Coping with uncertainty is a necessary part of ordinary life and is crucial to an understanding of how the mind works. For example, it is a vital element in developing artificial intelligence that will not be undermined by its own rigidities. There have been many approaches to the problem of uncertain inference, ranging from probability to inductive logic to nonmonotonic logic. Thisbook seeks to provide a clear exposition of these approaches within a unified framework. The principal market for the book (...) will be students and professionals in philosophy, computer science, and AI. Among the special features of the book are a chapter on evidential probability, which has not received a basic exposition before; chapters on nonmonotonic reasoning and theory replacement, matters rarely addressed in standard philosophical texts; and chapters on Mill's methods and statistical inference that cover material sorely lacking in the usual treatments of AI and computer science. (shrink)

In this paper, I consider the relationship between Inference to the Best Explanation and Bayesianism, both of which are well-known accounts of the nature of scientific inference. In Sect. 2, I give a brief overview of Bayesianism and IBE. In Sect. 3, I argue that IBE in its most prominently defended forms is difficult to reconcile with Bayesianism because not all of the items that feature on popular lists of “explanatory virtues”—by means of which IBE ranks competing explanations—have confirmational import. (...) Rather, some of the items that feature on these lists are “informational virtues”—properties that do not make a hypothesis \ more probable than some competitor \ given evidence E, but that, roughly-speaking, give that hypothesis greater informative content. In Sect. 4, I consider as a response to my argument a recent version of compatibilism which argues that IBE can provide further normative constraints on the objectively correct probability function. I argue that this response does not succeed, owing to the difficulty of defending with any generality such further normative constraints. Lastly, in Sect. 5, I propose that IBE should be regarded, not as a theory of scientific inference, but rather as a theory of when we ought to “accept” H, where the acceptability of H is fixed by the goals of science and concerns whether H is worthy of commitment as research program. In this way, IBE and Bayesianism, as I will show, can be made compatible, and thus the Bayesian and the proponent of IBE can be friends. (shrink)

Information theory and statistical mechanics II.

We argue that David Lewis’s principal principle implies a version of the principle of indifference. The same is true for similar principles that need to appeal to the concept of admissibility. Such principles are thus in accord with objective Bayesianism, but in tension with subjective Bayesianism. 1 The Argument2 Some Objections Met.

Those who model doxastic states with a set of probability functions, rather than a single function, face a pressing challenge: can they provide a plausible decision theory compatible with their view? Adam Elga and others claim that they cannot, and that the set of functions model should be rejected for this reason. This paper aims to answer this challenge. The key insight is that the set of functions model can be seen as an instance of the supervaluationist approach to vagueness (...) more generally. We can then generate our decision theory by applying the general supervaluationist semantics to decision-theoretic claims. The result: if an action is permissible according to all functions in the set, it’s determinately permissible; if impermissible according to all, determinately impermissible; and – crucially – if permissible according to some, but not all, it’s indeterminate whether it’s permissible. This proposal handles with ease some difficult cases ) on which alternative decision theories falter. One reason this view has been overlooked in the literature thus far is that all parties to the debate presuppose that an acceptable decision theory must classify each action as either permissible or impermissible. But I will argue that this thought is misguided. Seeing the set of functions model as an instance of supervaluationism provides a compelling motivation for the claim that there can be indeterminacy in the rationality of some actions. (shrink)

The _Principle of Indifference_ was once regarded as a linchpin of probabilistic reasoning, but has now fallen into disrepute as a result of the so-called _problem of multiple of partitions_. In ‘Evidential symmetry and mushy credence’ Roger White suggests that we have been too quick to jettison this principle and argues that the problem of multiple partitions rests on a mistake. In this paper I will criticise White’s attempt to revive POI. In so doing, I will argue that what underlies (...) the problem of multiple partitions is a fundamental tension between POI and the very idea of _evidential incomparability_. (shrink)

Sometimes different partitions of the same space each seem to divide that space into propositions that call for equal epistemic treatment. Famously, equal treatment in the form of equal point-valued credence leads to incoherence. Some have argued that equal treatment in the form of equal interval-valued credence solves the puzzle. This paper shows that, once we rule out intervals with extreme endpoints, this proposal also leads to incoherence.

