Dutch Book arguments have been presented for static belief systems and for belief change by conditionalization. An argument is given here that a rule for belief change which under certain conditions violates probability kinematics will leave the agent open to a Dutch Book.

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

How can self-locating propositions be integrated into normal patterns of belief revision? Puzzles such as Sleeping Beauty seem to show that such propositions lead to violation of ordinary principles for reasoning with subjective probability, such as Conditionalization and Reflection. I show that sophisticated forms of Conditionalization and Reflection are not only compatible with self-locating propositions, but also indispensable in understanding how they can function as evidence in Sleeping Beauty and similar cases.

We provide a solution to the well-known “Shooting-Room” paradox, developed by John Leslie in connection with his Doomsday Argument. In the “Shooting-Room” paradox, the death of an individual is contingent upon an event that has a 1/36 chance of occurring, yet the relative frequency of death in the relevant population is 0.9. There are two intuitively plausible arguments, one concluding that the appropriate subjective probability of death is 1/36, the other that this probability is 0.9. How are these two values (...) to be reconciled? We show that only the first argument is valid for a standard, countably additive probability distribution. However, both lines of reasoning are legitimate if probabilities are non-standard. The subjective probability of death rises from 1/36 to 0.9 by conditionalizing on an event that is not measurable, or whose probability is zero. Thus we can sometimes meaningfully ascribe conditional probabilities even when the event conditionalized upon is not of positive finite (or even infinitesimal) measure. (shrink)

How do temporal and eternal beliefs interact? I argue that acquiring a temporal belief should have no effect on eternal beliefs for an important range of cases. Thus, I oppose the popular view that new norms of belief change must be introduced for cases where the only change is the passing of time. I defend this position from the purported counter-examples of the Prisoner and Sleeping Beauty. I distinguish two importantly different ways in which temporal beliefs can be acquired and (...) draw some general conclusions about their impact on eternal beliefs. (shrink)

Colin Howson (1995 ) offers a counter-example to the rule of conditionalization. I will argue that the counter-example doesn't hit its target. The problem is that Howson mis-describes the total evidence the agent has. In particular, Howson overlooks how the restriction that the agent learn 'E and nothing else' interacts with the de se evidence 'I have learnt E'.

If an agent believes that the probability of E being true is 1/2, should she accept a bet on E at even odds or better? Yes, but only given certain conditions. This paper is about what those conditions are. In particular, we think that there is a condition that has been overlooked so far in the literature. We discovered it in response to a paper by Hitchcock (2004) in which he argues for the 1/3 answer to the Sleeping Beauty problem. (...) Hitchcock argues that this credence follows from calculating her fair betting odds, plus the assumption that Sleeping Beauty’s credences should track her fair betting odds. We will show that this last assumption is false. Sleeping Beauty’s credences should not follow her fair betting odds due to a peculiar feature of her epistemic situation. (shrink)

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

Bayesian, Jeffrey and Field conditionals are compared and it is shown why the last two cannot be reduced to the first. Maximum relative entropy is used in two kinds of justification of the Field conditional and the dispensability of entropy principles in general is discussed.

Van Fraassen (1989) argues that Inference to the Best Explanation is incoherent in the sense that adopting it as a rule for belief change will make one susceptible to a dynamic Dutch book. The present paper argues against this. A strategy is described that allows us to infer to the best explanation free of charge.

This is a 'state of the art' collection of essays on the relation between probabilities, especially conditional probabilities, and conditionals. It provides new negative results which sharply limit the ways conditionals can be related to conditional probabilities. There are also positive ideas and results which will open up new areas of research. The collection is intended to honour Ernest W. Adams, whose seminal work is largely responsible for creating this area of inquiry. As well as describing, evaluating, and applying Adams' (...) work the contributions extend his ideas in directions he may or may not have anticipated, but that he certainly inspired. In addition to a wide range of philosophers of science, the volume should interest computer scientists and linguists. (shrink)

In his recent book, Knowledge in a Social World, Alvin Goldman claims to have established that if a reasoner starts with accurate estimates of the reliability of new evidence and conditionalizes on this evidence, then this reasoner is objectively likely to end up closer to the truth. In this paper, I argue that Goldman's result is not nearly as philosophically significant as he would have us believe. First, accurately estimating the reliability of evidence – in the sense that Goldman requires (...) – is not quite as easy as it might sound. Second, being objectively likely to end up closer to the truth – in the sense that Goldman establishes – is not quite as valuable as it might sound. (shrink)

