An examination of topics involved in statistical reasoning with imprecise probabilities. The book discusses assessment and elicitation, extensions, envelopes and decisions, the importance of imprecision, conditional previsions and coherent statistical models.

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

A rational person doesn’t believe just anything. There are limits on what it is rational to believe. How wide are these limits? That’s the main question that interests me here. But a secondary question immediately arises: What factors impose these limits? A first stab is to say that one’s evidence determines what it is epistemically permissible for one to believe. Many will claim that there are further, non-evidentiary factors relevant to the epistemic rationality of belief. I will be ignoring the (...) details of alternative answers in order to focus on the question of what kind of rational constraints one’s evidence puts on belief. Our main question concerns how far epistemic permission and obligation can come apart.1 Suppose I am epistemically permitted to believe P, i.e., it would not be irrational for me to believe it. Am I thereby obliged to believe P, or are other options rationally available to me?2 Might I be equally rational in remaining agnostic about P, or even believing not-P? Or could even a slightly stronger or weaker degree of confidence be just as reasonable? (shrink)

In Bayesian epistemology, the problem of the priors is this: How should we set our credences (or degrees of belief) in the absence of evidence? That is, how should we set our prior or initial credences, the credences with which we begin our credal life? David Lewis liked to call an agent at the beginning of her credal journey a superbaby. The problem of the priors asks for the norms that govern these superbabies. -/- The Principle of Indifference gives a (...) very restrictive answer. It demands that such an agent divide her credences equally over all possibilities. That is, according to the Principle of Indifference, only one initial credence function is permissible, namely, the uniform distribution. In this paper, we offer a novel argument for the Principle of Indifference. I call it the Argument from Accuracy. (shrink)

Jon Williamson's Objective Bayesian Epistemology relies upon a calibration norm to constrain credal probability by both quantitative and qualitative evidence. One role of the calibration norm is to ensure that evidence works to constrain a convex set of probability functions. This essay brings into focus a problem for Williamson's theory when qualitative evidence specifies non-convex constraints.

We discuss several features of coherent choice functions—where the admissible options in a decision problem are exactly those that maximize expected utility for some probability/utility pair in fixed set S of probability/utility pairs. In this paper we consider, primarily, normal form decision problems under uncertainty—where only the probability component of S is indeterminate and utility for two privileged outcomes is determinate. Coherent choice distinguishes between each pair of sets of probabilities regardless the “shape” or “connectedness” of the sets of probabilities. (...) We axiomatize the theory of choice functions and show these axioms are necessary for coherence. The axioms are sufficient for coherence using a set of probability/almost-state-independent utility pairs. We give sufficient conditions when a choice function satisfying our axioms is represented by a set of probability/state-independent utility pairs with a common utility. (shrink)

It has been claimed that, in response to certain kinds of evidence, agents ought to adopt imprecise credences: doxastic states that are represented by sets of credence functions rather than single ones. In this paper I argue that, given some plausible constraints on accuracy measures, accuracy-centered epistemologists must reject the requirement to adopt imprecise credences. I then show that even the claim that imprecise credences are permitted is problematic for accuracy-centered epistemology. It follows that if imprecise credal states are permitted (...) or required in the cases that their defenders appeal to, then the requirements of rationality can outstrip what would be warranted by an interest in accuracy. (shrink)

In ‘A Non-Pragmatic Vindication of Probabilism’, Jim Joyce attempts to ‘depragmatize’ de Finetti’s prevision argument for the claim that our partial beliefs ought to satisfy the axioms of probability calculus. In this paper, I adapt Joyce’s argument to give a non-pragmatic vindication of various versions of David Lewis’ Principal Principle, such as the version based on Isaac Levi's account of admissibility, Michael Thau and Ned Hall's New Principle, and Jenann Ismael's Generalized Principal Principle. Joyce enumerates properties that must be had (...) by any measure of the distance from a set of partial beliefs to the set of truth values; he shows that, on any such measure, and for any set of partial beliefs that violates the probability axioms, there is a set that satisfies those axioms that is closer to every possible set of truth values. I replace truth values by objective chances in his argument; I show that for any set of partial beliefs that violates the probability axioms or a version of the Principal Principle, there is a set that satisfies them that is closer to every possible set of objective chances. (shrink)