In Bayesian epistemology, the problem of the priors is this: How should we set our credences (or degrees of belief) in the absence of evidence? That is, how should we set our prior or initial credences, the credences with which we begin our credal life? David Lewis liked to call an agent at the beginning of her credal journey a superbaby. The problem of the priors asks for the norms that govern these superbabies. -/- The Principle of Indifference gives a (...) very restrictive answer. It demands that such an agent divide her credences equally over all possibilities. That is, according to the Principle of Indifference, only one initial credence function is permissible, namely, the uniform distribution. In this paper, we offer a novel argument for the Principle of Indifference. I call it the Argument from Accuracy. (shrink)

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)

Dilation occurs when an interval probability estimate of some event E is properly included in the interval probability estimate of E conditional on every event F of some partition, which means that one’s initial estimate of E becomes less precise no matter how an experiment turns out. Critics maintain that dilation is a pathological feature of imprecise probability models, while others have thought the problem is with Bayesian updating. However, two points are often overlooked: (1) knowing that E is stochastically (...) independent of F (for all F in a partition of the underlying state space) is sufficient to avoid dilation, but (2) stochastic independence is not the only independence concept at play within imprecise probability models. In this paper we give a simple characterization of dilation formulated in terms of deviation from stochastic independence, propose a measure of dilation, and distinguish between proper and improper dilation. Through this we revisit the most sensational examples of dilation, which play up independence between dilator and dilatee, and find the sensationalism undermined by either fallacious reasoning with imprecise probabilities or improperly constructed imprecise probability models. (shrink)

One recent topic of debate in Bayesian epistemology has been the question of whether imprecise credences can be rational. I argue that one account of imprecise credences, the orthodox treatment as defended by James M. Joyce, is untenable. Despite Joyce’s claims to the contrary, a puzzle introduced by Roger White shows that the orthodox account, when paired with Bas C. van Fraassen’s Reflection Principle, can lead to inconsistent beliefs. Proponents of imprecise credences, then, must either provide a compelling reason to (...) reject Reflection or admit that the rational credences in White’s case are precise. (shrink)

In this paper, I respond to recent attempts by philosophers to deny the existence of something that is both real and significant: reasonable disagreements between epistemic peers. In their arguments against the possibility of such disagreements, skeptical philosophers typically invoke one or more of the following: indifference reasoning , equal weight principles , and uniqueness theses . I take up each of these in turn, finding ample reason to resist them. The arguments for indifference reasoning and equal weight principles tend (...) to overlook the possibility of a certain kind of agnostic credal state which I call deep agnosticism , the possibility of which derails the arguments. The arguments for uniqueness theses tend to invoke a flawed understanding of the evidential support relation. When these problems and misunderstandings are brought into the light and corrected, the threat to reasonable disagreement vanishes. (shrink)

The article begins by describing two longstanding problems associated with direct inference. One problem concerns the role of uninformative frequency statements in inferring probabilities by direct inference. A second problem concerns the role of frequency statements with gerrymandered reference classes. I show that past approaches to the problem associated with uninformative frequency statements yield the wrong conclusions in some cases. I propose a modification of Kyburg’s approach to the problem that yields the right conclusions. Past theories of direct inference have (...) postponed treatment of the problem associated with gerrymandered reference classes by appealing to an unexplicated notion of projectability . I address the lacuna in past theories by introducing criteria for being a relevant statistic . The prescription that only relevant statistics play a role in direct inference corresponds to the sort of projectability constraints envisioned by past theories. (shrink)

At least one of these conceptions of probability underlies any theory of statistical inference (or, to use Neyman's phrase, 'inductive behavior'). ...

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.

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 symmetry of our evidential situation. If our confidence is best modeled by a standard probability function this means that we are to distribute our subjective probability or credence sharply and evenly over possibilities among which our evidence does not discriminate. Once thought to be the central principle of probabilistic reasoning by great..

The Eternal Coin is a fair coin has existed forever, and will exist forever, in a region causally isolated from you. It is tossed every day. How confident should you be that the Coin lands heads today, conditional on (i) the hypothesis that it has landed Heads on every past day, or (ii) the hypothesis that it will land Heads on every future day? I argue for the extremely counterintuitive claim that the correct answer to both questions is 1.

Scientific reasoning is—and ought to be—conducted in accordance with the axioms of probability. This Bayesian view—so called because of the central role it accords to a theorem first proved by Thomas Bayes in the late eighteenth ...

in Probability is the Very Guide of Life: The Philosophical Uses of Chance, eds. Henry Kyburg, Jr. and Mariam Thalos, Open Court. Abridged version in Proceedings of the International Society for Bayesian Analysis 2002.

In addition to being uncertain about what the world is like, one can also be uncertain about one’s own spatial or temporal location in the world. My aim is to pose a problem arising from the interaction between these two sorts of uncertainty, solve the problem, and draw two lessons from the solution.