This paper critically analyzes Sherrilyn Roush’s (Tracking truth: knowledge, evidence and science, 2005) definition of evidence and especially her powerful defence that in the ideal, a claim should be probable to be evidence for anything. We suggest that Roush treats not one sense of ‘evidence’ but three: relevance, leveraging and grounds for knowledge; and that different parts of her argument fare differently with respect to different senses. For relevance, we argue that probable evidence is sufficient but not necessary for Roush’s (...) own two criteria of evidence to be met. With respect to grounds for knowledge, we agree that high probability evidence is indeed ideal for the central reason Roush gives: When believing a hypothesis on the basis of e it is desirable that e be probable. But we maintain that her further argument that Bayesians need probable evidence to warrant the method they recommend for belief revision rests on a mistaken interpretation of Bayesian conditionalization. Moreover, we argue that attempts to reconcile Roush’s arguments with Bayesianism fail. For leveraging, which we agree is a matter of great importance, the requirement that evidence be probable suffices for leveraging to the probability of the hypothesis if either one of Roush’s two criteria for evidence are met. Insisting on both then seems excessive. To finish, we show how evidence, as Roush defines it, can fail to track the hypothesis. This can remedied by adding a requirement that evidence be probable, suggesting another rationale for taking probable evidence as ideal—but only for a grounds-for-knowledge sense of evidence. (shrink)

Bayesian decision theory can be viewed as the core of psychological theory for idealized agents. To get a complete psychological theory for such agents, you have to supplement it with input and output laws. On a Bayesian theory that employs strict conditionalization, the input laws are easy to give. On a Bayesian theory that employs Jeffrey conditionalization, there appears to be a considerable problem with giving the input laws. However, Jeffrey conditionalization can be reformulated so that the problem disappears, and (...) in fact the reformulated version is more natural and easier to work with on independent grounds. (shrink)

According to Bayesian epistemology, the epistemically rational agent updates her beliefs by conditionalization: that is, her posterior subjective probability after taking account of evidence X, pnew, is to be set equal to her prior conditional probability pold(·|X). Bayesians can be challenged to provide a justification for their claim that conditionalization is recommended by rationality—whence the normative force of the injunction to conditionalize? There are several existing justifications for conditionalization, but none directly addresses the idea that conditionalization will be epistemically rational (...) if and only if it can reasonably be expected to lead to epistemically good outcomes. We apply the approach of cognitive decision theory to provide a justification for conditionalization using precisely that idea. We assign epistemic utility functions to epistemically rational agents; an agent’s epistemic utility is to depend both upon the actual state of the world and on the agent’s credence distribution over possible states. We prove that, under independently motivated conditions, conditionalization is the unique updating rule that maximizes expected epistemic utility. (shrink)

Jeffrey updating is a natural extension of Bayesian updating to cases where the evidence is uncertain. But, the resulting degrees of belief appear to be sensitive to the order in which the uncertain evidence is acquired, a rather un-Bayesian looking effect. This order dependence results from the way in which basic Jeffrey updating is usually extended to sequences of updates. The usual extension seems very natural, but there are other plausible ways to extend Bayesian updating that maintain order-independence. I will (...) explore three models of sequential updating, the usual extension and two alternatives. I will show that the alternative updating schemes derive from extensions of the usual rigidity requirement, which is at the heart of Jeffrey updating. Finally, I will establish necessary and sufficient conditions for order-independent updating, and show that extended rigidity is closely related to these conditions. (shrink)

Jeffrey updating is a natural extension of Bayesian updating to cases where the evidence is uncertain. But, the resulting degrees of belief appear to be sensitive to the order in which the uncertain evidence is acquired, a rather un-Bayesian looking effect. This order dependence results from the way in which basic Jeffrey updating is usually extended to sequences of updates. The usual extension seems very natural, but there are other plausible ways to extend Bayesian updating that maintain order-independence. I will (...) explore three models of sequential updating, the usual extension and two alternatives. I will show that the alternative updating schemes derive from extensions of the usual rigidity requirement, which is at the heart of Jeffrey updating. Finally, I will establish necessary and sufficient conditions for order-independent updating, and show that extended rigidity is closely related to these conditions. (shrink)

The objectivity of Bayesian induction relies on the ability of evidence to produce a convergence to agreement among agents who initially disagree about the plausibilities of hypotheses. I will describe three sorts of Bayesian convergence. The first reduces the objectivity of inductions about simple "occurrent events" to the objectivity of posterior probabilities for theoretical hypotheses. The second reveals that evidence will generally induce converge to agreement among agents on the posterior probabilities of theories only if the convergence is 0 or (...) 1. The third establishes conditions under which evidence will very probably compel posterior probabilities of theories to converge to 0 or 1. (shrink)