In “A Nonpragmatic Vindication of Probabilism”, Jim Joyce argues that our credences should obey the axioms of the probability calculus by showing that, if they don't, there will be alternative credences that are guaranteed to be more accurate than ours. But it seems that accuracy is not the only goal of credences: there is also the goal of matching one's credences to one's evidence. I will consider four ways in which we might make this latter goal precise: on the first, (...) the norms to which this goal gives rise act as ‘side constraints’ on our choice of credences; on the second, matching credences to evidence is a goal that is weighed against accuracy to give the overall cognitive value of credences; on the third, as on the second, proximity to the evidential goal and proximity to the goal of accuracy are both sources of value, but this time they are incomparable; on the fourth, the evidential goal is not an independent goal at all, but rather a byproduct of the goal of accuracy. All but the fourth way of making the evidential goal precise are pluralist about credal virtue: there is the virtue of being accurate and there is the virtue of matching the evidence and neither reduces to the other. The fourth way is monist about credal virtue: there is just the virtue of being accurate. The pluralist positions lead to problems for Joyce's argument; the monist position avoids them. I endorse the latter. (shrink)

Why should a Bayesian bother performing an experiment, one the result of which might well upset his own favored credence function? The Ramsey-Good theorem provides a decision theoretic answer. Provided you base your decision on expected utility, and the the experiment is cost-free, performing the experiment and then choosing has at least as much expected utility as choosing without further ado. Furthermore, doing the experiment is strictly preferable just in case at least one possible outcome of the experiment could alter (...) your choice of options. That is, you should pursue the information just in case it could make a difference to what you do. The problem remains, however, why one should do an experiment when nothing of much practical consequence hangs in the balance. One possibility here is to invoke the pure cognitive utility of states. It is shown that one constraint on cognitive value (what I call cogency) guarantees the value of learning: from a cognitive point of view you are justified in performing an experiment if the outcome of the experiment could make a different to what you think. A bonus of the basic framework is that it provides a new decision theoretic argument for the rationality of conditionalization. (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)

In this paper, I explore the question of whether the expected consequences of holding a belief can affect the rationality of doing so. Special attention is given to various ways in which one might attempt to exert some measure of control over what one believes and the normative status of the beliefs that result from the successful execution of such projects. I argue that the lessons which emerge from thinking about the case ofbelief have important implications for the way we (...) should think about the rationality of a number of other propositional attitudes,such as regret, desire, and fear. Finally,I suggest that a lack of clarity with respect to the relevant issues has given rise to a number of rather serious philosophical mistakes. (shrink)

In this paper, I explore the question of whether the expectedconsequences of holding a belief can affect the rationality ofdoing so. Special attention is given to various ways in whichone might attempt to exert some measure of control over whatone believes and the normative status of the beliefs thatresult from the successful execution of such projects. I arguethat the lessons which emerge from thinking about the case ofbelief have important implications for the way we should thinkabout the rationality of a (...) number of other propositional attitudes,such as regret, desire, and fear. Finally,I suggest that a lack of clarity with respect to the relevant issueshas given rise to a number of rather serious philosophical mistakes. (shrink)

The pragmatic character of the Dutch book argument makes it unsuitable as an "epistemic" justification for the fundamental probabilist dogma that rational partial beliefs must conform to the axioms of probability. To secure an appropriately epistemic justification for this conclusion, one must explain what it means for a system of partial beliefs to accurately represent the state of the world, and then show that partial beliefs that violate the laws of probability are invariably less accurate than they could be otherwise. (...) The first task can be accomplished once we realize that the accuracy of systems of partial beliefs can be measured on a gradational scale that satisfies a small set of formal constraints, each of which has a sound epistemic motivation. When accuracy is measured in this way it can be shown that any system of degrees of belief that violates the axioms of probability can be replaced by an alternative system that obeys the axioms and yet is more accurate in every possible world. Since epistemically rational agents must strive to hold accurate beliefs, this establishes conformity with the axioms of probability as a norm of epistemic rationality whatever its prudential merits or defects might be. (shrink)

Additionally, the text shows how to develop computationally feasible methods to mesh with this framework.

