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

Any theory of confirmation must answer the following question: what is the purpose of its conception of confirmation for scientific inquiry? In this article, we argue that no Bayesian conception of confirmation can be used for its primary intended purpose, which we take to be making a claim about how worthy of belief various hypotheses are. Then we consider a different use to which Bayesian confirmation might be put, namely, determining the epistemic value of (...) experimental outcomes, and thus to decide which experiments to carry out. Interestingly, Bayesian confirmation theorists rule out that confirmation be used for this purpose. We conclude that Bayesian confirmation is a means with no end. 1 Introduction2 Bayesian Confirmation Theory3 Bayesian Confirmation and Belief4 Confirmation and the Value of Experiments5 Conclusion. (shrink)

The likelihood principle (LP) is a core issue in disagreements between Bayesian and frequentist statistical theories. Yet statements of the LP are often ambiguous, while arguments for why a Bayesian must accept it rely upon unexamined implicit premises. I distinguish two propositions associated with the LP, which I label LP1 and LP2. I maintain that there is a compelling Bayesian argument for LP1, based upon strict conditionalization, standard Bayesian decision theory, and a proposition I call the (...) practical relevance principle. In contrast, I argue that there is no similarly compelling argument for or against LP2. I suggest that these conclusions lead to a restrictedly pluralistic view of Bayesian confirmation measures. (shrink)

Bayesian confirmation theories might be the best standing theories of confirmation to date, but they are certainly not paradox-free. Here I recognize that BCTs’ appeal mainly comes from the fact that they capture some of our intuitions about confirmation better than those the- ories that came before them and that the superiority of BCTs is suffi- ciently justified by those advantages. Instead, I will focus on Sylvan and Nola’s claim that it is desirable that our best (...) theory of confirmation be as paradox-free as possible. For this reason, I will show that, as they respond to different interests, the project of the BCTs is not incompatible with Sylvan and Nola’s project of a paradox-free confirmation logic. In fact, it will turn out that, provided we are ready to embrace some degree of non-classicality, both projects complement each other nicely. (shrink)

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

This article reveals one general scheme for creating counter examples to Bayesian confirmation theory. The reason of the problems is that: in daily life the degree of confirmation is affected not only by probability but also by some non-probabilistic factors, e.g., structural similarity, quantity of evidence, and marginal utility, while Bayesian confirmation theory considers only probabilities to measure the degree of confirmation. This article resolves these problems after some detail analyses, and proposes a new (...) confirmation measure based on these factors. (shrink)

Does the Bayesian theory of confirmation put real constraints on our inductive behavior? Or is it just a framework for systematizing whatever kind of inductive behavior we prefer? Colin Howson (Hume's Problem) has recently championed the second view. I argue that he is wrong, in that the Bayesian apparatus as it is usually deployed does constrain our judgments of inductive import, but also that he is right, in that the source of Bayesianism's inductive prescriptions is not the (...) Bayesian machinery itself, but rather what David Lewis calls the ``Principal Principle''. (shrink)

Following Nancy Cartwright and others, I suggest that most (if not all) theories incorporate, or depend on, one or more idealizing assumptions. I then argue that such theories ought to be regimented as counterfactuals, the antecedents of which are simplifying assumptions. If this account of the logic form of theories is granted, then a serious problem arises for Bayesians concerning the prior probabilities of theories that have counterfactual form. If no such probabilities can be assigned, the the posterior probabilities will (...) be undefined, as the latter are defined in terms of the former. I argue here that the most plausible attempts to address the problem of probabilities of conditionals fail to help Bayesians, and, hence, that Bayesians are faced with a new problem. In so far as these proposed solutions fail, I argue that Bayesians must give up Bayesianism or accept the counterintuitive view that no theories that incorporate any idealizations have ever really been confirmed to any extent whatsoever. Moreover, as it appears that the latter horn of this dilemma is highly implausible, we are left with the conclusion that Bayesianism should be rejected, at least as it stands. (shrink)