We report two issues concerning diverging sets of Bayesian (conditional) probabilities-divergence of "posteriors"-that can result with increasing evidence. Consider a set P of probabilities typically, but not always, based on a set of Bayesian "priors." Fix E, an event of interest, and X, a random variable to be observed. With respect to P, when the set of conditional probabilities for E, given X, strictly contains the set of unconditional probabilities for E, for each possible outcome X = x, call this (...) phenomenon dilation of the set of probabilities (Seidenfeld and Wasserman 1993). Thus, dilation contrasts with the asymptotic merging of posterior probabilities reported by Savage (1954) and by Blackwell and Dubins (1962). (1) In a wide variety of models for Robust Bayesian inference the extent to which X dilates E is related to a model specific index of how far key elements of P are from a distribution that makes X and E independent. (2) At a fixed confidence level, (1-α), Classical interval estimates A n for, e.g., a Normal mean θ have length O(n -1/2 ) (for sample size n). Of course, the confidence level correctly reports the (prior) probability that θ ∈ A n ,P(A n )=1-α , independent of the prior for θ . However, as shown by Pericchi and Walley (1991), if an ε -contamination class is used for the prior on the parameter θ , there is asymptotic (posterior) dilation for the A n , given the data. If, however, the intervals A ′ n are chosen with length $O(\sqrt{\log (\text{n})/\text{n})}$ , then there is no asymptotic dilation. We discuss the asymptotic rates of dilation for ClassClassical and Bayesian interval estimates and relate these to Bayesian hypothesis testing. (shrink)

I re-examine Coherence Arguments (Dutch Book Arguments, No Arbitrage Arguments) for diachronic constraints on Bayesian reasoning. I suggest to replace the usual game–theoretic coherence condition with a new decision–theoretic condition ('Diachronic Sure Thing Principle'). The new condition meets a large part of the standard objections against the Coherence Argument and frees it, in particular, from a commitment to additive utilities. It also facilitates the proof of the Converse Dutch Book Theorem. I first apply the improved Coherence Argument to van Fraassen's (...) (1984) Reflection principle. I then point out the failure of a Coherence Argument that is intended to support Conditionalization as a naive, universal, update rule. I also point out that Reflection is incompatible with the universal use of Conditionalization thus interpreted. The Coherence Argument therefore defeats the naive view on Bayesian learning that it was originally designed to justify. (shrink)

This paper argues that Ramsey's view of the calculus of subjective probabilities as, in effect, logical axioms is the correct view, with powerful heuristic value. This heuristic value is seen particularly in the analysis of the role of conditionalization in the Bayesian theory, where a semantic criterion of synchronic coherence is employed as the test of soundness, which the traditional formulation of conditionalization fails. On the other hand, there is a generally sound rule which supports conditionalization in appropriate contexts, though (...) these contexts are not universal. This sound Bayesian rule is seen to be analogous in certain respects to the deductive rule of modus ponens. (shrink)

This paper discusses the Bayesian updating rules of ordinary and Jeffrey conditionalisation. Their justification has been a topic of interest for the last quarter century, and several strategies proposed. None has been accepted as conclusive, and it is argued here that this is for a good reason; for by extending the domain of the probability function to include propositions describing the agent's present and future degrees of belief one can systematically generate a class of counterexamples to the rules. Dynamic Dutch (...) Book and other arguments for them are examined critically. A concluding discussion attempts to put these results in perspective within the Bayesian approach. (shrink)

Inductive probabilistic reasoning is understood as the application of inference patterns that use statistical background information to assign (subjective) probabilities to single events. The simplest such inference pattern is direct inference: from “70% of As are Bs” and “a is an A” infer that a is a B with probability 0.7. Direct inference is generalized by Jeffrey’s rule and the principle of cross-entropy minimization. To adequately formalize inductive probabilistic reasoning is an interesting topic for artificial intelligence, as an autonomous system (...) acting in a complex environment may have to base its actions on a probabilistic model of its environment, and the probabilities needed to form this model can often be obtained by combining statistical background information with particular observations made, i.e., by inductive probabilistic reasoning. In this paper a formal framework for inductive probabilistic reasoning is developed: syntactically it consists of an extension of the language of first-order predicate logic that allows to express statements about both statistical and subjective probabilities. Semantics for this representation language are developed that give rise to two distinct entailment relations: a relation ⊨ that models strict, probabilistically valid, inferences, and a relation that models inductive probabilistic inferences. The inductive entailment relation is obtained by implementing cross-entropy minimization in a preferred model semantics. A main objective of our approach is to ensure that for both entailment relations complete proof systems exist. This is achieved by allowing probability distributions in our semantic models that use non-standard probability values. A number of results are presented that show that in several important aspects the resulting logic behaves just like a logic based on real-valued probabilities alone. (shrink)

