The received model of degrees of belief represents them as probabilities. Over the last half century, many philosophers have been convinced that this model fails because it cannot make room for the idea that an agent’s degrees of belief should respect the available evidence. In its place they have advocated a model that represents degrees of belief using imprecise probabilities (sets of probability functions). This paper presents a model of degrees of belief based on Dempster–Shafer belief functions and then presents (...) arguments for belief functions over imprecise probabilities as a model of evidence-respecting degrees of belief. The arguments cover three kinds of issue: theoretical virtues (simplicity, interpretability and flexibility); motivations; and problem cases (dilation and belief inertia). (shrink)

Epistemic rationality is typically taken to be immodest at least in this sense: a rational epistemic state should always take itself to be doing at least as well, epistemically and by its own light, than any alternative epistemic state. If epistemic states are probability functions and their alternatives are other probability functions defined over the same collection of proposition, we can capture the relevant sense of immodesty by claiming that epistemic utility functions are (strictly) proper. In this paper I examine (...) what happens if we allow for the alternatives to an epistemic state to include probability functions with different domains. I first prove an impossibility result: on minimal assumptions, I show that there is no way of vindicating strong immodesty principles to the effect that any probability function should take itself to be doing at least as well than any alternative probability function, regardless of its domain. I then consider alternative, weaker generalizations of the traditional immodesty principle and prove some characterization results for some classes of epistemic utility functions satisfying each of the relevant principles. (shrink)

A decision theory is self-recommending if, when you ask it which decision theory you should use, it considers itself to be among the permissible options. I show that many alternatives to expected utility theory are not self-recommending, and I argue that this tells against them.

Rational credence should be coherent in the sense that your attitudes should not leave you open to a sure loss. Rational credence should be such that you can learn when confronted with relevant evidence. Rational credence should not be sensitive to irrelevant differences in the presentation of the epistemic situation. We explore the extent to which orthodox probabilistic approaches to rational credence can satisfy these three desiderata and find them wanting. We demonstrate that an imprecise probability approach does better. Along (...) the way we shall demonstrate that the problem of “belief inertia” is not an issue for a large class of IP credences, and provide a solution to van Fraassen’s box factory puzzle. (shrink)

According to van Fraassen, inference to the best explanation (IBE) is incompatible with Bayesianism. To argue to the contrary, many philosophers have suggested hybrid models of scientific reasoning with both explanationist and probabilistic elements. This paper offers another such model with two novel features. First, its Bayesian component is imprecise. Second, the domain of credence functions can be extended.

This paper aims to open a dialogue between philosophers working in decision theory and operations researchers and engineers working on decision-making under deep uncertainty. Specifically, we assess the recommendation to follow a norm of robust satisficing when making decisions under deep uncertainty in the context of decision analyses that rely on the tools of Robust Decision-Making developed by Robert Lempert and colleagues at RAND. We discuss two challenges for robust satisficing: whether the norm might derive its plausibility from an implicit (...) appeal to probabilistic representations of uncertainty of the kind that deep uncertainty is supposed to preclude; and whether there is adequate justification for adopting a satisficing norm, as opposed to an optimizing norm that is sensitive to considerations of robustness. We discuss decision-theoretic and voting-theoretic motivations for robust satisficing, and use these motivations to select among candidate formulations of the robust satisficing norm. (shrink)

According to a simple Bayesian argument from evil, the evil we observe is less likely given theism than given atheism, and therefore lowers the probability of theism. I consider the most common skeptical theist response to this argument, according to which our cognitive limitations make the probability of evil given theism inscrutable. I argue that if skeptical theists are right about this, then the probability of theism given evil is itself largely inscrutable, and that if this is so, we ought (...) to be agnostic about whether God exists. (shrink)