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

Bayesian confirmation theory is a leading theory to decide the confirmation/refutation of a hypothesis based on probability calculus. While it may be much discussed in philosophy of science, is it actually practiced in terms of hypothesis testing by scientists? Since the assignment of some of the probabilities in the theory is open to debate and the risk of making the wrong decision is unknown, many scientists do not use the theory in hypothesis testing. Instead, they use alternative (...) statistical tests that can measure the risk or the reliability in decision making, circumventing some of the theoretical problems in practice. Therefore, the theory is not very popular in hypothesis testing among scientists at present. However, there are some proponents of Bayesian hypothesis testing, and software packages are made available to accelerate utilization by scientists. Time will tell whether Bayesian confirmation theory can become both a leading theory and a widely practiced method. In addition, this theory can be used to model the belief of scientists when testing hypotheses. (shrink)

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

Following Nancy Cartwright and others, I suggest that most theories incorporate, or depend on, one or more idealizing assumptions. I then argue that such theories ought to be regimented as counterfactuals, the antecedents of which are simplifying assumptions. If this account of the logical form of theories is granted, then a serious problem arises for Bayesians concerning the prior probabilities of theories that have counterfactual form. If no such probabilities can be assigned, then posterior probabilities will be undefined, as the (...) latter are defined in terms of the former. I argue here that the most plausible attempts to address the problem of probabilities of conditionals fail to help Bayesians, and, hence, that Bayesians are faced with a new problem. In so far as these proposed solutions fail, I argue that Bayesians must give up Bayesianism or accept the counterintuitive view that no theories that incorporate any idealizations have ever really been confirmed to any extent whatsoever. Moreover, as it appears that the latter horn of this dilemma is highly implausible, we are left with the conclusion that Bayesianism should be rejected, at least as it stands. (shrink)

Kevin T. Kelly and Clark Glymour. Why Bayesian Confirmation Does Not Capture the Logic of Scientific Justification.

Bayesian epistemology postulates a probabilistic analysis of many sorts of ordinary and scientific reasoning. Huber ([2005]) has provided a novel criticism of Bayesianism, whose core argument involves a challenging issue: confirmation by uncertain evidence. In this paper, we argue that under a properly defined Bayesian account of confirmation by uncertain evidence, Huber's criticism fails. By contrast, our discussion will highlight what we take as some new and appealing features of Bayesian confirmation theory. 1. Introduction2. (...) Uncertain Evidence and Bayesian Confirmation3. Bayesian Confirmation by Uncertain Evidence: Test Cases and Basic Principles. (shrink)

This paper suggests a new principle for confirmation theory which is called the principle of explanatory surplus. This principle is shown to be non-Bayesian in character, and to lead to a treatment of simplicity in science. Two cases of the principle of explanatory surplus are considered. The first (number of parameters) is illustrated by curve-fitting examples, while the second (number of theoretical assumptions) is illustrated by the examples of Newton's Laws and Adler's Theory of the Inferiority Complex.

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

Bayesian confirmation theory—abbreviated to in these notes—is the predominant approach to confirmation in late twentieth century philosophy of science. It has many critics, but no rival theory can claim anything like the same following. The popularity of the Bayesian approach is due to its flexibility, its apparently effortless handling of various technical problems, the existence of various a priori arguments for its validity, and its injection of subjective and contextual elements into the process of confirmation (...) in just the places where critics of earlier approaches had come to think that subjectivity and sensitivity to context were necessary. (shrink)

We explore the grammar of Bayesian confirmation by focusing on some likelihood principles, including the Weak Law of Likelihood. We show that none of the likelihood principles proposed so far is satisfied by all incremental measures of confirmation, and we argue that some of these measures indeed obey new, prima facie strange, antilikelihood principles. To prove this, we introduce a new measure that violates the Weak Law of Likelihood while satisfying a strong antilikelihood condition. We conclude by (...) hinting at some relevant links between the likelihood principles considered here and other properties of Bayesian confirmation recently explored in the literature. (shrink)