Thats logic. LEWIS CARROLL, Through the Looking Glass 1-1. The fundamental problem of this work is the question of the nature of scientific inference.

Bayes' Theorem is a simple mathematical formula used for calculating conditional probabilities. It figures prominently in subjectivist or Bayesian approaches to epistemology, statistics, and inductive logic. Subjectivists, who maintain that rational belief is governed by the laws of probability, lean heavily on conditional probabilities in their theories of evidence and their models of empirical learning. Bayes' Theorem is central to these enterprises both because it simplifies the calculation of conditional probabilities and because it clarifies significant features of subjectivist position. Indeed, (...) the Theorem's central insight — that a hypothesis is confirmed by any body of data that its truth renders probable — is the cornerstone of all subjectivist methodology. (shrink)

In this paper, I argue for a view largely favorable to the Thirder view: when Sleeping Beauty wakes up on Monday, her credence in the coin’s landing heads is less than 1/2. Let’s call this “the Lesser view.” For my argument, I (i) criticize Strict Conditionalization as the rule for changing de se credences; (ii) develop a new rule; and (iii) defend it by Gaifman’s Expert Principle. Finally, I defend the Lesser view by making use of this new rule.

Since Ramsey, much discussion of the relation between probability and belief has taken for granted that there are degrees of belief, i.e., that there is a real-valued function, B, that characterizes the degree of belief that an agent has in each statement of his language. It is then supposed that B is a probability. It is then often supposed that as the agent accumulates evidence, this function should be updated by conditioning: BE(·) should be B(·E)/B(E). Probability is also important in (...) classical statistics, where it is generally supposed that probabilities are frequencies, and that inference proceeds by controlling error and not by conditioning. I will focus on the tension between these two approaches to probability, and in the main part of the paper show where and when Bayesian conditioning conflicts with error based statistics and how to resolve these conflicts. (shrink)

One of the fundamental problems of epistemology is to say when the evidence in an agent’s possession justifies the beliefs she holds. In this paper and its sequel, we defend the Bayesian solution to this problem by appealing to the following fundamental norm: Accuracy An epistemic agent ought to minimize the inaccuracy of her partial beliefs. In this paper, we make this norm mathematically precise in various ways. We describe three epistemic dilemmas that an agent might face if she attempts (...) to follow Accuracy, and we show that the only inaccuracy measures that do not give rise to such dilemmas are the quadratic inaccuracy measures. In the sequel, we derive the main tenets of Bayesianism from the relevant mathematical versions of Accuracy to which this characterization of the legitimate inaccuracy measures gives rise, but we also show that Jeffrey conditionalization has to be replaced by a different method of update in order for Accuracy to be satisfied. (shrink)

One of the fundamental problems of epistemology is to say when the evidence in an agent’s possession justifies the beliefs she holds. In this paper and its prequel, we defend the Bayesian solution to this problem by appealing to the following fundamental norm: Accuracy An epistemic agent ought to minimize the inaccuracy of her partial beliefs. In the prequel, we made this norm mathematically precise; in this paper, we derive its consequences. We show that the two core tenets of Bayesianism (...) follow from the norm, while the characteristic claim of the Objectivist Bayesian follows from the norm along with an extra assumption. Finally, we consider Richard Jeffrey’s proposed generalization of conditionalization. We show not only that his rule cannot be derived from the norm, unless the requirement of Rigidity is imposed from the start, but further that the norm reveals it to be illegitimate. We end by deriving an alternative updating rule for those cases in which Jeffrey’s is usually supposed to apply. (shrink)

The article responds to some of the points raised by B. van Fraassen concerning probability kinematics and direct inference within the framework of the approach to the revision of probability judgment proposed by Levi in The Enterprise of Knowledge. In particular, the critical importance of the question of direct inference is emphasized and explained.

