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.

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)

How should one react when one has a belief, but knows that other people—who have roughly the same evidence as one has, and seem roughly as likely to react to it correctly—disagree? This paper argues that the disagreement of other competent inquirers often requires one to be much less confident in one’s opinions than one would otherwise be.

AN INVESTIGATION OF THE LAWS OF THOUGHT. CHAPTER I. NATURE AND DESIGN OF THIS WORK. . HPHE design of the following treatise is to investigate the ...

:A normative Bayesian theory of deliberation and judgement requires a procedure for merging the evidence of a collection of agents. In order to provide such a procedure, one needs to ask what the evidence is that grounds Bayesian probabilities. After finding fault with several views on the nature of evidence, it is argued that evidence is whatever is rationally taken for granted. This view is shown to have consequences for an account of merging evidence, and it is argued that standard (...) axioms for merging need to be altered somewhat. (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.

The question of how the probabilistic opinions of different individuals should be aggregated to form a group opinion is controversial. But one assumption seems to be pretty much common ground: for a group of Bayesians, the representation of group opinion should itself be a unique probability distribution, 410–414, [45]; Bordley Management Science, 28, 1137–1148, [5]; Genest et al. The Annals of Statistics, 487–501, [21]; Genest and Zidek Statistical Science, 114–135, [23]; Mongin Journal of Economic Theory, 66, 313–351, [46]; Clemen and (...) Winkler Risk Analysis, 19, 187–203, [7]; Dietrich and List [14]; Herzberg Theory and Decision, 1–19, [28]). We argue that this assumption is not always in order. We show how to extend the canonical mathematical framework for pooling to cover pooling with imprecise probabilities by employing set-valued pooling functions and generalizing common pooling axioms accordingly. As a proof of concept, we then show that one IP construction satisfies a number of central pooling axioms that are not jointly satisfied by any of the standard pooling recipes on pain of triviality. Following Levi, 3–11, [39]), we also argue that IP models admit of a much better philosophical motivation as a model of rational consensus. (shrink)

In this paper, I begin by defending permissivism: the claim that, sometimes, there is more than one way to rationally respond to a given body of evidence. Then I argue that, if we accept permissivism, certain worries that arise as a result of learning that our beliefs were caused by the communities we grew up in, the schools we went to, or other irrelevant influences dissipate. The basic strategy is as follows: First, I try to pinpoint what makes irrelevant influences (...) worrying and I come up with two candidate principles. I then argue that one principle should be rejected because it is inconsistent with permissivism. The principle we should accept implies that it is sometimes rational to maintain our beliefs, even upon learning that they were caused by irrelevant influences. (shrink)

How should a group with different opinions (but the same values) make decisions? In a Bayesian setting, the natural question is how to aggregate credences: how to use a single credence function to naturally represent a collection of different credence functions. An extension of the standard Dutch-book arguments that apply to individual decision-makers recommends that group credences should be updated by conditionalization. This imposes a constraint on what aggregation rules can be like. Taking conditionalization as a basic constraint, we gather (...) lessons from the established work on credence aggregation, and extend this work with two new impossibility results. We then explore contrasting features of two kinds of rules that satisfy the constraints we articulate: one kind uses fixed prior credences, and the other uses geometric averaging, as opposed to arithmetic averaging. We also prove a new characterisation result for geometric averaging. Finally we consider applications to neighboring philosophical issues, including the epistemology of disagreement. (shrink)

Dilation occurs when an interval probability estimate of some event E is properly included in the interval probability estimate of E conditional on every event F of some partition, which means that one’s initial estimate of E becomes less precise no matter how an experiment turns out. Critics maintain that dilation is a pathological feature of imprecise probability models, while others have thought the problem is with Bayesian updating. However, two points are often overlooked: (1) knowing that E is stochastically (...) independent of F (for all F in a partition of the underlying state space) is sufficient to avoid dilation, but (2) stochastic independence is not the only independence concept at play within imprecise probability models. In this paper we give a simple characterization of dilation formulated in terms of deviation from stochastic independence, propose a measure of dilation, and distinguish between proper and improper dilation. Through this we revisit the most sensational examples of dilation, which play up independence between dilator and dilatee, and find the sensationalism undermined by either fallacious reasoning with imprecise probabilities or improperly constructed imprecise probability models. (shrink)

