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)

How can the propositional attitudes of several individuals be aggregated into overall collective propositional attitudes? Although there are large bodies of work on the aggregation of various special kinds of propositional attitudes, such as preferences, judgments, probabilities and utilities, the aggregation of propositional attitudes is seldom studied in full generality. In this paper, we seek to contribute to …lling this gap in the literature. We sketch the ingredients of a general theory of propositional attitude aggregation and prove two new theorems. (...) Our …rst theorem simultaneously characterizes some prominent aggregation rules in the cases of probability, judgment and preference aggregation, including linear opinion pooling and Arrovian dictatorships. Our second theorem abstracts even further from the speci…c kinds of attitudes in question and describes the properties of a large class of aggregation rules applicable to a variety of belief-like attitudes. Our approach integrates some previously disconnected areas of investigation. (shrink)

The article proceeds upon the assumption that the beliefs and degrees of belief of rational agents satisfy a number of constraints, including: consistency and deductive closure for belief sets, conformity to the axioms of probability for degrees of belief, and the Lockean Thesis concerning the relationship between belief and degree of belief. Assuming that the beliefs and degrees of belief of both individuals and collectives satisfy the preceding three constraints, I discuss what further constraints may be imposed on the aggregation (...) of beliefs and degrees of belief. Some possibility and impossibility results are presented. The possibility results suggest that the three proposed rationality constraints are compatible with reasonable aggregation procedures for belief and degree of belief. (shrink)

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)

This paper explores the fact that linear opinion pooling can be represented as a Bayesian update on the opinions of others. It uses this fact to propose a new interpretation of the pooling weights. Relative to certain modelling assumptions the weights can be equated with the so-called truth-conduciveness known from the context of Condorcet's jury theorem. This suggests a novel way to elicit the weights.

This paper focuses on radical pooling, or the question of how to aggregate credences when there is a fundamental disagreement about which is the relevant logical space for inquiry. The solution advanced is based on the notion of consensus as common ground, where agents can find it by suspending judgment on logical possibilities. This is exemplified with cases of scientific revolution. On a formal level, the proposal uses algebraic joins and imprecise probabilities; which is shown to be compatible with the (...) principles of marginalization, rigidity, reverse bayesianism, and minimum divergence commonly endorsed in these contexts. Furthermore, I extend results from previous work by to show that pooling sets of imprecise probabilities can satisfy important pooling axioms. (shrink)

Much of our information comes to us indirectly, in the form of conclusions others have drawn from evidence they gathered. When we hear these conclusions, how can we modify our own opinions so as to gain the benefit of their evidence? In this paper we study the method known as geometric pooling. We consider two arguments in its favour, raising several objections to one, and proposing an amendment to the other.

This paper provides an introductory review of the theory of judgment aggregation. It introduces the paradoxes of majority voting that originally motivated the field, explains several key results on the impossibility of propositionwise judgment aggregation, presents a pedagogical proof of one of those results, discusses escape routes from the impossibility and relates judgment aggregation to some other salient aggregation problems, such as preference aggregation, abstract aggregation and probability aggregation. The present illustrative rather than exhaustive review is intended to give readers (...) new to the field of judgment aggregation a sense of this rapidly growing research area. (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)

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)

Formal epistemologists criticise the Conciliatory View of peer disagreement for being non-commutative with conditionalisation, path dependent and does not preserve the independence between propositions. Failing to commute with conditionalisation, one may switch the order between conciliating and conditionalising and obtain different outcomes. Failing to be path independent, the outcome of conciliation varies with the order of the acquisition of new testimonies. Failing to preserve the independence between propositions, one may suffer from a sure-loss and hence be deemed irrational. The three (...) formal deficiencies urge people to abandon the Conciliatory View. This paper aims to show that one may save the Conciliatory View by conciliating with nonlinear functions. Research in the study of opinion pooling shows that the three deficiencies are not problems of the Conciliatory View, but problems of linear averaging. Hence, one can get rid of these formal deficiencies by making conciliation with nonlinear averaging functions. After showing how the three deficiencies can be avoided, I will explore the features of nonlinear averaging functions and argue that they have properties that correctly capture people’s intuition concerning disagreement. The conclusion, therefore, is to suggest epistemologists develop a more fine-grained taxonomy for cases of disagreement. With a deliberate categorisation of different kinds of disagreement, epistemologists can pick the proper averaging rule to apply in each specific case, and get rid of possible formal deficiencies. (shrink)

