Suppose several individuals (e.g., experts on a panel) each assign probabilities to some events. How can these individual probability assignments be aggregated into a single collective probability assignment? This article reviews several proposed solutions to this problem. We focus on three salient proposals: linear pooling (the weighted or unweighted linear averaging of probabilities), geometric pooling (the weighted or unweighted geometric averaging of probabilities), and multiplicative pooling (where probabilities are multiplied rather than averaged). We present axiomatic characterisations of each class of (...) pooling functions (most of them classic, but one new) and argue that linear pooling can be justified procedurally, but not epistemically, while the other two pooling methods can be justified epistemically. The choice between them, in turn, depends on whether the individuals' probability assignments are based on shared information or on private information. We conclude by mentioning a number of other pooling methods. (shrink)

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)

Epistemic utility theory seeks to establish epistemic norms by combining principles from decision theory and social choice theory with ways of determining the epistemic utility of agents’ attitudes. Recently, Moss, 1053–69, 2011) has applied this strategy to the problem of finding epistemic compromises between disagreeing agents. She shows that the norm “form compromises by maximizing average expected epistemic utility”, when applied to agents who share the same proper epistemic utility function, yields the result that agents must form compromises by splitting (...) the difference between their credence functions. However, this “split the difference” norm is in conflict with conditionalization, since applications of the two norms don’t commute. A common response in the literature seems to be to abandon the procedure of splitting the difference in favor of compromise strategies that avoid non-commutativity. This would also entail abandoning Moss’ norm. I explore whether a different response is feasible. If agents can use epistemic utility-based considerations to agree on an order in which they will apply the two norms, they might be able to avoid diachronic incoherence. I show that this response can’t save Moss’ norm, because the agreements concerning the order of compromising and updating it generates are not stable over time, and hence cannot avoid diachronic incoherence. I also show that a variant of Moss’ norm, which requires that the weights given to each agent’s epistemic utility change in a way that ensures commutativity, cannot be justified on epistemological grounds. (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)

The Herald of Free Enterprise, a ferry operating in the English Channel, sank on March 6, 1987, drowning nearly two hundred people. The official inquiry found that the company running the ferry was extremely sloppy, with poor routines of checking and management. “From top to bottom the body corporate was infected with the disease of sloppiness.”1 But the courts did not penalize anyone in what might seem to be an appropriate measure, failing to identify individuals in the company or on (...) the ship itself who were seriously enough at fault. As one commentator put it, “The primary requirement of finding an individual who was liable . . . stood in the way of attaching any significance to the organizational sloppiness that had been found by the official inquiry.”2 In a case like this it can make good sense to hold that while the individuals involved may not bear a high degree of personal responsibility, together as a corporate enterprise they should carry full responsibility for what occurred. Although the members may not fully satisfy the conditions for being held personally responsible—although there are mitigating circumstances that excuse them in some mea-. (shrink)

It is commonly assumed that when we assign different credences to a proposition, a perfect compromise between our opinions simply ‘splits the difference’ between our credences. I introduce and defend an alternative account, namely that a perfect compromise maximizes the average of the expected epistemic values that we each assign to alternative credences in the disputed proposition. I compare the compromise strategy I introduce with the traditional strategy of compromising by splitting the difference, and I argue that my strategy is (...) a reasonable characterization of epistemic compromise. (shrink)

Leitgeb and Pettigrew argue that (1) agents should minimize the expected inaccuracy of their beliefs and (2) inaccuracy should be measured via the Brier score. They show that in certain diachronic cases, these claims require an alternative to Jeffrey Conditionalization. I claim that this alternative is an irrational updating procedure and that the Brier score, and quadratic scoring rules generally, should be rejected as legitimate measures of inaccuracy.