A number of cases involving self-locating beliefs have been discussed in the Bayesian literature. I suggest that many of these cases, such as the sleeping beauty case, are entangled with issues that are independent of self-locating beliefs per se. In light of this, I propose a division of labor: we should address each of these issues separately before we try to provide a comprehensive account of belief updating. By way of example, I sketch some ways of extending Bayesianism in order (...) to accommodate these issues. Then, putting these other issues aside, I sketch some ways of extending Bayesianism in order to accommodate self-locating beliefs. Finally, I propose a constraint on updating rules, the "Learning Principle", which rules out certain kinds of troubling belief changes, and I use this principle to assess some of the available options. (shrink)

This paper examines three accounts of the sleeping beauty case: an account proposed by Adam Elga, an account proposed by David Lewis, and a third account defended in this paper. It provides two reasons for preferring the third account. First, this account does a good job of capturing the temporal continuity of our beliefs, while the accounts favored by Elga and Lewis do not. Second, Elga’s and Lewis’ treatments of the sleeping beauty case lead to highly counterintuitive consequences. The proposed (...) account also leads to counterintuitive consequences, but they’re not as bad as those of Elga’s account, and no worse than those of Lewis’ account. (shrink)

Physicist J. Richard Gott uses the Copernican principle that “we are not special” to make predictions about the future lifetime of the human race, based on how long the human race has been in existence so far. We show that the predictions which can be derived from Gott’s argument are less strong than one might be inclined to believe, that Gott’s argument illegitimately assumes that the human race will not last forever, that certain versions of Gott’s argument are incompatible with (...) Bayesian conditionalization, and that Gott’s argument is self-refuting. (shrink)

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

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)

Let two events A, B be given. We consider probability distributions over the partition P = {A ∩ B, A ∩ ¯ B, ¯ A ∩ B, ¯ A ∩ ¯ B}. By a “constraint” is meant a probabilistically coherent set of statements each of the form Prob(E) = x, where E is a subset of P . Let C be the class of constraints. By a “method of ampliative inference” is meant any total function M from C to the (...) class of probability distributions over P , provided that M has the following property. (shrink)

A glance at the sky raises my probability of rain to .7. As it happens, the conditional probabilities of each state given rain remain the same, and similarly for their conditional probabilities given no rain. As Jeffrey (1983, Ch. 11) points out, my new distribution P2 is therefore fixed by the law of total probability. For example, P2(RC) = P2(RC | R)P2(R)+P2(RC | ¯.

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.

When preferences are such that there is no unique additive prior, the issue of which updating rule to use is of extreme importance. This paper presents an axiomatization of the rule which requires updating of all the priors by Bayes rule. The decision maker has conditional preferences over acts. It is assumed that preferences over acts conditional on event E happening, do not depend on lotteries received on Ec, obey axioms which lead to maxmin expected utility representation with multiple priors, (...) and have common induced preferences over lotteries. The paper shows that when all priors give positive probability to an event E, a certain coherence property between conditional and unconditional preferences is satisfied if and only if the set of subjective probability measures considered by the agent given E is obtained by updating all subjective prior probability measures using Bayes rule. (shrink)

The goal of a partial belief is to be accurate, or close to the truth. By appealing to this norm, I seek norms for partial beliefs in self-locating and non-self-locating propositions. My aim is to find norms that are analogous to the Bayesian norms, which, I argue, only apply unproblematically to partial beliefs in non-self-locating propositions. I argue that the goal of a set of partial beliefs is to minimize the expected inaccuracy of those beliefs. However, in the self-locating framework, (...) there are two equally legitimate definitions of expected inaccuracy. And, while each gives rise to the same synchronic norm for partial beliefs, they give rise to different, inconsistent diachronic norms. I conclude that both norms are rationally permissible. En passant, I note that this entails that both Halfer and Thirder solutions to the well-known Sleeping Beauty puzzle are rationally permissible. (shrink)

It is often objected that the Everett interpretation of QM cannot make sense of quantum probabilities, in one or both of two ways: either it can’t make sense of probability at all, or it can’t explain why probability should be governed by the Born rule. David Deutsch has attempted to meet these objections. He argues not only that rational decision under uncertainty makes sense in the Everett interpretation, but also that under reasonable assumptions, the credences of a rational agent in (...) an Everett world should be constrained by the Born rule. David Wallace has developed and defended Deutsch’s proposal, and greatly clarified its conceptual basis. In particular, he has stressed its reliance on the distinguishing symmetry of the Everett view, viz., that all possible outcomes of a quantum measurement are treated as equally real. The argument thus tries to make a virtue of what has usually been seen as the main obstacle to making sense of probability in the Everett world. In this note I outline some objections to the Deutsch-Wallace argument, and to related proposals by Hilary Greaves about the epistemology of Everettian QM. (In the latter case, my arguments include an appeal to an Everettian analogue of the Sleeping Beauty problem.) The common thread to these objections is that the symmetry in question remains a very significant obstacle to making sense of probability in the Everett interpretation. (shrink)