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)

Evidentialism is a view about the conditions under which a person is epistemically justified in having a particular doxastic attitude toward a proposition. Evidentialism holds that the justified attitudes are determined entirely by the person's evidence. This is the traditional view of justification. It is now widely opposed. The essays included in this volume develop and defend the tradition.Evidentialism has many assets. In addition to providing an intuitively plausible account of epistemic justification, it helps to resolve the problem of the (...) criterion, helps to disentangle epistemic and ethical evaluations, and illuminates the relationship between epistemic evaluations of beliefs and the evaluation of the methods used to form beliefs. These issues are all addressed in the essays presented here. External world skepticism poses the classic problem for an epistemological theory. The final essay in this volume argues that evidentialism is uniquely well qualified to make sense of skepticism and to respond to its challenge.Evidentialism is a version of epistemic internalism. Recent epistemology has included many attacks on internalism and has seen the development of numerous externalist theories. The essays included here respond to those attacks and raise objections to externalist theories, especially the principal rival, reliabilism. Internalism generally has been criticized for having unacceptable deontological implications, for failing to connect epistemic justification to truth, and for failing to provide an adequate account of what makes basic beliefs justified. Each of these charges is answered in these essays. The collection includes two previously unpublished essays and new afterwords to five of the reprinted essays; it will be the definitive resource on evidentialism for all epistemologists. (shrink)

To the extent that we have reasons to avoid these “bad B -properties”, these arguments provide reasons not to have an incoherent credence function b — and perhaps even reasons to have a coherent one. But, note that these two traditional arguments for probabilism involve what might be called “pragmatic” reasons (not) to be (in)coherent. In the case of the Dutch Book argument, the “bad” property is pragmatically bad (to the extent that one values money). But, it is not clear (...) whether the DBA pinpoints any epistemic defect of incoherent agents. The same can be said for Representation Theorem arguments, since they involve the structure of an agent’s preferences. (shrink)

Evidentialism is a view about the conditions under which a person is epistemically justified in having a particular doxastic attitude toward a proposition. Evidentialism holds that the justified attitudes are determined entirely by the person's evidence. This is the traditional view of justification. It is now widely opposed. The essays included in this volume develop and defend the tradition. Evidentialism has many assets. In addition to providing an intuitively plausible account of epistemic justification, it helps to resolve the problem of (...) the criterion, helps to disentangle epistemic and ethical evaluations, and illuminates the relationship between epistemic evaluations of beliefs and the evaluation of the methods used to form beliefs. These issues are all addressed in the essays presented here. External world skepticism poses the classic problem for an epistemological theory. The final essay in this volume argues that evidentialism is uniquely well qualified to make sense of skepticism and to respond to its challenge. Evidentialism is a version of epistemic internalism. Recent epistemology has included many attacks on internalism and has seen the development of numerous externalist theories. The essays included here respond to those attacks and raise objections to externalist theories, especially the principal rival, reliabilism. Internalism generally has been criticized for having unacceptable deontological implications, for failing to connect epistemic justification to truth, and for failing to provide an adequate account of what makes basic beliefs justified. Each of these charges is answered in these essays. The collection includes two previously unpublished essays and new afterwords to five of the reprinted essays; it will be the definitive resource on evidentialism for all epistemologists. (shrink)

The most immediately appealing model for formal constraints on degrees of belief is provided by probability theory, which tells us, for instance, that the probability of P can never be greater than that of (P v Q). But while this model has much intuitive appeal, many have been concerned to provide arguments showing that ideally rational degrees of belief would conform to the calculus of probabilities. The arguments most frequently used to make this claim plausible are the so-called "Dutch Book" (...) arguments. (shrink)

We review de Finetti’s two coherence criteria for determinate probabilities: coherence1defined in terms of previsions for a set of events that are undominated by the status quo – previsions immune to a sure-loss – and coherence2 defined in terms of forecasts for events undominated in Brier score by a rival forecast. We propose a criterion of IP-coherence2 based on a generalization of Brier score for IP-forecasts that uses 1-sided, lower and upper, probability forecasts. However, whereas Brier score is a strictly (...) proper scoring rule for eliciting determinate probabilities, we show that there is no real-valuedstrictly proper IP-score. Nonetheless, with respect to either of two decision rules – Γ-maximin or E-admissibility-+-Γ-maximin – we give a lexicographic strictly proper IP-scoring rule that is based on Brier score. (shrink)

While in principle probabilistic logics might be applied to solve a range of problems, in practice they are rarely applied at present. This is perhaps because they seem disparate, complicated, and computationally intractable. However, we shall argue in this programmatic paper that several approaches to probabilistic logic into a simple unifying framework: logically complex evidence can be used to associate probability intervals or probabilities with sentences.