When do probability distribution functions (PDFs) about future climate misrepresent uncertainty? How can we recognise when such misrepresentation occurs and thus avoid it in reasoning about or communicating our uncertainty? And when we should not use a PDF, what should we do instead? In this paper we address these three questions. We start by providing a classification of types of uncertainty and using this classification to illustrate when PDFs misrepresent our uncertainty in a way that may adversely affect decisions. We (...) then discuss when it is reasonable and appropriate to use a PDF to reason about or communicate uncertainty about climate. We consider two perspectives on this issue. On one, which we argue is preferable, available theory and evidence in climate science basically excludes using PDFs to represent our uncertainty. On the other, PDFs can legitimately be provided when resting on appropriate expert judgement and recognition of associated risks. Once we have specified the border between appropriate and inappropriate uses of PDFs, we explore alternatives to their use. We briefly describe two formal alternatives, namely imprecise probabilities and possibilistic distribution functions, as well as informal possibilistic alternatives. We suggest that the possibilistic alternatives are preferable. -/- . (shrink)

Although much has been written about divine knowledge, and some on divine beliefs, virtually nothing has been written about divine credences. In this essay we comparatively assess four views on divine credences: (1) God has only beliefs, not credences; (2) God has both beliefs and credences; (3) God has only credences, not beliefs; and (4) God has neither credences nor beliefs, only knowledge. We weigh the costs and benefits of these four views and draw connections to current discussions in philosophical (...) theology. (shrink)

We defend three controversial claims about preference, credence, and choice. First, all agents (not just rational ones) have complete preferences. Second, all agents (again, not just rational ones) have real-valued credences in every proposition in which they are confident to any degree. Third, there is almost always some unique thing we ought to do, want, or believe.

Epistemologists often appeal to the idea that a normative theory must provide useful, usable, guidance to argue for one normative epistemology over another. I argue that this is a mistake. Guidance considerations have no role to play in theory choice in epistemology. I show how this has implications for debates about the possibility and scope of epistemic dilemmas, the legitimacy of idealisation in Bayesian epistemology, uniqueness versus permissivism, sharp versus mushy credences, and internalism versus externalism.

Epistemic states of uncertainty play important roles in ethical and political theorizing. Theories that appeal to a “veil of ignorance,” for example, analyze fairness or impartiality in terms of certain states of ignorance. It is important, then, to scrutinize proposed conceptions of ignorance and explore promising alternatives in such contexts. Here, I study Lerner’s probabilistic egalitarian theorem in the setting of imprecise probabilities. Lerner’s theorem assumes that a social planner tasked with distributing income to individuals in a population is “completely (...) ignorant” about which utility functions belong to which individuals. Lerner models this ignorance with a certain uniform probability distribution, and shows that, under certain further assumptions, income should be equally distributed. Much of the criticism of the relevance of Lerner’s result centers on the representation of ignorance involved. Imprecise probabilities provide a general framework for reasoning about various forms of uncertainty including, in particular, ignorance. To what extent can Lerner’s conclusion be maintained in this setting? (shrink)

The main aim of this book is to introduce the topic of limited awareness, and changes in awareness, to those interested in the philosophy of decision-making and uncertain reasoning.

How much does rationality constrain what we should believe on the basis of our evidence? According to this book, not very much. For most people and most bodies of evidence, there is a wide range of beliefs that rationality permits them to have in response to that evidence. The argument, which takes inspiration from William James' ideas in 'The Will to Believe', proceeds from two premises. The first is a theory about the basis of epistemic rationality. It's called epistemic utility (...) theory, and it says that what it is epistemically rational for you to believe is what it would be rational for you to choose if you were given the chance to pick your beliefs and, when picking them, you were to care only about their epistemic value. So, to say which beliefs are permitted, we must say how to measure epistemic value, and which decision rule to use when picking your beliefs. The second premise is a claim about attitudes to epistemic risk, and it says that rationality permits many different such attitudes. These attitudes can show up in epistemic utility theory in two ways: in the way you measure epistemic value; and in the decision rule you use to pick beliefs. This book explores the latter. The result is permissivism about epistemic rationality: different attitudes to epistemic risk lead to different choices of prior beliefs; given most bodies of evidence, different priors lead to different posteriors; and even once we fix your attitudes to epistemic risk, if they are at all risk-inclined, there is a range of different priors and therefore different posteriors they permit. (shrink)