How can different individuals' probability assignments to some events be aggregated into a collective probability assignment? Classic results on this problem assume that the set of relevant events -- the agenda -- is a sigma-algebra and is thus closed under disjunction (union) and conjunction (intersection). We drop this demanding assumption and explore probabilistic opinion pooling on general agendas. One might be interested in the probability of rain and that of an interest-rate increase, but not in the probability of rain or (...) an interest-rate increase. We characterize linear pooling and neutral pooling for general agendas, with classic results as special cases for agendas that are sigma-algebras. As an illustrative application, we also consider probabilistic preference aggregation. Finally, we compare our results with existing results on binary judgment aggregation and Arrovian preference aggregation. This paper is the first of two self-contained, but technically related companion papers inspired by binary judgment-aggregation theory. (shrink)

A group is often construed as one agent with its own probabilistic beliefs (credences), which are obtained by aggregating those of the individuals, for instance through averaging. In their celebrated “Groupthink”, Russell et al. (2015) require group credences to undergo Bayesian revision whenever new information is learnt, i.e., whenever individual credences undergo Bayesian revision based on this information. To obtain a fully Bayesian group, one should often extend this requirement to non-public or even private information (learnt by not all or (...) just one individual), or to non-representable information (not representable by any event in the domain where credences are held). I pro- pose a taxonomy of six types of ‘group Bayesianism’. They differ in the information for which Bayesian revision of group credences is required: public representable information, private representable information, public non-representable information, etc. Six corre- sponding theorems establish how individual credences must (not) be aggregated to ensure group Bayesianism of any type, respectively. Aggregating through standard averaging is never permitted; instead, different forms of geometric averaging must be used. One theorem—that for public representable information—is essentially Russell et al.’s central result (with minor corrections). Another theorem—that for public non-representable information—fills a gap in the theory of externally Bayesian opinion pooling. (shrink)

Judgment aggregation studies how collective opinions arise from the aggregation of individual ones. This article surveys a variety of aggregation rules (possible ways of aggregating individual judgments into collective ones). Aggregation by majority opinion is known to satisfy some but not all the desiderata for an aggregation rule. More general impossibility results show that not all the natural desiderata can be satisfied by a single aggregation rule. To interpret these results, we focus here on some applications of judgment aggregation models (...) in Epistemology. In particular, we explore their possible role in an account of collective belief and collective reason. The focus on these applications allows us to give precise answers concerning which of the natural desiderata must be abandoned. (shrink)

Supra-Bayesianism is the Bayesian response to learning the opinions of others. Probability pooling constitutes an alternative response. One natural question is whether there are cases where probability pooling gives the supra-Bayesian result. This has been called the problem of Bayes-compatibility for pooling functions. It is known that in a common prior setting, under standard assumptions, linear pooling cannot be nontrivially Bayes-compatible. We show by contrast that geometric pooling can be nontrivially Bayes-compatible. Indeed, we show that, under certain assumptions, geometric and (...) Bayes-compatible pooling are equivalent. Granting supra-Bayesianism its usual normative status, one upshot of our study is thus that, in a certain class of epistemic contexts, geometric pooling enjoys a normative advantage over linear pooling as a social learning mechanism. We discuss the philosophical ramifications of this advantage, which we show to be robust to variations in our statement of the Bayes-compatibility problem. (shrink)