This paper examines the famous doctrine that independent prediction garners more support than accommodation. The standard arguments for the doctrine are found to be invalid, and a more realistic position is put forward, that whether evidence supports or not a hypothesis depends on the prior probability of the hypothesis, and is independent of whether it was proposed before or after the evidence. This position is implicit in the subjective Bayesian theory of confirmation, and the paper ends with a (...) brief account of this theory, and answer to the principal objections to it. (shrink)

There is currently no viable alternative to the Bayesian analysis of scientific inference, yet the available versions of Bayesianism fail to do justice to several aspects of the testing and confirmation of scientific hypotheses. Bayes or Bust? provides the first balanced treatment of the complex set of issues involved in this nagging conundrum in the philosophy of science. Both Bayesians and anti-Bayesians will find a wealth of new insights on topics ranging from Bayes's original paper to contemporary formal (...) learning theory. In a paper published posthumously in 1763, the Reverend Thomas Bayes made a seminal contribution to the understanding of "analogical or inductive reasoning." Building on his insights, modem Bayesians have developed an account of scientific inference that has attracted numerous champions as well as numerous detractors. Earman argues that Bayesianism provides the best hope for a comprehensive and unified account of scientific inference, yet the presently available versions of Bayesianisin fail to do justice to several aspects of the testing and confirming of scientific theories and hypotheses. By focusing on the need for a resolution to this impasse, Earman sharpens the issues on which a resolution turns. John Earman is Professor of History and Philosophy of Science at the University of Pittsburgh. (shrink)

Bayesian confirmation theory is rife with confirmation measures. Many of them differ from each other in important respects. It turns out, though, that all the standard confirmation measures in the literature run counter to the so-called “Reverse Matthew Effect” (“RME” for short). Suppose, to illustrate, that H1 and H2 are equally successful in predicting E in that p(E | H1)/p(E) = p(E | H2)/p(E) > 1. Suppose, further, that initially H1 is less probable than H2 in (...) that p(H1) < p(H2). Then by RME it follows that the degree to which E confirms H1 is greater than the degree to which it confirms H2. But by all the standard confirmation measures in the literature, in contrast, it follows that the degree to which E confirms H1 is less than or equal to the degree to which it confirms H2. It might seem, then, that RME should be rejected as implausible. Festa (2012), however, argues that there are scientific contexts in which RME holds. If Festa’s argument is sound, it follows that there are scientific contexts in which none of the standard confirmation measures in the literature is adequate. Festa’s argument is thus interesting, important, and deserving of careful examination. I consider five distinct respects in which E can be related to H, use them to construct five distinct ways of understanding confirmation measures, which I call “Increase in Probability”, “Partial Dependence”, “Partial Entailment”, “Partial Discrimination”, and “Popper Corroboration”, and argue that each such way runs counter to RME. The result is that it is not at all clear that there is a place in Bayesian confirmation theory for RME. (shrink)

There is currently no viable alternative to the Bayesian analysis of scientific inference, yet the available versions of Bayesianism fail to do justice to several aspects of the testing and confirmation of scientific hypotheses. Bayes or Bust? provides the first balanced treatment of the complex set of issues involved in this nagging conundrum in the philosophy of science. Both Bayesians and anti-Bayesians will find a wealth of new insights on topics ranging from Bayes’s original paper to contemporary formal (...) learning theory.In a paper published posthumously in 1763, the Reverend Thomas Bayes made a seminal contribution to the understanding of "analogical or inductive reasoning." Building on his insights, modem Bayesians have developed an account of scientific inference that has attracted numerous champions as well as numerous detractors. Earman argues that Bayesianism provides the best hope for a comprehensive and unified account of scientific inference, yet the presently available versions of Bayesianisin fail to do justice to several aspects of the testing and confirming of scientific theories and hypotheses. By focusing on the need for a resolution to this impasse, Earman sharpens the issues on which a resolution turns. John Earman is Professor of History and Philosophy of Science at the University of Pittsburgh. (shrink)

