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.

A number of Bayesians claim that, if one has no evidence relevant to a proposition P, then one's credence in P should be spread over the interval [0, 1]. Against this, I argue: first, that it is inconsistent with plausible claims about comparative levels of confidence; second, that it precludes inductive learning in certain cases. Two motivations for the view are considered and rejected. A discussion of alternatives leads to the conjecture that there is an in-principle limitation on formal representations (...) of belief: they cannot be both fully accurate and maximally specific. (shrink)

The epistemic state of complete ignorance is not a probability distribution. In it, we assign the same, unique, ignorance degree of belief to any contingent outcome and each of its contingent, disjunctive parts. That this is the appropriate way to represent complete ignorance is established by two instruments, each individually strong enough to identify this state. They are the principle of indifference (PI) and the notion that ignorance is invariant under certain redescriptions of the outcome space, here developed into the (...) ‘principle of invariance of ignorance' (PII). Both instruments are so innocuous as almost to be platitudes. Yet the literature in probabilistic epistemology has misdiagnosed them as paradoxical or defective since they generate inconsistencies when conjoined with the assumption that an epistemic state must be a probability distribution. To underscore the need to drop this assumption, I express PII in its most defensible form as relating symmetric descriptions and show that paradoxes still arise if we assume the ignorance state to be a probability distribution. *Received February 2007; revised July 2007. †To contact the author, please write to: Department of History and Philosophy of Science, University of Pittsburgh, Pittsburgh, PA 15260; e-mail: [email protected]. (shrink)

Many have claimed that unspecific evidence sometimes demands unsharp, indeterminate, imprecise, vague, or interval-valued probabilities. Against this, a variant of the diachronic Dutch Book argument shows that perfectly rational agents always have perfectly sharp probabilities.

The Philosophy of Science, if the phrase were to be understood in the comprehensive sense which most naturally offers itself to our thoughts, ...

Can self-locating beliefs be relevant to non-self-locating claims? Traditional Bayesian modeling techniques have trouble answering this question because their updating rule fails when applied to situations involving contextsensitivity. This essay develops a fully general framework for modeling stories involving context-sensitive claims. The key innovations are a revised conditionalization rule and a principle relating models of the same story with different modeling languages. The essay then applies the modeling framework to the Sleeping Beauty Problem, showing that when Beauty awakens her degree (...) of belief in heads should be one-third. This demonstrates that it can be rational for an agent who gains only self-locating beliefs between two times to alter her degree of belief in a non-self-locating claim. (shrink)

During the last three decades, there has been a growing realization among physicists and cosmologists that the relation between particle physics and cosmology may constitute yet another successful example of the unity of science. However, there are important conceptual problems in the unification of the two disciplines, e.g. in connection with the cosmological constant and the conjecture of inflation. The present article will outline some of these problems, and argue that the victory for the unity of science in the context (...) of cosmology and particle physics is still far from obvious. (shrink)

During the last three decades, there has been a growing realization among physicists and cosmologists that the relation between particle physics and cosmology may constitute yet another successful example of the unity of science. However, there are important conceptual problems in the unification of the two disciplines, e.g. in connection with the cosmological constant and the conjecture of inflation. The present article will outline some of these problems, and argue that the victory for the unity of science in the context (...) of cosmology and particle physics is still far from obvious. (shrink)

In this paper I take a sceptical view of the standard cosmological model and its variants, mainly on the following grounds: (i) The method of mathematical modelling that characterises modern natural philosophy-as opposed to Aristotle's-goes well with the analytic, piecemeal approach to physical phenomena adopted by Galileo, Newton and their followers, but it is hardly suited for application to the whole world. (ii) Einstein's first cosmological model (1917) was not prompted by the intimations of experience but by a desire to (...) satisfy Mach's Principle. (iii) The standard cosmological model-a Friedmann-Lematre-Robertson-Walker spacetime expanding with or without end from an initial singularity-is supported by the phenomena of redshifted light from distant sources and very nearly isotropic thermal background radiation provided that two mutually inconsistent physical theories are jointly brought to bear on these phenomena, viz the quantum theory of elementary particles and Einstein's theory of gravity. (iv) While the former is certainly corroborated by high-energy experiments conducted under conditions allegedly similar to those prevailing in the early world, precise tests of the latter involve applications of the Schwarzschild solution or the PPN formalism for which there is no room in a Friedmann-Lematre-Robertson-Walker spacetime. (shrink)