Judgment-aggregation theory has always focused on the attainment of rational collective judgments. But so far, rationality has been understood in static terms: as coherence of judgments at a given time, defined as consistency, completeness, and/or deductive closure. This paper asks whether collective judgments can be dynamically rational, so that they change rationally in response to new information. Formally, a judgment aggregation rule is dynamically rational with respect to a given revision operator if, whenever all individuals revise their judgments in light (...) of some information (a learnt proposition), then the new aggregate judgments are the old ones revised in light of this information, i.e., aggregation and revision commute. We prove an impossibility theorem: if the propositions on the agenda are non-trivially connected, no judgment aggregation rule with standard properties is dynamically rational with respect to any revision operator satisfying some basic conditions on revision. Our theorem is the dynamic-rationality counterpart of some well-known impossibility theorems for static rationality. We also explore how dynamic rationality might be achieved by relaxing some of the conditions on the aggregation rule and/or the revision operator. Notably, premise-based aggregation rules are dynamically rational with respect to so-called premise-based revision operators. (shrink)

Social choice theory is the study of collective decision processes and procedures. It is not a single theory, but a cluster of models and results concerning the aggregation of individual inputs (e.g., votes, preferences, judgments, welfare) into collective outputs (e.g., collective decisions, preferences, judgments, welfare). Central questions are: How can a group of individuals choose a winning outcome (e.g., policy, electoral candidate) from a given set of options? What are the properties of different voting systems? When is a voting system (...) democratic? How can a collective (e.g., electorate, legislature, collegial court, expert panel, or committee) arrive at coherent collective preferences or judgments on some issues, on the basis of its members' individual preferences or judgments? How can we rank different social alternatives in an order of social welfare? Social choice theorists study these questions not just by looking at examples, but by developing general models and proving theorems. (shrink)

We introduce a family of rules for adjusting one's credences in response to learning the credences of others. These rules have a number of desirable features. 1. They yield the posterior credences that would result from updating by standard Bayesian conditionalization on one's peers' reported credences if one's likelihood function takes a particular simple form. 2. In the simplest form, they are symmetric among the agents in the group. 3. They map neatly onto the familiar Condorcet voting results. 4. They (...) preserve shared agreement about independence in a wide range of cases. 5. They commute with conditionalization and with multiple peer updates. Importantly, these rules have a surprising property that we call synergy - peer testimony of credences can provide mutually supporting evidence raising an individual's credence higher than any peer's initial prior report. At first, this may seem to be a strike against them. We argue, however, that synergy is actually a desirable feature and the failure of other updating rules to yield synergy is a strike against them. (shrink)

This thesis is a collection of three self-contained papers on related themes in the area of formal and social epistemology. The first paper explores the possibility of measuring the coherence of a set with multiplicative averaging. It has been pointed out that all the existing probabilistic measures of coherence are flawed for taking the relevance between a set of propositions as the primary factor which determines the coherence of the set. What I show in this paper is that a group (...) of measures, namely the confirmation-based ones, can be saved from this problem if we adopt a nonlinear averaging function to measure the coherence of a set. The second paper discusses how people should conciliate in disagreements. Some epistemologists take linear averaging as the only way of conciliating and claim that conciliating leads to fallacious results. In the paper, I show that the problem is not conciliating, but taking linear averaging as the only way to conciliate. Since there is no reason for us to insist on conciliating with linear averaging, we can adopt nonlinear averaging functions for conciliating and thereby avoid the formal deficiencies. The third paper focuses on the pragmatic results of taking conciliating as a general strategy in disagreements. There is a potential dilemma about conciliating: if everyone always conciliates, it is likely for an epistemic bubble to arise. If everyone refuses to conciliate, an epistemic echo chamber may appear. A possible way of solving the dilemma is to develop a diachronic strategy which tells people how to both conciliate and update their estimate of their interlocutors’ reliability. Although the three papers differ in the subject, they jointly offer some unifying reflections on the way we approach philosophical problems with formal tools. From 5 the first two papers, we see that a formal analysis of a philosophical position is incomplete if philosophers fail to consider a sufficiently wide range of formal tools. The third paper, on the contrary, shows that we should change the ordinary way of modelling a notion if required. This thesis concludes by proposing a pluralistic view of formal methods in formal epistemology. (shrink)