This chapter defines and defends time-slice epistemology, according to which there are no essentially diachronic norms of rationality. The chapter begins by distinguishing two notions of time-slice epistemology, and ends by defending time-slice theories of action under indeterminacy, i.e. theories about how you should act when the outcome of your decision depends on some indeterminate claim. In a recent chapter, J. Robert G. Williams defends a theory of action under indeterminacy which is subject to several objections. An alternative theory is (...) proposed in its place. The resulting discussion highlights a more general moral about action under indeterminacy, namely that time-slice theories are supported by strong analogies with ethical theories. In particular, our understanding of agents torn between interpretations of a decision situation should be guided by our theories of agents torn between incommensurable values. (shrink)

This major work challenges some widely held positions in epistemology - those of Peirce and Popper on the one hand and those of Quine and Kuhn on the other.

Permissivism about rationality is the view that there is sometimes more than one rational response to a given body of evidence. In this paper I discuss the relationship between permissivism, deference to rationality, and peer disagreement. I begin by arguing that—contrary to popular opinion—permissivism supports at least a moderate version of conciliationism. I then formulate a worry for permissivism. I show that, given a plausible principle of rational deference, permissive rationality seems to become unstable and to collapse into unique rationality. (...) I conclude with a formulation of a way out of this problem on behalf of the permissivist. (shrink)

Recent authors have drawn attention to a new kind of defeating evidence commonly referred to as higher-order evidence. Such evidence works by inducing doubts that one’s doxastic state is the result of a flawed process – for instance, a process brought about by a reason-distorting drug. I argue that accommodating defeat by higher-order evidence requires a two-tiered theory of justification, and that the phenomenon gives rise to a puzzle. The puzzle is that at least in some situations involving higher-order defeaters (...) the correct epistemic rules issue conflicting recommendations. For instance, a subject ought to believe p, but she ought also to suspend judgment in p. I discuss three responses. The first resists the puzzle by arguing that there is only one correct epistemic rule, an Über-rule. The second accepts that there are genuine epistemic dilemmas. The third appeals to a hierarchy or ordering of correct epistemic rules. I spell out problems for all of these responses. I conclude that the right lesson to draw from the puzzle is that a state can be epistemically rational or justified even if one has what looks to be strong evidence to think that it is not. As such, the considerations put forth constitute a non question-begging argument for a kind of externalism. (shrink)

In this essay, I present a straightforward counterexample to the Uniqueness Thesis, which holds, roughly speaking, that there is a unique rational response to any particular body of evidence.

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

How should you take into account the opinions of an advisor? When you completely defer to the advisor's judgment, then you should treat the advisor as a guru. Roughly, that means you should believe what you expect she would believe, if supplied with your extra evidence. When the advisor is your own future self, the resulting principle amounts to a version of the Reflection Principle---a version amended to handle cases of information loss. When you count an advisor as an epistemic (...) peer, you should give her conclusions the same weight as your own. Denying that view---call it the ``equal weight view''---leads to absurdity: the absurdity that you could reasonably come to believe yourself to be an epistemic superior to an advisor simply by noting cases of disagreement with her, and taking it that she made most of the mistakes. Accepting the view seems to lead to another absurdity: that one should suspend judgment about everything that one's smart and well-informed friends disagree on, which means suspending judgment about almost everything interesting. But despite appearances, the equal weight view does not have this absurd consequence. Furthermore, the view can be generalized to handle cases involving not just epistemic peers, but also epistemic superiors and inferiors. (shrink)

It has been claimed that epistemic peers, upon discovering that they disagree on some issue, should give up their opposing views and ‘split the difference’. The present paper challenges this claim by showing, with the help of computer simulations, that what the rational response to the discovery of peer disagreement is—whether it is sticking to one’s belief or splitting the difference—depends on factors that are contingent and highly context-sensitive.Keywords: Peer disagreement; Computer simulations; Opinion dynamics; Hegselmann–Krause model; Social epistemology.