The Sleeping Beauty Problem attracts so much attention because it connects to a wide variety of unresolved issues in formal epistemology, decision theory, and the philosophy of science. The problem raises unanswered questions concerning relative frequencies, objective chances, the relation between self-locating and non-self-locating information, the relation between self-location and updating, Dutch Books, accuracy arguments, memory loss, indifference principles, the existence of multiple universes, and many-worlds interpretations of quantum mechanics. After stating the problem, this article surveys its connections to all (...) of these areas. (shrink)

Inflationary cosmology has been widely accepted due to its successful predictions: for a “generic” initial state, inflation produces a homogeneous, flat, bubble with an appropriate spectrum of density perturbations. However, the discovery that inflation is “generically eternal,” leading to a vast multiverse of inflationary bubbles with different low-energy physics, threatens to undermine this account. There is a “predictability crisis” in eternal inflation, because extracting predictions apparently requires a well-defined measure over the multiverse. This has led to discussions of anthropic predictions (...) based on a measure over the multiverse, and an assumption that we are typical observers. I will give a pessimistic assessment of attempts to make predictions in this sense, emphasizing in particular problems that arise even if a unique measure can be found. (shrink)

A longstanding issue in attempts to understand the Everett (Many-Worlds) approach to quantum mechanics is the origin of the Born rule: why is the probability given by the square of the amplitude? Following Vaidman, we note that observers are in a position of self-locating uncertainty during the period between the branches of the wave function splitting via decoherence and the observer registering the outcome of the measurement. In this period it is tempting to regard each branch as equiprobable, but we (...) argue that the temptation should be resisted. Applying lessons from this analysis, we demonstrate (using methods similar to those of Zurek's envariance-based derivation) that the Born rule is the uniquely rational way of apportioning credence in Everettian quantum mechanics. In doing so, we rely on a single key principle: changes purely to the environment do not affect the probabilities one ought to assign to measurement outcomes in a local subsystem. We arrive at a method for assigning probabilities in cases that involve both classical and quantum self-locating uncertainty. This method provides unique answers to quantum Sleeping Beauty problems, as well as a well-defined procedure for calculating probabilities in quantum cosmological multiverses with multiple similar observers. (shrink)

In a recent, thought-provoking paper Adam Elga argues against unsharp – e.g., indeterminate, fuzzy and unreliable – probabilities. Rationality demands sharpness, he contends, and this means that decision theories like Levi's, Gärdenfors and Sahlin's, and Kyburg's, though they employ different decision rules, face a common, and serious, problem. This article defends the rule to maximize minimum expected utility against Elga's objection.

This paper aims to cast light on the reasons that explain the shift of opinion—from scepticism to realism—concerning the reality of atoms and molecules in the beginning of the twentieth century, in light of Jean Perrin’s theoretical and experimental work on the Brownian movement. The story told has some rather interesting repercussions for the rationality of accepting the reality of explanatory posits. Section 2 presents the key philosophical debate concerning the role and status of explanatory hypotheses c. 1900, focusing on (...) the work of Duhem, Stallo, Ostwald, Poincaré and Boltzmann. Section 3 examines in detail Perrin’s theoretical account of the molecular origins of Brownian motion, reconstructs the structure and explains the strength of Perrin’s argument for the reality of molecules. Section 4 draws three important lessons for the current debate over scientific realism. (shrink)

Traditional procedures for rational updating fail when it comes to self-locating opinions, such as your credences about where you are and what time it is. This paper develops an updating procedure for rational agents with self-locating beliefs. In short, I argue that rational updating can be factored into two steps. The first step uses information you recall from your previous self to form a hypothetical credence distribution, and the second step changes this hypothetical distribution to reflect information you have genuinely (...) learned as time has passed. While the second step resembles traditional procedures of updating by conditionalization, the first is best understood by analogy to traditional models of how agents transmit self-locating opinions through ordinary interpersonal communication. (shrink)

Newton’s equations of motion tell us that a mass at rest at the apex of a dome with the shape specified here can spontaneously move. It has been suggested that this indeterminism should be discounted since it draws on an incomplete rendering of Newtonian physics, or it is “unphysical,” or it employs illicit idealizations. I analyze and reject each of these reasons. †To contact the author, please write to: Department of History and Philosophy of Science, University of Pittsburgh, Pittsburgh, PA (...) 15260; e‐mail: [email protected]. (shrink)

While there is no universal logic of induction, the probability calculus succeeds as a logic of induction in many contexts through its use of several notions concerning inductive inference. They include Addition, through which low probabilities represent disbelief as opposed to ignorance; and Bayes property, which commits the calculus to a ‘refute and rescale’ dynamics for incorporating new evidence. These notions are independent and it is urged that they be employed selectively according to needs of the problem at hand. It (...) is shown that neither is adapted to inductive inference concerning some indeterministic systems. (shrink)

