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.

Richard Pettigrew offers an extended investigation into a particular way of justifying the rational principles that govern our credences. The main principles that he justifies are the central tenets of Bayesian epistemology, though many other related principles are discussed along the way. Pettigrew looks to decision theory in order to ground his argument. He treats an agent's credences as if they were a choice she makes between different options, gives an account of the purely epistemic utility enjoyed by different sets (...) of credences, and then appeals to the principles of decision theory to show that, when epistemic utility is measured in this way, the credences that violate the principles listed above are ruled out as irrational. The account of epistemic utility set out here is the veritist's: the sole fundamental source of epistemic utility for credences is their accuracy. Thus, Pettigrew conducts an investigation in the version of epistemic utility theory known as accuracy-first epistemology. (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)

This is the classic work upon which modern-day game theory is based. What began as a modest proposal that a mathematician and an economist write a short paper together blossomed, when Princeton University Press published Theory of Games and Economic Behavior. In it, John von Neumann and Oskar Morgenstern conceived a groundbreaking mathematical theory of economic and social organization, based on a theory of games of strategy. Not only would this revolutionize economics, but the entirely new field of scientific inquiry (...) it yielded--game theory--has since been widely used to analyze a host of real-world phenomena from arms races to optimal policy choices of presidential candidates, from vaccination policy to major league baseball salary negotiations. And it is today established throughout both the social sciences and a wide range of other sciences. (shrink)

The literature of economic theory, like that of philosophy, abounds in prefaces and prolegomena. Methodology and analysis of concepts take an important place in a science which has not found the sure path of development. But there is no sure path for methodology either. The selfconscious methodology of social science has been largely a borrowing from that of physical science, where procedures have developed to a stage of considerable maturity. But the analogy falls down where guidance is most needed, at (...) the points where social science is most likely to develop new concepts and new types of structure. Philosophers have not been lacking, indeed, to belittle the entire enterprise, and to deny the possibility of anything that could strictly be called social science. (shrink)

Constrained coherence is compared to coherence and its role in the behavioural interpretation of coherence is discussed. The equivalence of these two notions is proven for coherent conditional previsions, showing that the same course of reasoning applies to several similar concepts developed in the realm of imprecise probability theory.

There are decision problems where the preferences that seem rational to many people cannot be accommodated within orthodox decision theory in the natural way. In response, a number of alternatives to the orthodoxy have been proposed. In this paper, I offer an argument against those alternatives and in favour of the orthodoxy. I focus on preferences that seem to encode sensitivity to risk. And I focus on the alternative to the orthodoxy proposed by Lara Buchak’s risk-weighted expected utility theory. I (...) will show that the orthodoxy can be made to accommodate all of the preferences that Buchak’s theory can accommodate. (shrink)

Beliefs come in different strengths. An agent's credence in a proposition is a measure of the strength of her belief in that proposition. Various norms for credences have been proposed. Traditionally, philosophers have tried to argue for these norms by showing that any agent who violates them will be lead by her credences to make bad decisions. In this article, we survey a new strategy for justifying these norms. The strategy begins by identifying an epistemic utility function and a decision-theoretic (...) norm; we then show that the decision-theoretic norm applied to the epistemic utility function yields the norm for credences that we wish to justify. We survey results already obtained using this strategy, and we suggest directions for future research. (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)

I introduce a mathematical account of expectation based on a qualitative criterion of coherence for qualitative comparisons between gambles (or random quantities). The qualitative comparisons may be interpreted as an agent’s comparative preference judgments over options or more directly as an agent’s comparative expectation judgments over random quantities. The criterion of coherence is reminiscent of de Finetti’s quantitative criterion of coherence for betting, yet it does not impose an Archimedean condition on an agent’s comparative judgments, it does not require the (...) binary relation reflecting an agent’s comparative judgments to be reflexive, complete or even transitive, and it applies to an absolutely arbitrary collection of gambles, free of structural conditions (e.g., closure, measurability, etc.). Moreover, unlike de Finetti’s criterion of coherence, the qualitative criterion respects the principle of weak dominance, a standard of rational decision making that obliges an agent to reject a gamble that is possibly worse and certainly no better than another gamble available for choice. Despite these weak assumptions, I establish a qualitative analogue of de Finetti’s Fundamental Theorem of Prevision, from which it follows that any coherent system of comparative expectations can be extended to a weakly ordered coherent system of comparative expectations over any collection of gambles containing the initial set of gambles of interest. The extended weakly ordered coherent system of comparative expectations satisfies familiar additivity and scale invariance postulates (i.e., independence) when the extended collection forms a linear space. In the course of these developments, I recast de Finetti’s quantitative account of coherent prevision in the qualitative framework adopted in this article. I show that comparative previsions satisfy qualitative analogues of de Finetti’s famous bookmaking theorem and his Fundamental Theorem of Prevision.The results of this article complement those of another article (Pedersen, Strictly coherent preferences, no holds barred, Manuscript, 2013). I explain how those results entail that any coherent weakly ordered system of comparative expectations over a unital linear space can be represented by an expectation function taking values in a (possibly non-Archimedean) totally ordered field extension of the system of real numbers. The ordered field extension consists of formal power series in a single infinitesimal, a natural and economical representation that provides a relief map tracing numerical non-Archimedean features to qualitative non-Archimedean features. (shrink)