This paper analyzes concepts of independence and assumptions of convexity in the theory of sets of probability distributions. The starting point is Kyburg and Pittarelli’s discussion of “convex Bayesianism” (in particular their proposals concerning E-admissibility, independence, and convexity). The paper offers an organized review of the literature on independence for sets of probability distributions; new results on graphoid properties and on the justification of “strong independence” (using exchangeability) are presented. Finally, the connection between Kyburg and Pittarelli’s results and recent developments (...) on the axiomatization of non-binary preferences, and its impact on “complete” independence, are described. (shrink)

How much should your confidence in your beliefs be shaken when you learn that others – perhaps ‘epistemic peers’ who seem as well-qualified as you are – hold beliefs contrary to yours? This article describes motivations that push different philosophers towards opposite answers to this question. It identifies a key theoretical principle that divides current writers on the epistemology of disagreement. It then examines arguments bearing on that principle, and on the wider issue. It ends by describing some outstanding questions (...) that thinking about this issue raises. (shrink)

Sometimes we get evidence of our own epistemic malfunction. This can come from finding out we’re fatigued, or have been drugged, or that other competent and well-informed thinkers disagree with our beliefs. This sort of evidence seems to seems to behave differently from ordinary evidence about the world. In particular, getting such evidence can put agents in a position where the most rational response involves violating some epistemic ideal.

How much should your confidence in your beliefs be shaken when you learn that others – perhaps 'epistemic peers' who seem as well-qualified as you are – hold beliefs contrary to yours? This article describes motivations that push different philosophers towards opposite answers to this question. It identifies a key theoretical principle that divides current writers on the epistemology of disagreement. It then examines arguments bearing on that principle, and on the wider issue. It ends by describing some outstanding questions (...) that thinking about this issue raises. (shrink)

Responding rationally to the information that others disagree with one’s beliefs requires assessing the epistemic credentials of the opposing beliefs. Conciliatory accounts of disagreement flow in part from holding that these assessments must be independent from one’s own initial reasoning on the disputed matter. I argue that this claim, properly understood, does not have the untoward consequences some have worried about. Moreover, some of the difficulties it does engender must be faced by many less conciliatory accounts of disagreement.

Epistemic decision theory (EDT) employs the mathematical tools of rational choice theory to justify epistemic norms, including probabilism, conditionalization, and the Principal Principle, among others. Practitioners of EDT endorse two theses: (1) epistemic value is distinct from subjective preference, and (2) belief and epistemic value can be numerically quantified. We argue the first thesis, which we call epistemic puritanism, undermines the second.

Two theses figure centrally in work on the epistemology of disagreement: Equal Weight (‘EW’) and Uniqueness (‘U’). According to EW, you should give precisely as much weight to the attitude of a disagreeing epistemic peer as you give to your own attitude. U has it that, for any given proposition and total body of evidence, some doxastic attitude is the one the evidence makes rational (justifies) toward that proposition. Although EW has received considerable discussion, the case for U has not (...) been critically evaluated. Endorsing U, we argue, commits one to the highly controversial thesis that whatever fixes your rational attitudes can do so only by fixing what evidence you have. This commitment imposes a relatively demanding requirement on justified belief in U, one that we argue is not satisfied by what is currently the strongest available case for U, due to Roger White [2005]. Our challenge to U makes more trouble for its proponents than do the worries about U expressed by Gideon Rosen [2001] and Thomas Kelly [2010]. Moreover, if Kelly [2010] is correct in thinking that EW “carries with it a commitment to” U—a claim which we accept for reasons similar to Kelly’s but is beyond this paper’s scope (but see Ballantyne and Coffman [forthcoming])—then our challenge to U bears importantly on EW: to the extent that our challenge to U succeeds, EW also suffers. (shrink)

Various authors have recently argued that you cannot rationally stick to your belief in the face of known disagreement with an epistemic peer, that is, a person you take to have the same evidence and judgmental skills as you do. For, they claim, because there is but one rational response to any body of evidence, a disagreement with an epistemic peer indicates that at least one of you is not responding rationally to the evidence. Given that you take your peer (...) to have the same judgmental skills as you do, and thus regard her to be equally good at assessing the evidence as you are, you will have as much reason for thinking that it is you who is not responding rationally to the evidence as for thinking that it is her. You thus have reason for thinking that your belief on the disputed matter is not a rational response to the evidence. Hence, you cannot rationally stick to your belief. (shrink)