The duality of truth and falsity in a Boolean algebra of propositions is used to generate a duality of belief and disbelief. To each additive probability measure that represents belief there corresponds a dual additive measure that represents disbelief. The dual measure has its own peculiar calculus, in which, for example, measures are added when propositions are combined under conjunction. A Venn diagram of the measure has the contradiction as its total space. While additive measures are not self-dual, the epistemic (...) state of complete ignorance is represented by the unique, monotonic, non-additive measure that is self-dual in its contingent propositions. Convex sets of additive measures fail to represent complete ignorance since they are not self-dual. (shrink)

Bayesian probabilistic explication of inductive inference conflates neutrality of supporting evidence for some hypothesis H (“not supporting H”) with disfavoring evidence (“supporting not-H”). This expressive inadequacy leads to spurious results that are artifacts of a poor choice of inductive logic. I illustrate how such artifacts have arisen in simple inductive inferences in cosmology. In the inductive disjunctive fallacy, neutral support for many possibilities is spuriously converted into strong support for their disjunction. The Bayesian “doomsday argument” is shown to rely entirely (...) on a similar artifact, for the result disappears in a reanalysis that employs fragments of inductive logic able to represent evidential neutrality. Finally, the mere supposition of a multiverse is not yet enough to warrant the introduction of probabilities without some factual analog of a randomizer over the multiverses. (shrink)

A Bayesian account of the virtue of unification is given. On this account, the ability of a theory to unify disparate phenomena consists in the ability of the theory to render such phenomena informationally relevant to each other. It is shown that such ability contributes to the evidential support of the theory, and hence that preference for theories that unify the phenomena need not, on a Bayesian account, be built into the prior probabilities of theories.

The extension of the method of science to the study of the astronomical universe as a whole is an impressive and important feature of recent intellectual history. In a field where previously myth and metaphysics ranged at will with speculative abandon but with little in the way of progressively cumulative information or insight, by contrast, scientific cosmology has already produced significant results of an observational sort and opened up vistas of disciplined theoretical vision that are a token of even more (...) extensive disclosures to come. With the aid of high-powered telescopes, observationally grounded descriptive generalizations have been established about the structure, distribution and motions of the bodies constituting the basic astronomical units out of which the universe is composed. Such units are, in contemporary investigations, identified as the extragalactic nebulae. Preliminary surveys revealed a uniformity of distribution of these nebulae throughout space and in all directions. Also original surveys, accomplished primarily by Hubble, suggested an empirical law stating a linear proportionality between the distances to nebulae and the red-shift in their spectra. If such red-shift is interpreted as a velocity effect, it may be taken as marking a mutual recession of nebulae from one another, and leads thereby to the general idea of an “expanding universe.' Meanwhile, theoretical cosmology in the hands of the mathematical physicists has kept pace with observational advances by offering a variety of ”models” or explanatory theoretical devices for the systematic ordering and incorporation of the fruits of observation. Beginning with the ground-breaking paper by Einstein in 1917 where an extension was made of relativistic principles to the cosmologic domain, a variety of further attempts were made along classical relativistic lines by Einstein himself, de Sitter, Robertson, Tolman and Lamaître while somewhat different modes of approach were subsequently opened in several directions by Milne, Eddington, Hoyle, and others. While nothing in the way of a stable or wholly acceptable theory has yet made its appearance, nevertheless fruitful and promising lines of research have thus been inaugurated. (shrink)

Cosmological inflation is widely considered an integral and empirically successful component of contemporary cosmology. It was originally motivated by its solution of certain so-called fine-tuning problems of the hot big bang model, particularly what are known as the horizon problem and the flatness problem. Although the physics behind these problems is clear enough, the nature of the problems depends on the sense in which the hot big bang model is fine-tuned and how the alleged fine-tuning is problematic. Without clear explications (...) of these, it remains unclear precisely what problems inflationary theory is meant to be solving and whether it does in fact solve them. I analyze the structure of these problems and consider various interpretations that may substantiate the alleged fine-tuning. On the basis of this analysis I argue that at present there is no unproblematic interpretation available for which it can be said that inflation solves the big bang model’s alleged fine-tuning problems. (shrink)