One can have no prior credence whatsoever (not even zero) in a temporally indexical claim. This fact saves the principle of conditionalization from potential counterexample and undermines the Elga and Arntzenius/Dorr arguments for the thirder position and Lewis' argument for the halfer position on the Sleeping Beauty Problem, thereby supporting the double-halfer position.

There is a substantial literature on the Bayesian approach, and the application of Bayes'' theorem, to legal matters. However, I have found no discussion that explores fully the issue of how a Bayesian juror might be led from an initial "presumption of innocence" to the judgment (required for conviction in criminal cases) that the suspect is "guilty beyond a reasonable doubt". I shall argue here that a Bayesian juror, if she acts in accord with what the law prescribes, will virtually (...) never reach such a judgment. In section I, I discuss Bayesianism, Bayes'' rule and the relation between them. Section II addresses the two legal notions key to my worries about Bayesian jurors: the presumption of innocence and the reasonable doubt criterion. Section III explores what a Bayesian is to make of these notions, and how the legal system requires her to reason in their light. If I am right, there emerges a conflict between current legal prescriptions and the Bayesian approach. Section IV explores the import of this conflict and how it might be resolved. (shrink)

Following the standard practice in sociology, cultural anthropology and history, sociologists, historians of science and some philosophers of science define scientific communities as groups with shared beliefs, values and practices. In this paper it is argued that in real cases the beliefs of the members of such communities often vary significantly in important ways. This has rather dire implications for the convergence defense against the charge of the excessive subjectivity of subjective Bayesianism because that defense requires that communities of Bayesian (...) inquirers share a significant set of modal beliefs. The important implication is then that given the actual variation in modal beliefs across individuals, either Bayesians cannot claim that actual theories have been objectively confirmed or they must accept that such theories have been confirmed relative only to epistemically insignificant communities. (shrink)

The proportional weight view in epistemology of disagreement generalizes the equal weight view and proposes that we assign to judgments of different people weights that are proportional to their epistemic qualifications. It is shown that if the resulting degrees of confidence are to constitute a probability function, they must be the weighted arithmetic means of individual degrees of confidence, while if the resulting degrees of confidence are to obey the Bayesian rule of conditionalization, they must be the weighted geometric means (...) of individual degrees of confidence. The double bind entails that the proportional weight view (and its moderate adjustment in favor of one’s own judgment) is inconsistent with Bayesianism. (shrink)

The orthodoxy that conditional probabilities reflect what are for a subject evidential bearings is seconded. This significance suggests that there should be principles equating rationally revised probabilities on new information with probabilities reached by conditionalizing on this information. Several principles, two of which are endorsed, are considered. A book is made against a violator of these, and it is argued that there must be something wrong with a person against whom such books can be made. Appendices comment on Popper-functions, elaborate (...) on bets and odds, and relate dutch books and strategies to conditions of inconsistency (Ramsey's idea) and imperfection. (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 epistemology’ became an epistemological movement in the 20th century, though its two main features can be traced back to the eponymous Reverend Thomas Bayes (c. 1701-61). Those two features are: (1) the introduction of a formal apparatus for inductive logic; (2) the introduction of a pragmatic self-defeat test (as illustrated by Dutch Book Arguments) for epistemic rationality as a way of extending the justification of the laws of deductive logic to include a justification for the laws of inductive logic. (...) The formal apparatus itself has two main elements: the use of the laws of probability as coherence constraints on rational degrees of belief (or degrees of confidence) and the introduction of a rule of probabilistic inference, a rule or principle of conditionalization. (shrink)