Many policy decisions take input from collections of scientific models. Such decisions face significant and often poorly understood uncertainty. We rework the so-called confidence approach to tackle decision-making under severe uncertainty with multiple models, and we illustrate the approach with a case study: insurance pricing using hurricane models. The confidence approach has important consequences for this case and offers a powerful framework for a wide class of problems. We end by discussing different ways in which model ensembles can feed information (...) into the approach, appropriate to different collections of models. (shrink)

We argue that all comparative expressions in natural language obey a principle that we call Comparability: if x and y are at least as F as themselves, then either x is at least as F as y or y is at least as F as x. This principle has been widely rejected among philosophers, especially by ethicists, and its falsity has been claimed to have important normative implications. We argue that Comparability is needed to explain the goodness of several patterns (...) of inference that seem manifestly valid, that the purported failures of Comparability would have absurd consequences, and that the influential arguments against Comparability are less compelling than they may have initially seemed. (shrink)

Sometimes epistemologists theorize about belief, a tripartite attitude on which one can believe, withhold belief, or disbelieve a proposition. In other cases, epistemologists theorize about credence, a fine-grained attitude that represents one’s subjective probability or confidence level toward a proposition. How do these two attitudes relate to each other? This article explores the relationship between belief and credence in two categories: descriptive and normative. It then explains the broader significance of the belief-credence connection and concludes with general lessons from the (...) debate thus far. (shrink)

Presenting the first comprehensive, in-depth study of hyperintensionality, this book equips readers with the basic tools needed to appreciate some of current and future debates in the philosophy of language, semantics, and metaphysics. After introducing and explaining the major approaches to hyperintensionality found in the literature, the book tackles its systematic connections to normativity and offers some contributions to the current debates. The book offers undergraduate and graduate students an essential introduction to the topic, while also helping professionals in related (...) fields get up to speed on open research-level problems. (shrink)

In a quantum universe with a strong arrow of time, it is standard to postulate that the initial wave function started in a particular macrostate---the special low-entropy macrostate selected by the Past Hypothesis. Moreover, there is an additional postulate about statistical mechanical probabilities according to which the initial wave function is a ''typical'' choice in the macrostate. Together, they support a probabilistic version of the Second Law of Thermodynamics: typical initial wave functions will increase in entropy. Hence, there are two (...) sources of randomness in such a universe: the quantum-mechanical probabilities of the Born rule and the statistical mechanical probabilities of the Statistical Postulate. I propose a new way to understand time's arrow in a quantum universe. It is based on what I call the Thermodynamic Theories of Quantum Mechanics. According to this perspective, there is a natural choice for the initial quantum state of the universe, which is given by not a wave function but by a density matrix. The density matrix plays a microscopic role: it appears in the fundamental dynamical equations of those theories. The density matrix also plays a macroscopic / thermodynamic role: it is exactly the projection operator onto the Past Hypothesis subspace. Thus, given an initial subspace, we obtain a unique choice of the initial density matrix. I call this property "the conditional uniqueness" of the initial quantum state. The conditional uniqueness provides a new and general strategy to eliminate statistical mechanical probabilities in the fundamental physical theories, by which we can reduce the two sources of randomness to only the quantum mechanical one. I also explore the idea of an absolutely unique initial quantum state, in a way that might realize Penrose's idea of a strongly deterministic universe. (shrink)

