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.

(This is for the series Elements of Decision Theory published by Cambridge University Press and edited by Martin Peterson) -/- Our beliefs come in degrees. I believe some things more strongly than I believe others. I believe very strongly that global temperatures will continue to rise during the coming century; I believe slightly less strongly that the European Union will still exist in 2029; and I believe much less strongly that Cardiff is east of Edinburgh. My credence in something is (...) a measure of the strength of my belief in it; it represents my level of confidence in it. These are the states of mind we report when we say things like ‘I’m 20% confident I switch off the gas before I left' or ‘I’m 99.9% confident that it is raining outside'. -/- There are laws that govern these credences. For instance, I shouldn't be more confident that sea levels will rise by over 2 metres in the next 100 years than I am that they'll rise by over 1 metre, since the latter is true if the former is. This book is about a particular way we might try to establish these laws of credence: the Dutch Book arguments (For briefer overviews of these arguments, see Alan Hájek’s entry in the Oxford Handbook of Rational and Social Choice and Susan Vineberg’s entry in the Stanford Encyclopaedia.) -/- We begin, in Chapter 2, with the standard formulation of the various Dutch Book arguments that we'll consider: arguments for Probabilism, Countable Additivity, Regularity, and the Principal Principle. In Chapter 3, we subject this standard formulation to rigorous stress-testing, and make some small adjustments so that it can withstand various objections. What we are left with is still recognisably the orthodox Dutch Book argument. In Chapter 4, we set out the Dutch Strategy argument for Conditionalization. In Chapters 5 and 6, we consider two objections to Dutch Book arguments that cannot be addressed by making small adjustments. Instead, we must completely redesign those arguments, replacing them with ones that share a general approach but few specific details. In Chapter 7, we consider a further objection to which I do not have a response. In Chapter 8, we'll ask what happens to the Dutch Book arguments if we change certain features of the basic framework in which we've been working: first, we ask how Dutch Book arguments fare when we consider credences in self-locating propositions, such as It is Monday; second, we lift the assumption that the background logic is classical and explore Dutch Book arguments for non-classical logics; third, we lift the assumption that an agent's credal state can be represented by a single assignment of numerical values to the propositions she considers. In Chapter 9, we present the mathematical results that underpin these arguments. (shrink)

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)

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)

Taking Joyce’s (1998; 2009) recent argument(s) for probabilism as our point of departure, we propose a new way of grounding formal, synchronic, epistemic coherence requirements for (opinionated) full belief. Our approach yields principled alternatives to deductive consistency, sheds new light on the preface and lottery paradoxes, and reveals novel conceptual connections between alethic and evidential epistemic norms.