Kuhn’s famous thesis that there is ‘no unique algorithm’ for choosing between rival scientific theories is analysed using the machinery of social choice theory. It is shown that the problem of theory choice as posed by Kuhn is formally identical to a standard social choice problem. This suggests that analogues of well-known results from the social choice literature, such as Arrow’s impossibility theorem, may apply to theory choice. If an analogue of Arrow’s theorem does hold for theory choice this would (...) refute Kuhn’s thesis, but it would also pose a threat to the rationality of science, a threat that is if anything more worrying than that posed by Kuhn. Various possible ‘escape routes’ from Arrow’s impossibility result are examined, in particular Amartya Sen’s idea of ‘enriching the informational basis’. It is shown that Sen’s idea can be applied to the problem of theory choice in science. This in turn sheds light on two well-known approaches to inductive inference in philosophy of science: Bayesianism and statistical model selection. (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)

Jim Joyce argues for two amendments to probabilism. The first is the doctrine that credences are rational, or not, in virtue of their accuracy or “closeness to the truth” (1998). The second is a shift from a numerically precise model of belief to an imprecise model represented by a set of probability functions (2010). We argue that both amendments cannot be satisfied simultaneously. To do so, we employ a (slightly-generalized) impossibility theorem of Seidenfeld, Schervish, and Kadane (2012), who show that (...) there is no strictly proper scoring rule for imprecise probabilities. -/- The question then is what should give way. Joyce, who is well aware of this no-go result, thinks that a quantifiability constraint on epistemic accuracy should be relaxed to accommodate imprecision. We argue instead that another Joycean assumption — called strict immodesty— should be rejected, and we prove a representation theorem that characterizes all “mildly” immodest measures of inaccuracy. (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)

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)

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)

This paper argues for the existence of a fourth positive generic value relation that can hold between two items beyond ‘better than’, ‘worse than’, and ‘equally good’: namely ‘on a par’.

Naive versions of decision theory take probabilities and utilities as primitive and use expected value to give norms on rational decision. However, standard decision theory takes rational preference as primitive and uses it to construct probability and utility. This paper shows how to justify a version of the naive theory, by taking dominance as the most basic normatively required preference relation, and then extending it by various conditions under which agents should be indifferent between acts. The resulting theory can make (...) all the decisions of classical expected utility theory, plus more in cases where expected utilities are infinite or undefined. Although the theory requires similarly strong assumptions to classical expected utility theory, versions of the theory can be developed with slightly weaker assumptions, without having to prove a new representation theorem for the weaker theory. This alternate foundation is particularly useful if probability is prior to preference, as suggested by the recent program to base probabilism on accuracy and alethic considerations rather than pragmatic ones. (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.

According to accuracy-first epistemology, accuracy is the fundamental epistemic good. Epistemic norms — Probabilism, Conditionalization, the Principal Principle, etc. — have their binding force in virtue of helping to secure this good. To make this idea precise, accuracy-firsters invoke Epistemic Decision Theory (EpDT) to determine which epistemic policies are the best means toward the end of accuracy. Hilary Greaves and others have recently challenged the tenability of this programme. Their arguments purport to show that EpDT encourages obviously epistemically irrational behavior. (...) We develop firmer conceptual foundations for EpDT. First, we detail a theory of praxic and epistemic good. Then we show that, in light of their very different good-making features, EpDT will evaluate epistemic states and epistemic acts according to different criteria. So, in general, rational preference over states and acts won’t agree. Finally, we argue that based on direction-of-fit considerations, it’s preferences over the former that matter for normative epistemology, and that EpDT, properly spelt out, arrives at the correct verdicts in a range of putative problem cases. (shrink)

We provide self-contained proof of a theorem relating probabilistic coherence of forecasts to their non-domination by rival forecasts with respect to any proper scoring rule. The theorem recapitulates insights achieved by other investigators, and clarifi es the connection of coherence and proper scoring rules to Bregman divergence.

We investigate differences between a simple Dominance Principle applied to sums of fair prices for variables and dominance applied to sums of forecasts for variables scored by proper scoring rules. In particular, we consider differences when fair prices and forecasts correspond to finitely additive expectations and dominance is applied with infinitely many prices and/or forecasts.

This thesis develops a general theory of value which explains and supports the claim that goods differ in kind. It is claimed that goods differ in kind in two related ways: insofar as they are worthy of different modes of valuation, such as respect, use, and appreciation, and insofar as the social relations within which they are properly produced, distributed, and enjoyed differ. Ideals, which are conceptions of higher modes of life, inform the ways in which we understand goods as (...) belonging to different kinds. This theory undermines consequentialist theories of practical reasoning by showing that such theories cannot properly take account of the heterogeneity of values. It proves superior to theories of homogeneous value, such as hedonism, in accounting for the full range of our experiences of value, which include such phenomena as degradation, grief, and pride. The theory of value forms the basis for an investigation of the nature of economic goods, as distinguished from other sorts of goods such as personal and political goods. By understanding the differences between economic goods and other kinds of goods, it is possible to determine which goods are properly traded on the market, or otherwise treated as commodities. The practice of commercial surrogate parenting and the application of cost-benefit analysis to safety and environmental issues is criticized in the light of this investigation. (shrink)