It is well known that aggregating the degree-of-belief functions of different subjects by linear pooling or averaging is subject to a commutativity dilemma: other than in trivial cases, conditionalizing the individual degree-of-belief functions on a piece of evidence E followed by linearly aggregating them does not yield the same result as rst aggregating them linearly and then conditionalizing the resulting social degree- of-belief function on E. In the present paper we suggest a novel way out of this dilemma: adapting the (...) method of update or learning such that linear pooling com- mutes with it. As it turns out, the resulting update scheme – imaging on the evidence – is well-known from areas such as the study of conditionals and cau- sal decision theory, and a formal result from which the required commutativity property is derivable was supplied already by Gärdenfors in a different con- text. We end up determining under which conditions imaging would seem to be right method of update, and under which conditions, therefore, group update would not be affected by the commutativity dilemma. (shrink)

It is well known that aggregating the degree-of-belief functions of different subjects by linear pooling or averaging is subject to a commutativity dilemma: other than in trivial cases, conditionalizing the individual degree-of-belief functions on a piece of evidence E followed by linearly aggregating them does not yield the same result as rst aggregating them linearly and then conditionalizing the resulting social degree- of-belief function on E. In the present paper we suggest a novel way out of this dilemma: adapting the (...) method of update or learning such that linear pooling com- mutes with it. As it turns out, the resulting update scheme – imaging on the evidence – is well-known from areas such as the study of conditionals and cau- sal decision theory, and a formal result from which the required commutativity property is derivable was supplied already by Gärdenfors in a different con- text. We end up determining under which conditions imaging would seem to be right method of update, and under which conditions, therefore, group update would not be affected by the commutativity dilemma. (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 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)

In this paper we propose and analyze a game-theoretic model of the epistemology of peer disagreement. In this model, the peers' rationality is evaluated in terms of their probability of ending the disagreement with a true belief. We find that different strategies---in particular, one based on the Steadfast View and one based on the Conciliatory View---are rational depending on the truth-sensitivity of the individuals involved in the disagreement. Interestingly, the Steadfast and the Conciliatory Views can even be rational simultaneously in (...) some circumstances. We tentatively provide some reasons to favor the Conciliatory View in such cases. We argue that the game-theoretic perspective is a fruitful one in this debate, and this fruitfulness has not been exhausted by the present paper. (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)

How should the opinion of a group be related to the opinions of the group members? In this article, we will defend a package of four norms – coherence, locality, anonymity and unanimity. Existing results show that there is no tenable procedure for aggregating outright beliefs or for aggregating credences that meet these criteria. In response, we consider the prospects for aggregating credal pairs – pairs of prior probabilities and evidence. We show that there is a method of aggregating credal (...) pairs that possesses all four virtues. (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.

Incorporated groups include businesses, universities, churches and the like. Organized to act as single centers of agency, they also routinely satisfy the three conditions that make an agent fit to be held responsible: they face significant choices, can recognize the relative value of different options, and are able to choose in sensitivity to such values. But is it redundant to hold a corporate agent responsible for something, when certain members are also held responsible for the individual parts they play? No (...) it is not, for it is often possible for a corporate entity to be fully fit to be held responsible, when this is not true of the individual members; they may be able to make excuses that are not available at the corporate level. Does the case made for corporate responsibility extend to unincorporated collectivities like nations or religions? Not strictly but it does explain why it may be sensible to treat those collectivities as if they had corporate responsibility in certain domains. (shrink)

De Finetti introduced the concept of coherent previsions and conditional previsions through a gambling argument and through a parallel argument based on a quadratic scoring rule. He shows that the two arguments lead to the same concept of coherence. When dealing with events only, there is a rich class of scoring rules which might be used in place of the quadratic scoring rule. We give conditions under which a general strictly proper scoring rule can replace the quadratic scoring rule while (...) preserving the equivalence of de Finetti’s two arguments. In proving our results, we present a strengthening of the usual minimax theorem. We also present generalizations of de Finetti’s fundamental theorem of prevision to deal with conditional previsions. (shrink)