Is Bayesian decision theory a panacea for many of the problems in epistemology and the philosophy of science, or is it philosophical snake-oil? For years a debate had been waged amongst specialists regarding the import and legitimacy of this body of theory. Mark Kaplan had written the first accessible and non-technical book to address this controversy. Introducing a new variant on Bayesian decision theory the author offers a compelling case that, while no panacea, decision theory does in fact have the (...) most profound consequences for the way in which philosophers think about inquiry, criticism and rational belief. The new variant on Bayesian theory is presented in such a way that a non-specialist will be able to understand it. The book also offers new solutions to some classic paradoxes. It focuses on the intuitive motivations of the Bayesian approach to epistemology and addresses the philosophical worries to which it has given rise. (shrink)

I hear the patter of little feet around the house, I expect Bruce. What I expect is a cat, a particular cat. If I heard such a patter in another house, I might expect a cat but no particular cat. What I expect then seems to be a Meinongian incomplete cat. I expect winter, expect stormy weather, expect to shovel snow, expect fatigue---a season, a phenomenon, an activity, a state. I expect that someday mankind will inhabit at least five planets. (...) This time what I expect is a state of affairs. If we let surface grammar be our guide, the objects of expectation seem quite a miscellany. The same goes for belief, since expectation is one kind of belief. The same goes for desire: I could want Bruce, want a cat but no particular cat, want winter, want stormy weather, want to shovel snow, want fatigue, or want that someday mankind will inhabit at least five planets. The same goes for other attitudes to the extent that they consist partly of beliefs or desires or lacks thereof. But the seeming diversity of objects might be an illusion. Perhaps the objects of attitudes are uniform in category, and it is our ways of speaking elliptically about these uniform objects that are diverse. That indeed is our consensus. We mostly think that the attitudes uniformly have propositions as their objects. That is why we speak habitually of "propositional attitudes.". (shrink)

This major work challenges some widely held positions in epistemology - those of Peirce and Popper on the one hand and those of Quine and Kuhn on the other.

Is Bayesian decision theory a panacea for many of the problems in epistemology and the philosophy of science, or is it philosophical snake-oil? For years a debate had been waged amongst specialists regarding the import and legitimacy of this body of theory. Mark Kaplan had written the first accessible and non-technical book to address this controversy. Introducing a new variant on Bayesian decision theory the author offers a compelling case that, while no panacea, decision theory does in fact have the (...) most profound consequences for the way in which philosophers think about inquiry, criticism and rational belief. The new variant on Bayesian theory is presented in such a way that a non-specialist will be able to understand it. The book also offers new solutions to some classic paradoxes. It focuses on the intuitive motivations of the Bayesian approach to epistemology and addresses the philosophical worries to which it has given rise. (shrink)

an observation to formulate a theory, it is no surprise that the resulting theory accurately captures that observation. However, when the theory makes a novel prediction—when it predicts an observation that was not used in its formulation—this seems to provide more substantial confirmation of the theory. This paper presents a new approach to the vexed problem of understanding the epistemic difference between prediction and accommodation. In fact, there are several problems that need to be disentangled; in all of them, the (...) key is the concept of overfitting. We float the hypothesis that accommodation is a defective methodology only when the methods used to accommodate the data fail to guard against the risk of overfitting. We connect our analysis with the proposals that other philosophers have made. We also discuss its bearing on the conflict between instrumentalism and scientific realism. Introduction Predictivisms—a taxonomy Observations Formulating the problem What might Annie be doing wrong? Solutions Observations explained Mayo on severe tests The miracle argument and scientific realism Concluding comments. (shrink)

Traditional analyses of the curve fitting problem maintain that the data do not indicate what form the fitted curve should take. Rather, this issue is said to be settled by prior probabilities, by simplicity, or by a background theory. In this paper, we describe a result due to Akaike [1973], which shows how the data can underwrite an inference concerning the curve's form based on an estimate of how predictively accurate it will be. We argue that this approach throws light (...) on the theoretical virtues of parsimoniousness, unification, and non ad hocness, on the dispute about Bayesianism, and on empiricism and scientific realism. * Both of us gratefully acknowledge support from the Graduate School at the University of Wisconsin-Madison, and NSF grant DIR-8822278 (M.F.) and NSF grant SBE-9212294 (E.S.). Special thanks go to A. W. F. Edwards.William Harper. Martin Leckey. Brian Skyrms, and especially Peter Turney for helpful comments on an earlier draft. (shrink)

The law of gravitation, an example of physical law The relation of mathematics to physics The great conservation principles Symmetry in physical law The distinction of past and future Probability and uncertainty: the quantum mechanical view of nature Seeking new laws.