According to the traditional Bayesian view of credence, its structure is that of precise probability, its objects are descriptive propositions about the empirical world, and its dynamics are given by conditionalization. Each of the three essays that make up this thesis deals with a different variation on this traditional picture. The first variation replaces precise probability with sets of probabilities. The resulting imprecise Bayesianism is sometimes motivated on the grounds that our beliefs should not be more precise than the evidence (...) calls for. One known problem for this evidentially motivated imprecise view is that in certain cases, our imprecise credence in a particular proposition will remain the same no matter how much evidence we receive. In the first essay I argue that the problem is much more general than has been appreciated so far, and that it’s difficult to avoid without compromising the initial evidentialist motivation. The second variation replaces descriptive claims with moral claims as the objects of credence. I consider three standard arguments for probabilism with respect to descriptive uncertainty—representation theorem arguments, Dutch book arguments, and accuracy arguments—in order to examine whether such arguments can also be used to establish probabilism with respect to moral uncertainty. In the second essay, I argue that by and large they can, with some caveats. First, I don’t examine whether these arguments can be given sound non-cognitivist readings, and any conclusions therefore only hold conditional on cognitivism. Second, decision-theoretic representation theorems are found to be less convincing in the moral case, because there they implausibly commit us to thinking that intertheoretic comparisons of value are always possible. Third and finally, certain considerations may lead one to think that imprecise probabilism provides a more plausible model of moral epistemology. The third variation considers whether, in addition to conditionalization, agents may also change their minds by becoming aware of propositions they had not previously entertained, and therefore not previously assigned any probability. More specifically, I argue that if we wish to make room for reflective equilibrium in a probabilistic moral epistemology, we must allow for awareness growth. In the third essay, I sketch the outline of such a Bayesian account of reflective equilibrium. Given that this account gives a central place to awareness growth, and that the rationality constraints on belief change by awareness growth are much weaker than those on belief change by conditionalization, it follows that the rationality constraints on the credences of agents who are seeking reflective equilibrium are correspondingly weaker. (shrink)

It has been claimed that, in response to certain kinds of evidence, agents ought to adopt imprecise credences: doxastic states that are represented by sets of credence functions rather than single ones. In this paper I argue that, given some plausible constraints on accuracy measures, accuracy-centered epistemologists must reject the requirement to adopt imprecise credences. I then show that even the claim that imprecise credences are permitted is problematic for accuracy-centered epistemology. It follows that if imprecise credal states are permitted (...) or required in the cases that their defenders appeal to, then the requirements of rationality can outstrip what would be warranted by an interest in accuracy. (shrink)

Traditional Bayesianism requires that an agent’s degrees of belief be represented by a real-valued, probabilistic credence function. However, in many cases it seems that our evidence is not rich enough to warrant such precision. In light of this, some have proposed that we instead represent an agent’s degrees of belief as a set of credence functions. This way, we can respect the evidence by requiring that the set, often called the agent’s credal state, includes all credence functions that are in (...) some sense compatible with the evidence. One known problem for this evidentially motivated imprecise view is that in certain cases, our imprecise credence in a particular proposition will remain the same no matter how much evidence we receive. In this article I argue that the problem is much more general than has been appreciated so far, and that it’s difficult to avoid it without compromising the initial evidentialist motivation. _1_ Introduction _2_ Precision and Its Problems _3_ Imprecise Bayesianism and Respecting Ambiguous Evidence _4_ Local Belief Inertia _5_ From Local to Global Belief Inertia _6_ Responding to Global Belief Inertia _7_ Conclusion. (shrink)

This paper offers a probabilistic treatment of the conditions for argument cogency as endorsed in informal logic: acceptability, relevance, and sufficiency. Treating a natural language argument as a reason-claim-complex, our analysis identifies content features of defeasible argument on which the RSA conditions depend, namely: change in the commitment to the reason, the reason’s sensitivity and selectivity to the claim, one’s prior commitment to the claim, and the contextually determined thresholds of acceptability for reasons and for claims. Results contrast with, and (...) may indeed serve to correct, the informal understanding and applications of the RSA criteria concerning their conceptual dependence, their function as update-thresholds, and their status as obligatory rather than permissive norms, but also show how these formal and informal normative approachs can in fact align. (shrink)

Two compelling principles, the Reasonable Range Principle and the Preservation of Irrelevant Evidence Principle, are necessary conditions that any response to peer disagreements ought to abide by. The Reasonable Range Principle maintains that a resolution to a peer disagreement should not fall outside the range of views expressed by the peers in their dispute, whereas the Preservation of Irrelevant Evidence Principle maintains that a resolution strategy should be able to preserve unanimous judgments of evidential irrelevance among the peers. No standard (...) Bayesian resolution strategy satisfies the PIE Principle, however, and we give a loss aversion argument in support of PIE and against Bayes. The theory of imprecise probability allows one to satisfy both principles, and we introduce the notion of a set-based credal judgment to frame and address a range of subtle issues that arise in peer disagreements. (shrink)