In his recent book The Advancement of Science, Philip Kitcher endorses eliminative induction, or the view that confirmation of hypotheses proceeds by the elimination of alternatives. My intention here is to critically examine Kitcher's eliminativist view of confirmation, and his rejection of the widely held Bayesian position, according to which an hypothesis H is confirmed by evidence E just in case the probability of H conditional on E is greater than the simple unconditional probability of H [i.e. (...) p > p]. Here, I will maintain that while there are aspects of the eliminative approach which Bayesianism must incorporate, as has been argued by John Earman, the eliminativism advocated by Kitcher is unsatisfactory. In addition, I will suggest that Kitcher's reasons for rejecting Bayesian confirmation theory are unconvincing and depend on an overly restrictive version of the position and that indeed Bayesianism has resources needed by an eliminativist account of confirmation. (shrink)

In his recent bookThe Advancement of Science,Philip Kitcher endorses eliminative induction, or the view that confirmation of hypotheses proceeds by the elimination of alternatives. My intention here is to critically examine Kitcher's eliminativist view of confirmation, and his rejection of the widely held Bayesian position, according to which an hypothesis H is confirmed by evidence E just in case the probability of H conditional on E is greater than the simple unconditional probability of H [i.e. p(H/E) > (...) p(H)]. Here, I will maintain that while there are aspects of the eliminative approach which Bayesianism must incorporate, as has been argued by John Earman, the eliminativism advocated by Kitcher is unsatisfactory. In addition, I will suggest that Kitcher's reasons for rejecting Bayesian confirmation theory are unconvincing and depend on an overly restrictive version of the position and that indeed Bayesianism has resources needed by an eliminativist account of confirmation. (shrink)

Peter Brössel and Franz Huber in 2015 argued that the Bayesian concept of confirmation had no use. I will argue that it has both the uses they discussed—it can be used for making claims about how worthy of belief various hypotheses are, and it can be used to measure the epistemic value of experiments. Furthermore, it can be useful in explanations. More generally, I will argue that more coarse-grained concepts can be useful, even when we have more fine-grained (...) concepts available. (shrink)

In the world of philosophy of science, the dominant theory of confirmation is Bayesian. In the wider philosophical world, the idea of inference to the best explanation exerts a considerable influence. Here we place the two worlds in collision, using Bayesian confirmation theory to argue that explanatoriness is evidentially irrelevant.

Recently in epistemology a number of authors have mounted Bayesian objections to dogmatism. These objections depend on a Bayesian principle of evidential confirmation: Evidence E confirms hypothesis H just in case Pr(H|E) > Pr(H). I argue using Keynes' and Knight's distinction between risk and uncertainty that the Bayesian principle fails to accommodate the intuitive notion of having no reason to believe. Consider as an example an unfamiliar card game: at first, since you're unfamiliar with the game, (...) you assign credences based on the indifference principle. Later you learn how the game works and discover that the odds dictate you assign the very same credences. Examples like this show that if you initially have no reason to believe H, then intuitively E can give you reason to believe H even though Pr(H|E) ≤ Pr(H). I show that without the principle, the objections to dogmatism fail. (shrink)