How does it come about then, that great scientists such as Einstein, Schrödinger and De Broglie are nevertheless dissatisfied with the situation? Of course, all these objections are levelled not against the correctness of the formulae, but against their interpretation. [...] The lesson to be learned from what I have told of the origin of quantum mechanics is that probable refinements of mathematical methods will not suffice to produce a satisfactory theory, but that somewhere in our doctrine is hidden a (...) concept, unjustified by experience, which we must eliminate to open up the road. (Born [ 1954 ], pp. 8, 11) It is truly surprising how little difference all this makes. Most physicists use quantum mechanics every day in their working lives without needing to worry about the fundamental problem of its interpretation. (Weinberg [ 1992 ], p. 66) I endorse the view that it may be of no relevance to the acceptability of the Everett interpretation of quantum mechanics as a physical theory whether or not an informed observer can be uncertain about the outcome of a quantum measurement prior to its having occurred. However, I suggest that the very possibility of post-measurement, pre-observation uncertainty has an essential role to play in both confirmation theory and decision theory in a branching universe. This is supported by arguments which do not appeal to van Fraassen’s Reflection Principle. (shrink)

Vague subjective probability may be modeled by means of a set of probability functions, so that the represented opinion has only a lower and upper bound. The standard rule of conditionalization can be straightforwardly adapted to this. But this combination has difficulties which, though well known in the technical literature, have not been given sufficient attention in probabilist or Bayesian epistemology. Specifically, updating on apparently irrelevant bits of news can be destructive of one’s explicitly prior expectations. Stability of vague subjective (...) opinion appears to need a more complex model. (shrink)

Probabilism in epistemology does not have to be of the Bayesian variety. The probabilist represents a person''s opinion as a probability function; the Bayesian adds that rational change of opinion must take the form of conditionalizing on new evidence. I will argue that this is the correct procedure under certain special conditions. Those special conditions are important, and instantiated for example in scientific experimentation, but hardly universal. My argument will be related to the much maligned Reflection Principle (van Fraassen, 1984, (...) 1995), and partly inspired by the work of Brian Skyrms (1987). (shrink)

The right interpretation of subjective probability is implicit in the theories of upper and lower odds, and upper and lower previsions, developed, respectively, by Cedric Smith (1961) and Peter Walley (1991). On this interpretation you are free to assign contingent events the probability 1 (and thus to employ conditionalization as a method of probability revision) without becoming vulnerable to a weak Dutch book.

A simple rule of probability revision ensures that the final result ofa sequence of probability revisions is undisturbed by an alterationin the temporal order of the learning prompting those revisions.This Uniformity Rule dictates that identical learning be reflectedin identical ratios of certain new-to-old odds, and is grounded in the oldBayesian idea that such ratios represent what is learned from new experiencealone, with prior probabilities factored out. The main theorem of this paperincludes as special cases (i) Field's theorem on commuting probability-kinematical (...) revisions and (ii) the equivalence of two strategiesfor generalizing Jeffrey's solution to the old evidence problem tothe case of uncertain old evidence and probabilistic new explanation. (shrink)

A simple rule of probability revision ensures that the final result of a sequence of probability revisions is undisturbed by an alteration in the temporal order of the learning prompting those revisions. This Uniformity Rule dictates that identical learning be reflected in identical ratios of certain new-to-old odds, and is grounded in the old Bayesian idea that such ratios represent what is learned from new experience alone, with prior probabilities factored out. The main theorem of this paper includes as special (...) cases (i) Field's theorem on commuting probability-kinematical revisions and (ii) the equivalence of two strategies for generalizing Jeffrey's solution to the old evidence problem to the case of uncertain old evidence and probabilistic new explanation. (shrink)

According to Bayesian epistemology, the epistemically rational agent updates her beliefs by conditionalization: that is, her posterior subjective probability after taking account of evidence X, new, is to be set equal to her prior conditional probability old(.|X). Bayesians can be challenged to provide a justification for their claim that conditionalization is recommended by rationality --- whence the normative force of the injunction to conditionalize?

The justification of Bayes' rule by cognitive rationality principles is undertaken by extending the propositional axiom systems usually proposed in two contexts of belief change: revising and updating. Probabilistic belief change axioms are introduced, either by direct transcription of the set-theoretic ones, or in a stronger way but nevertheless in the spirit of the underlying propositional principles. Weak revising axioms are shown to be satisfied by a General Conditioning rule, extending Bayes' rule but also compatible with others, and weak updating (...) axioms by a General Imaging rule, extending Lewis' rule. Strong axioms (equivalent to the Miller–Popper axiom system) are necessary to justify Bayes' rule in a revising context, and justify in fact an extended Bayes' rule which applies, even if the message has zero probability. (shrink)

