as (...) their evidence. My central argument is that the degree-of-support interpretation lets us better model good reasoning in certain cases involving old evidence. Degree-of-belief interpretations make the wrong predictions not only about whether old evidence confirms new hypotheses, but about the values of the probabilities that enter into Bayes’ Theorem when we calculate the probability of hypotheses conditional on old evidence and new background information. (shrink)

David Stove was a philosopher strong on argument and polemic. His work on the logical intepretation of probability led to a defence of induction in The Rationality of Induction (1986). It resulted too in his denunciation of Popper, Kuhn, Lakatos and Feyeraband as irrationalists because of their "deductivism" (the thesis that the only logic is deductive logic). Stove also defended controversial views on the intelligence of women and on Darwinism. The article surveys his life and work.

John Maynard Keynes’s A Treatise on Probability is the seminal text for the logical interpretation of probability. According to his analysis, probabilities are evidential relations between a hypothesis and some evidence, just like the relations of deductive logic. While some philosophers had suggested similar ideas prior to Keynes, it was not until his Treatise that the logical interpretation of probability was advocated in a clear, systematic and rigorous way. I trace Keynes’s influence in the philosophy of probability through a heterogeneous (...) sample of thinkers who adopted his interpretation. This sample consists of Frederick C. Benenson, Roy Harrod, Donald C. Williams, Henry E. Kyburg and David Stove. The ideas of Keynes prove to be adaptable to their diverse theories of probability. My discussion indicates both the robustness of Keynes’s probability theory and the importance of its influence on the philosophers whom I describe. I also discuss the Problem of the Priors. I argue that none of those I discuss have obviously improved on Keynes’s theory with respect to this issue. (shrink)

Although written in Japanese, 確率としての信念（Belief as Probability） pursues a good explanation of Ramsey's subjective theory of probability.

In this contribution we will present a generalization of de Finetti's betting game in which a gambler is allowed to buy and sell unknown events' betting odds from more than one bookmaker. In such a framework, the sole coherence of the books the gambler can play with is not sucient, as in the original de Finetti's frame, to bar the gambler from a sure-win opportunity. The notion of joint coherence which we will introduce in this paper characterizes those coherent books (...) on which sure- win is impossible. Our main results provide geometric characterizations of the space of all books which are jointly coherent with a xed one. As a consequence we will also show that joint coherence is decidable. (shrink)

The objective Bayesian view of proof (or logical probability, or evidential support) is explained and defended: that the relation of evidence to hypothesis (in legal trials, science etc) is a strictly logical one, comparable to deductive logic. This view is distinguished from the thesis, which had some popularity in law in the 1980s, that legal evidence ought to be evaluated using numerical probabilities and formulas. While numbers are not always useful, a central role is played in uncertain reasoning by the (...) ‘proportional syllogism’, or argument from frequencies, such as ‘nearly all aeroplane flights arrive safely, so my flight is very likely to arrive safely’. Such arguments raise the ‘problem of the reference class’, arising from the fact that an individual case may be a member of many different classes in which frequencies differ. For example, if 15 per cent of swans are black and 60 per cent of fauna in the zoo is black, what should I think about the likelihood of a swan in the zoo being black? The nature of the problem is explained, and legal cases where it arises are given. It is explained how recent work in data mining on the relevance of features for prediction provides a solution to the reference class problem. (shrink)