Bayesian confirmation theory, as traditionally interpreted, treats the temporal relationship between the formulation of a hypothesis and the confirmation (or recognition) of evidence entailed by that hypothesis merely as a component of the psychology of discovery and acceptance of a hypothesis. The temporal order of these events is irrelevant to the logic of rational theory choice. A few years ago Richmond Campbell and Thomas Vinci offered a reinterpretation of Bayes' Theorem in defense of the view that the (...) temporal relationship between hypothesis and evidence really does matter. More specifically, they advocated the thesis that successful predictions implied by scientific hypothesis H will increase the degree of confirmation of H only if they are novel predictions in the following sense; Evidence E is heuristically novel with respect to hypothesis H if and only if H was not deliberately designed to explain E (if E has already been corroborated) or in anticipation of E (if E is regarded as likely to be corroborated at some future date). Campbell and Vinci argue that the traditional interpretation of Bayes' Theorem misconstrues the significance of predictive novelty by ignoring heuristic novelty. In this paper I review the formal component of their preferred interpretation and demonstrate that it fails to establish that heuristic novelty has any special effect on theory confirmation. That is, even on their revisionist interpretation, regardless of whether H was deliberately designed to explain known evidence E (or in anticipation of suspected E), or whether H was designed without any awareness of the entailment relationship between H and E, Bayes' Theorem will generate the same epistemic probability for hypothesis H. Cases where new evidence was not foreseen at the time a hypothesis first emerged do not carry any more epistemic weight than they would have if such evidence had been foreseen. * Research on this paper was supported by an NEH summer grant. My thanks also to Richmond Campbell and Thomas Vinci for their reply to my initial evaluation of their position (in private correspondence), which compelled me to reconsider their arguments more carefully. (shrink)

Book review of Paul Horwich, Probability and Evidence (Cambridge Philosophy Classics edition), Cambridge: Cambridge University Press, 2016, 147pp, £14.99 (paperback).

Charles Chihara has presented a problem he claims Bayesian confirmation theory cannot handle. Chihara gives examples in which he claims the change in belief cannot be construced as conditionalizing on new evidence. These are situations in which the agent suddenly thinks of new possibilities. I propose a solution that incorporates the important ideas of Bayesian theory. In particular, I present a principle which shows that the change of belief in Chihara's example is due to simple conditionalization.

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

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

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

According to the comparative Bayesian concept of confirmation, rationalized versions of creationism come out as empirically confirmed. From a scientific viewpoint, however, they are pseudo-explanations because with their help all kinds of experiences are explainable in an ex-post fashion, by way of ad-hoc fitting of an empirically empty theoretical framework to the given evidence. An alternative concept of confirmation that attempts to capture this intuition is the use novelty criterion of confirmation. Serious objections have been raised (...) against this criterion. In this paper I suggest solutions to these objections. Based on them, I develop an account of genuine confirmation that unifies the UN-criterion with a refined probabilistic confirmation concept that is explicated in terms of the confirmation of evidence-transcending content parts of the hypothesis. (shrink)

I show how scoring rules, interpreted as measuring the inaccuracy of a set of degrees of belief, may be exploited to construct confirmation measures as used in Bayesian confirmation theory. I construct two confirmation measures from two particular standard scoring rules. One of these measures is genuinely new, the second is trivially ordinally equivalent to the difference measure. These two measures are tested against three well-known measures of confirmation in a simple but illuminating case that (...) contains in a natural way the problem of irrelevant conjunction. The genuinely new measure emerges, arguably, as the best. (shrink)

Some authors claim that minimal models have limited epistemic value (Fumagalli, 2016; Grüne-Yanoff, 2009a). Others defend the epistemic benefits of modelling by invoking the role of robustness analysis for hypothesis confirmation (see, e.g., Levins, 1966; Kuorikoski et al., 2010) but such arguments find much resistance (see, e.g., Odenbaugh & Alexandrova, 2011). In this paper, we offer a Bayesian rationalization and defence of the view that robustness analysis can play a confirmatory role, and thereby shed light on the potential (...) of minimal models for hypothesis confirmation. We illustrate our argument by reference to a case study from macroeconomics. At the same time, we also show that there are cases in which robustness analysis is detrimental to confirmation. We characterize these cases and link them to recent investigations on evidential variety (Landes, 2020b, 2021; Osimani and Landes, forthcoming). We conclude that robustness analysis over minimal models can confirm, but its confirmatory value depends on concrete circumstances. (shrink)