Cosmology raises novel philosophical questions regarding the use of probabilities in inference. This work aims at identifying and assessing lines of arguments and problematic principles in probabilistic reasoning in cosmology. -/- The first, second, and third papers deal with the intersection of two distinct problems: accounting for selection effects, and representing ignorance or indifference in probabilistic inferences. These two problems meet in the cosmology literature when anthropic considerations are used to predict cosmological parameters by conditionalizing the distribution of, e.g., the (...) cosmological constant on the number of observers it allows for. However, uniform probability distributions usually appealed to in such arguments are an inadequate representation of indifference, and lead to unfounded predictions. It has been argued that this inability to represent ignorance is a fundamental flaw of any inductive framework using additive measures. In the first paper, I examine how imprecise probabilities fare as an inductive framework and avoid such unwarranted inferences. In the second paper, I detail how this framework allows us to successfully avoid the conclusions of Doomsday arguments in a way no Bayesian approach that represents credal states by single credence functions could. -/- There are in the cosmology literature several kinds of arguments referring to self- locating uncertainty. In the multiverse framework, different "pocket-universes" may have different fundamental physical parameters. We don’t know if we are typical observers and if we can safely assume that the physical laws we draw from our observations hold elsewhere. The third paper examines the validity of the appeal to the "Sleeping Beauty problem" and assesses the nature and role of typicality assumptions often endorsed to handle such questions. -/- A more general issue for the use of probabilities in cosmology concerns the inadequacy of Bayesian and statistical model selection criteria in the absence of well-motivated measures for different cosmological models. The criteria for model selection commonly used tend to focus on optimizing the number of free parameters, but they can select physically implausible models. The fourth paper examines the possibility for Bayesian model selection to circumvent the lack of well-motivated priors. (shrink)

This essay gives an account of epistemic deference for agents with imprecise credences. I look at the two main imprecise deference principles in the literature, known as Identity Reflection and Pointwise Reflection (Moss, 2021). I show that Pointwise Reflection is strictly weaker than Identity Reflection, and argue that, if you are certain you will update by conditionalisation, you should defer to your future self according to Identity Reflection. Then I give a more general justification for Pointwise and Identity Reflection from (...) the assumption that you defer to someone whenever you consider their doxastic state to be better than yours, in the sense of leading to better decision-making. (shrink)

Imprecise probabilities are an increasingly popular way of reasoning about rational credence. However they are subject to an apparent failure to display convincing inductive learning. This paper demonstrates that a small modification to the update rule for IP allows us to overcome this problem, albeit at the cost of satisfying only a weaker concept of coherence.

– We offer a new motivation for imprecise probabilities. We argue that there are propositions to which precise probability cannot be assigned, but to which imprecise probability can be assigned. In such cases the alternative to imprecise probability is not precise probability, but no probability at all. And an imprecise probability is substantially better than no probability at all. Our argument is based on the mathematical phenomenon of non-measurable sets. Non-measurable propositions cannot receive precise probabilities, but there is a natural (...) way for them to receive imprecise probabilities. The mathematics of non-measurable sets is arcane, but its epistemological import is far-reaching; even apparently mundane propositions are liable to be affected by non-measurability. The phenomenon of non-measurability dramatically reshapes the dialectic between critics and proponents of imprecise credence. Non-measurability offers natural rejoinders to prominent critics of imprecise credence. Non-measurability even reverses some of the critics’ arguments—by the very lights that have been used to argue against imprecise credences, imprecise credences are better than precise credences. (shrink)

Many philosophers argue that Keynes’s concept of the “weight of arguments” is an important aspect of argument appraisal. The weight of an argument is the quantity of relevant evidence cited in the premises. However, this dimension of argumentation does not have a received method for formalisation. Kyburg has suggested a measure of weight that uses the degree of imprecision in his system of “Evidential Probability” to quantify weight. I develop and defend this approach to measuring weight. I illustrate the usefulness (...) of this measure by employing it to develop an answer to Popper’s Paradox of Ideal Evidence. (shrink)