In concrete applications of probability, statistical investigation gives us knowledge of some probabilities, but we generally want to know many others that are not directly revealed by our data. For instance, we may know prob(P/Q) (the probability of P given Q) and prob(P/R), but what we really want is prob(P/Q&R), and we may not have the data required to assess that directly. The probability calculus is of no help here. Given prob(P/Q) and prob(P/R), it is consistent with the probability calculus (...) for prob(P/Q&R) to have any value between 0 and 1. Is there any way to make a reasonable estimate of the value of prob(P/Q&R)? A related problem occurs when probability practitioners adopt undefended assumptions of statistical independence simply on the basis of not seeing any connection between two propositions. This is common practice, but its justification has eluded probability theorists, and researchers are typically apologetic about making such assumptions. Is there any way to defend the practice? This paper shows that on a certain conception of probability — nomic probability — there are principles of “probable probabilities” that license inferences of the above sort. These are principles telling us that although certain inferences from probabilities to probabilities are not deductively valid, nevertheless the second-order probability of their yielding correct results is 1. This makes it defeasibly reasonable to make the inferences. Thus I argue that it is defeasibly reasonable to assume statistical independence when we have no information to the contrary. And I show that there is a function Y(r,s:a) such that if prob(P/Q) = r, prob(P/R) = s, and prob(P/U) = a (where U is our background knowledge) then it is defeasibly reasonable to expect that prob(P/Q&R) = Y(r,s:a). Numerous other defeasible inferences are licensed by similar principles of probable probabilities.. (shrink)

Bayesians often confuse insistence that probability judgment ought to be indeterminate (which is incompatible with Bayesian ideals) with recognition of the presence of imprecision in the determination or measurement of personal probabilities (which is compatible with these ideals). The confusion is discussed and illustrated by remarks in a recent essay by R. C. Jeffrey.

Handbook of the History of Logic, vol. 10, eds. Dov Gabbay, Stephan Hartmann, and John Woods, forthcoming.

What psychological and philosophical significance should we attach to recent efforts at computer simulations of human cognitive capacities? In answering this question, I find it useful to distinguish what I will call "strong" AI from "weak" or "cautious" AI. According to weak AI, the principal value of the computer in the study of the mind is that it gives us a very powerful tool. For example, it enables us to formulate and test hypotheses in a more rigorous and precise fashion. (...) But according to strong AI, the computer is not merely a tool in the study of the mind; rather, the appropriately programmed computer really is a mind, in the sense that computers given the right programs can be literally said to. (shrink)

We argue that David Lewis’s principal principle implies a version of the principle of indifference. The same is true for similar principles that need to appeal to the concept of admissibility. Such principles are thus in accord with objective Bayesianism, but in tension with subjective Bayesianism. 1 The Argument2 Some Objections Met.

the symmetry of our evidential situation. If our confidence is best modeled by a standard probability function this means that we are to distribute our subjective probability or credence sharply and evenly over possibilities among which our evidence does not discriminate. Once thought to be the central principle of probabilistic reasoning by great..

This paper examines the debate between permissive and impermissive forms of Bayesianism. It briefly discusses some considerations that might be offered by both sides of the debate, and then replies to some new arguments in favor of impermissivism offered by Roger White. First, it argues that White’s (Oxford studies in epistemology, vol 3. Oxford University Press, Oxford, pp 161–186, 2010) defense of Indifference Principles is unsuccessful. Second, it contends that White’s (Philos Perspect 19:445–459, 2005) arguments against permissive views do not (...) succeed. (shrink)

Maxwell's deduction of the probability distribution over the velocity of gas molecules—one of the most important passages in physics (Truesdell)—presents a riddle: a physical discovery of the first importance was made in a single inferential leap without any apparent recourse to empirical evidence. -/- Tychomancy proposes that Maxwell's derivation was not made a priori; rather, he inferred his distribution from non-probabilistic facts about the dynamics of intermolecular collisions. Further, the inference is of the same sort as everyday reasoning about the (...) physical probabilities attached to such canonical chance setups as tossed coins or rolled dice. The structure of this reasoning is investigated and some simple rules for inferring physical probabilities from symmetries and other causally relevant properties of physical systems are proposed. -/- Not only physics but evolutionary biology and population ecology, the science of measurement error, and climate modeling have benefited enormously from the same kind of reasoning, the book goes on to argue. Inferences from dynamics to probability are so obvious to us, however, that their methodological importance has been largely overlooked. (shrink)

It has been argued recently that dogmatism in epistemology is incompatible with Bayesianism. That is, it has been argued that dogmatism cannot be modelled using traditional techniques for Bayesian modelling. I argue that our response to this should not be to throw out dogmatism, but to develop better modelling techniques. I sketch a model for formal learning in which an agent can discover a posteriori fundamental epistemic connections. In this model, there is no formal objection to dogmatism.

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)