Sometimes appearances provide prima facie support that is later undercut. In (Weisberg 2009) I argued that this phenomenon is incompatible with Bayesianism because its update rules have a formal property known as “rigidity”. Here I generalize that argument to cover Dempster-Shafer the-ory and Ranking Theory. The update rules of these models, in virtue of their rigidity, are similarly incompatible with the undermining of perceptual support. I then critique three solutions that have been proposed to the problem. I conclude that perceptual (...) undermining poses a more pervasive and more severe problem than is currently thought. (shrink)

Inference to the Best Explanation (IBE) and Bayesianism are our two most prominent theories of scientific inference. Are they compatible? Van Fraassen famously argued that they are not, concluding that IBE must be wrong since Bayesianism is right. Writers since then, from both the Bayesian and explanationist camps, have usually considered van Fraassen's argument to be misguided, and have plumped for the view that Bayesianism and IBE are actually compatible. I argue that van Fraassen's argument is actually not so misguided, (...) and that it causes more trouble for compatibilists than is typically thought. Bayesianism in its dominant, subjectivist form, can only be made compatible with IBE if IBE is made subservient to conditionalization in a way that robs IBE of much of its substance and interest. If Bayesianism and IBE are to be fit together, I argue, a strongly objective Bayesianism is the preferred option. I go on to sketch this objectivist, IBE-based Bayesianism, and offer some preliminary suggestions for its development. (shrink)

Conditionalization and Jeffrey Conditionalization cannot simultaneously satisfy two widely held desiderata on rules for empirical learning. The first desideratum is confirmational holism, which says that the evidential import of an experience is always sensitive to our background assumptions. The second desideratum is commutativity, which says that the order in which one acquires evidence shouldn't affect what conclusions one draws, provided the same total evidence is gathered in the end. (Jeffrey) Conditionalization cannot satisfy either of these desiderata without violating the other. (...) This is a surprising problem, and I offer a diagnosis of its source. I argue that (Jeffrey) Conditionalization is inherently anti-holistic in a way that is just exacerbated by the requirement of commutativity. The dilemma is thus a superficial manifestation of (Jeffrey) Conditionalization's fundamentally anti-holistic nature. (shrink)

Van Fraassen famously endorses the Principle of Reflection as a constraint on rational credence, and argues that Reflection is entailed by the more traditional principle of Conditionalization. He draws two morals from this alleged entailment. First, that Reflection can be regarded as an alternative to Conditionalization – a more lenient standard of rationality. And second, that commitment to Conditionalization can be turned into support for Reflection. Van Fraassen also argues that Reflection implies Conditionalization, thus offering a new justification for Conditionalization. (...) I argue that neither principle entails the other, and thus neither can be used to motivate the other in the way van Fraassen says. There are ways to connect Conditionalization to Reflection, but these connections depend on poor assumptions about our introspective access, and are not tight enough to draw the sorts of conclusions van Fraassen wants. Upon close examination, the two principles seem to be getting at two quite independent epistemic norms. (shrink)

Conditionalization is an intuitive and popular epistemic principle. By contrast, the Reflection principle is well known to have some very unappealing consequences. But van Fraassen argues that Conditionalization entails Reflection, so that proponents of Conditionalization must accept Reflection and its consequences. Van Fraassen also argues that Reflection implies Conditionalization, thus offering a new justification for Conditionalization. I argue that neither principle entails the other, and thus neither can be used to motivate the other in the way van Fraassen says. I (...) also propose a replacement for Reflection that accounts for the intuitions that made Reflection appealing, but doesn’t lead to Reflection’s bad consequences. (shrink)

Knowledge and its Limits presents a systematic new conception of knowledge as a kind of mental stage sensitive to the knower's environment. It makes a major contribution to the debate between externalist and internalist philosophies of mind, and breaks radically with the epistemological tradition of analyzing knowledge in terms of true belief. The theory casts new light on such philosophical problems as scepticism, evidence, probability and assertion, realism and anti-realism, and the limits of what can be known. The arguments are (...) illustrated by rigorous models based on epistemic logic and probability theory. The result is a new way of doing epistemology and a notable contribution to the philosophy of mind. (shrink)

A theory of evidential probability is developed from two assumptions:(1) the evidential probability of a proposition is its probability conditional on the total evidence;(2) one's total evidence is one's total knowledge. Evidential probability is distinguished from both subjective and objective probability. Loss as well as gain of evidence is permitted. Evidential probability is embedded within epistemic logic by means of possible worlds semantics for modal logic; this allows a natural theory of higher-order probability to be developed. In particular, it is (...) emphasized that it is sometimes uncertain which propositions are part of one's total evidence; some surprising implications of this fact are drawn out. (shrink)