Many have claimed that epistemic rationality sometimes requires us to have imprecise credal states (i.e. credal states representable only by sets of credence functions) rather than precise ones (i.e. credal states representable by single credence functions). Some writers have recently argued that this claim conflicts with accuracy-centered epistemology, i.e., the project of justifying epistemic norms by appealing solely to the overall accuracy of the doxastic states they recommend. But these arguments are far from decisive. In this essay, we prove some (...) new results, which show that there is little hope for reconciling the rationality of credal imprecision with accuracy-centered epistemology. (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)

It is natural to think of precise probabilities as being special cases of imprecise probabilities, the special case being when one’s lower and upper probabilities are equal. I argue, however, that it is better to think of the two models as representing two different aspects of our credences, which are often vague to some degree. I show that by combining the two models into one model, and understanding that model as a model of vague credence, a natural interpretation arises that (...) suggests a hypothesis concerning how we can improve the accuracy of aggregate credences. I present empirical results in support of this hypothesis. I also discuss how this modeling interpretation of imprecise probabilities bears upon a philosophical objection that has been raised against them, the so-called inductive learning problem. (shrink)

Confronted with the possibility of severe environmental harms, such as catastrophic climate change, some researchers have suggested that we should abandon the principle at the heart of standard decision theory—the injunction to maximize expected utility—and embrace a different one: the Precautionary Principle. Arguably, the most sophisticated philosophical treatment of the Precautionary Principle is due to Steel. Steel interprets PP as a qualitative decision rule and appears to conclude that a quantitative decision-theoretic statement of PP is both impossible and unnecessary. In (...) this article, we propose a decision-theoretic formulation of PP in terms of lexical utilities. We show that this lexical model is largely faithful to Steel’s approach, but also that it corrects three problems with Steel’s account and clarifies the relationship between PP and standard decision theory. Using a range of examples, we illustrate how the lexical model can be used to explore a variety of issues related to precautionary reasoning. (shrink)

This dissertation aims to determine the optimal level of granularity for the variables used in probabilistic causal models. These causal models are useful for generating explanations in a number of scientific contexts. In Chapter 1, I argue that there is rarely a unique level of granularity at which a given phenomenon can be causally explained, thereby rejecting various causal exclusion arguments. In Chapter 2, I consider several recent proposals for measuring the explanatory power of causal explanations, and show that these (...) measures fail to track the comparative depth of explanations given at different levels of granularity. In Chapter 3, I offer a pragmatic account of how to partition the measure space of a causal variable so as to optimally explain its effect. My account uses the decision-theoretic notion of value of information, and indexes the relative depth of an explanation to a particular agent faced with a particular decision problem. Chapter 4 applies this same decisiontheoretic framework to answer the epistemic question of how to discover constitutive relationships in nature. In Chapter 5, I describe the formal details of the relationship between random variables that are meant to be coarse-grained and fine-grained representations of the same type of phenomenon. I use this formal framework to rebut a popular argument for the view that special science probabilities can be objective chances. Chapter 6 discusses challenges related to the causal interpretation of Bayes nets that use imprecise rather than precise probabilities. (shrink)

Gordon Belot has recently developed a novel argument against Bayesianism. He shows that there is an interesting class of problems that, intuitively, no rational belief forming method is likely to get right. But a Bayesian agent’s credence, before the problem starts, that she will get the problem right has to be 1. This is an implausible kind of immodesty on the part of Bayesians. My aim is to show that while this is a good argument against traditional, precise Bayesians, the (...) argument doesn’t neatly extend to imprecise Bayesians. As such, Belot’s argument is a reason to prefer imprecise Bayesianism to precise Bayesianism. (shrink)

The basic Bayesian model of credence states, where each individual’s belief state is represented by a single probability measure, has been criticized as psychologically implausible, unable to represent the intuitive distinction between precise and imprecise probabilities, and normatively unjustifiable due to a need to adopt arbitrary, unmotivated priors. These arguments are often used to motivate a model on which imprecise credal states are represented by sets of probability measures. I connect this debate with recent work in Bayesian cognitive science, where (...) probabilistic models are typically provided with explicit hierarchical structure. Hierarchical Bayesian models are immune to many classic arguments against single-measure models. They represent grades of imprecision in probability assignments automatically, have strong psychological motivation, and can be normatively justified even when certain arbitrary decisions are required. In addition, hierarchical models show much more plausible learning behavior than flat representations in terms of sets of measures, which—on standard assumptions about update—rule out simple cases of learning from a starting point of total ignorance. (shrink)