Disagreement is a ubiquitous feature of human life, and philosophers have dutifully attended to it. One important question related to disagreement is epistemological: How does a rational person change her beliefs (if at all) in light of disagreement from others? The typical methodology for answering this question is to endorse a steadfast or conciliatory disagreement norm (and not both) on a priori grounds and selected intuitive cases. In this paper, I argue that this methodology is misguided. Instead, a thoroughgoingly Bayesian (...) strategy is what's needed. Such a strategy provides conciliatory norms in appropriate cases and steadfast norms in appropriate cases. I argue, further, that the few extant efforts to address disagreement in the Bayesian spirit are laudable but uncompelling. A modelling, rather than a functional, approach gets us the right norms and is highly general, allowing the epistemologist to deal with (1) multiple epistemic interlocutors, (2) epistemic superiors and inferiors (i.e. not just epistemic peers), and (3) dependence between interlocutors. (shrink)

How can different individuals' probability functions on a given sigma-algebra of events be aggregated into a collective probability function? Classic approaches to this problem often require 'event-wise independence': the collective probability for each event should depend only on the individuals' probabilities for that event. In practice, however, some events may be 'basic' and others 'derivative', so that it makes sense first to aggregate the probabilities for the former and then to let these constrain the probabilities for the latter. We formalize (...) this idea by introducing a 'premise-based' approach to probabilistic opinion pooling, and show that, under a variety of assumptions, it leads to linear or neutral opinion pooling on the 'premises'. This paper is the second of two self-contained, but technically related companion papers inspired by binary judgment-aggregation theory. (shrink)

The aim of this article is to introduce the theory of judgment aggregation, a growing interdisciplinary research area. The theory addresses the following question: How can a group of individuals make consistent collective judgments on a given set of propositions on the basis of the group members' individual judgments on them? I begin by explaining the observation that initially sparked the interest in judgment aggregation, the so-called "doctinal" and "discursive paradoxes". I then introduce the basic formal model of judgment aggregation, (...) which allows me to present some illustrative variants of a generic impossibility result. I subsequently turn to the question of how this impossibility result can be avoided, going through several possible escape routes. Finally, I relate the theory of judgment aggregation to other branches of aggregation theory. Rather than offering a comprehensive survey of the theory of judgment aggregation, I hope to introduce the theory in a succinct and pedagogical way, providing an illustrative rather than exhaustive coverage of some of its key ideas and results. (shrink)

How can the propositional attitudes of several individuals be aggregated into overall collective propositional attitudes? Although there are large bodies of work on the aggregation of various special kinds of propositional attitudes, such as preferences, judgments, probabilities and utilities, the aggregation of propositional attitudes is seldom studied in full generality. In this paper, we seek to contribute to filling this gap in the literature. We sketch the ingredients of a general theory of propositional attitude aggregation and prove two new theorems. (...) Our first theorem simultaneously characterizes some prominent aggregation rules in the cases of probability, judgment and preference aggregation, including linear opinion pooling and Arrovian dictatorships. Our second theorem abstracts even further from the specific kinds of attitudes in question and describes the properties of a large class of aggregation rules applicable to a variety of belief-like attitudes. Our approach integrates some previously disconnected areas of investigation. (shrink)

This paper provides an introductory review of the theory of judgment aggregation. It introduces the paradoxes of majority voting that originally motivated the field, explains several key results on the impossibility of propositionwise judgment aggregation, presents a pedagogical proof of one of those results, discusses escape routes from the impossibility and relates judgment aggregation to some other salient aggregation problems, such as preference aggregation, abstract aggregation and probability aggregation. The present illustrative rather than exhaustive review is intended to give readers (...) new to the field of judgment aggregation a sense of this rapidly growing research area. (shrink)

This paper provides an introductory review of the theory of judgment aggregation. It introduces the paradoxes of majority voting that originally motivated the field, explains several key results on the impossibility of propositionwise judgment aggregation, presents a pedagogical proof of one of those results, discusses escape routes from the impossibility and relates judgment aggregation to some other salient aggregation problems, such as preference aggregation, abstract aggregation and probability aggregation. The present illustrative rather than exhaustive review is intended to give readers (...) new to the field of judgment aggregation a sense of this rapidly growing research area. (shrink)