Whilst Bayesian epistemology is widely regarded nowadays as our best theory of knowledge, there are still a relatively large number of incompatible and competing approaches falling under that umbrella. Very recently, Wallmann and Williamson wrote an interesting article that aims at showing that a subjective Bayesian who accepts the principal principle and uses a known physical chance as her degree of belief for an event A could end up having incoherent or very implausible beliefs if she subjectively chooses the probability (...) of an event F for which she has much poorer evidence. They also argued that their own version of objective Bayesianism is completely immune to that challenge. In this article, after having presented the strongest version of Wallmann’s and Williamson’s argument, I will show that if successful, it has far-reaching consequences and would not only invalidate moderate subjective Bayesianism and imprecise probalism but also a form of objective Bayesianism that relies on conditionalisation, the principal principle, reference classes, and the principle of indifference applied to the most basic partitions. I then argue that their argument can be defeated by adding the rule that it is always irrational to choose a probability that can be computed from the known probabilities associated to one’s other beliefs. I finally argue that the authors’ main intuition that probabilities have different degrees of reliability favours imprecise Bayesianism over precise Bayesianism. (shrink)

There is currently much discussion about how decision making should proceed when an agent's degrees of belief are imprecise; represented by a set of probability functions. I show that decision rules recently discussed by Sarah Moss, Susanna Rinard and Rohan Sud all suffer from the same defect: they all struggle to rationalize diachronic ambiguity aversion. Since ambiguity aversion is among the motivations for imprecise credence, this suggests that the search for an adequate imprecise decision rule is not yet over.

There is a growing interest in the foundations as well as the application of imprecise probability in contemporary epistemology. This dissertation is concerned with the application. In particular, the research presented concerns ways in which imprecise probability, i.e. sets of probability measures, may helpfully address certain philosophical problems pertaining to rational belief. The issues I consider are disagreement among epistemic peers, complete ignorance, and inductive reasoning with imprecise priors. For each of these topics, it is assumed that belief can be (...) modeled with imprecise probability, and thus there is a non-classical solution to be given to each problem. I argue that this is the case for peer disagreement and complete ignorance. However, I discovered that the approach has its shortcomings, too, specifically in regard to inductive reasoning with imprecise priors. Nevertheless, the dissertation ultimately illustrates that imprecise probability as a model of rational belief has a lot of promise, but one should be aware of its limitations also. (shrink)

This article considers the extent to which Bayesian networks with imprecise probabilities, which are used in statistics and computer science for predictive purposes, can be used to represent causal structure. It is argued that the adequacy conditions for causal representation in the precise context—the Causal Markov Condition and Minimality—do not readily translate into the imprecise context. Crucial to this argument is the fact that the independence relation between random variables can be understood in several different ways when the joint probability (...) distribution over those variables is imprecise, none of which provides a compelling basis for the causal interpretation of imprecise Bayes nets. I conclude that there are serious limits to the use of imprecise Bayesian networks to represent causal structure. (shrink)

We represent consensus formation processes based on iterated opinion pooling as a dynamic approach to common knowledge of posteriors :1236–1239, 1976; Geanakoplos and Polemarchakis in J Econ Theory 28:192–200, 1982). We thus provide a concrete and plausible Bayesian rationalization of consensus through iterated pooling. The link clarifies the conditions under which iterated pooling can be rationalized from a Bayesian perspective, and offers an understanding of iterated pooling in terms of higher-order beliefs.

One implication of Bell’s theorem is that there cannot in general be hidden variable models for quantum mechanics that both are noncontextual and retain the structure of a classical probability space. Thus, some hidden variable programs aim to retain noncontextuality at the cost of using a generalization of the Kolmogorov probability axioms. We generalize a theorem of Feintzeig to show that such programs are committed to the existence of a finite null cover for some quantum mechanical experiments, i.e., a finite (...) collection of probability zero events whose disjunction exhausts the space of experimental possibilities. (shrink)