Enjoying great popularity in decision theory, epistemology, and philosophy of science, Bayesianism as understood here is fundamentally concerned with epistemically ideal rationality. It assumes a tight connection between evidential probability and ideally rational credence, and usually interprets evidential probability in terms of such credence. Timothy Williamson challenges Bayesianism by arguing that evidential probabilities cannot be adequately interpreted as the credences of an ideal agent. From this and his assumption that evidential probabilities cannot be interpreted as the actual credences (...) of human agents either, he concludes that no interpretation of evidential probabilities in terms of credence is adequate. I argue to the contrary. My overarching aim is to show on behalf of Bayesians how one can still interpret evidential probabilities in terms of ideally rational credence and how one can maintain a tight connection between evidential probabilities and ideally rational credence even if the former cannot be interpreted in terms of the latter. By achieving this aim I illuminate the limits and prospects of Bayesianism. (shrink)

I defend the thesis that legal standards of proof are reducible to thresholds of probability. Many reject this thesis because it appears to permit finding defendants liable solely on the basis of statistical evidence. To the contrary, I argue – by combining Thomson's (1986) causal analysis of legal evidence with formal methods of causal inference – that legal standards of proof can be reduced to probabilities, but that deriving these probabilities involves more than just statistics.

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.

We suggest a rigorous theory of how objective single-case transition probabilities fit into our world. The theory combines indeterminism and relativity in the “branching space–times” pattern, and relies on the existing theory of causae causantes (originating causes). Its fundamental suggestion is that (at least in simple cases) the probabilities of all transitions can be computed from the basic probabilities attributed individually to their originating causes. The theory explains when and how one can reasonably infer from the (...) class='Hi'>probabilities of one “chance set-up” to the probabilities of another such set-up that is located far away. (shrink)

In response to Smith, I argue that probabilities cannot be rationally neglected. I show that Smith’s proposal for ignoring low-probability outcomes must, on pain of violating dominance reasoning, license taking arbitrarily high risk for arbitrarily little reward.

The fact that the standard probabilistic calculus does not define probabilities for sentences with embedded conditionals is a fundamental problem for the probabilistic theory of conditionals. Several authors have explored ways to assign probabilities to such sentences, but those proposals have come under criticism for making counterintuitive predictions. This paper examines the source of the problematic predictions and proposes an amendment which corrects them in a principled way. The account brings intuitions about counterfactual conditionals to bear on the (...) interpretation of indicatives and relies on the notion of causal (in)dependence. (shrink)

Quantum mechanics may be formulated as Sensible Quantum Mechanics (SQM) so that it contains nothing probabilistic except conscious perceptions. Sets of these perceptions can be deterministically realized with measures given by expectation values of positive-operator-valued awareness operators. Ratios of the measures for these sets of perceptions can be interpreted as frequency- type probabilities for many actually existing sets. These probabilities gener- ally cannot be given by the ordinary quantum “probabilities” for a single set of alternatives. Probabilism, or (...) ascribing probabilities to unconscious aspects of the world, may be seen to be an aesthemamorphic myth. (shrink)

In eternally inflating cosmology, infinitely many pocket universes are seeded. Attempts to show that universes like our observable universe are probable amongst them have failed, since no unique probability measure is recoverable. This lack of definite probabilities is taken to reveal a complete predictive failure. Inductive inference over the pocket universes, it would seem, is impossible. I argue that this conclusion of impossibility mistakes the nature of the problem. It confuses the case in which no inductive inference is possible, (...) with another in which a weaker inductive logic applies. The alternative, applicable inductive logic is determined by background conditions and is the same, non-probabilistic logic as applies to an infinite lottery. This inductive logic does not preclude all predictions, but does affirm that predictions useful to deciding for or against eternal inflation are precluded. (shrink)

We argue that indeterminate probabilities are not only rationally permissible for a Bayesian agent, but they may even be rationally required . Our first argument begins by assuming a version of interpretivism: your mental state is the set of probability and utility functions that rationalize your behavioral dispositions as well as possible. This set may consist of multiple probability functions. Then according to interpretivism, this makes it the case that your credal state is indeterminate. Our second argument begins with (...) our describing a world that plausibly has indeterminate chances. Rationality requires a certain alignment of your credences with corresponding hypotheses about the chances. Thus, if you hypothesize the chances to be indeterminate, your will inherit their indeterminacy in your corresponding credences. Our third argument is motivated by a dilemma. Epistemic rationality requires you to stay open-minded about contingent matters about which your evidence has not definitively legislated. Practical rationality requires you to be able to act decisively at least sometimes. These requirements can conflict with each other-for thanks to your open-mindedness, some of your options may have undefined expected utility, and if you are choosing among them, decision theory has no advice to give you. Such an option is playing Nover and Hájek’s Pasadena Game , and indeed any option for which there is a positive probability of playing the Pasadena Game. You can serve both masters, epistemic rationality and practical rationality, with an indeterminate credence to the prospect of playing the Pasadena game. You serve epistemic rationality by making your upper probability positive-it ensures that you are open-minded. You serve practical rationality by making your lower probability 0-it provides guidance to your decision-making. No sharp credence could do both. (shrink)

Beyond Chance and Credence introduces a new way of thinking of probabilities in science that combines physical and epistemic considerations. Myrvold shows that conceiving of probabilities in this way solves puzzles associated with the use of probability and statistical mechanics.

It is well known that a degree-of-belief function P is coherent if and only if it satisfies the probability calculus. In this paper, we show that the notion of coherence can be extended to higher order probabilities such as P(P(h)=p)=q, and that a higher order degree-of-belief function P is coherent if and only if it satisfies the probability calculus plus the following axiom: P(h)=p iff P(P(h)=p)=1. Also, a number of lemmata which extend an incomplete probability function to a complete (...) one are established. (shrink)

This paper rejects a traditional epistemic interpretation of conditional probability. Suppose some chance process produces outcomes X, Y,..., with probabilities P(X), P(Y),... If later observation reveals that outcome Y has in fact been achieved, then the probability of outcome X cannot normally be revised to P(X|Y) ['P&Y)/P(Y)]. This can only be done in exceptional circumstances - when more than just knowledge of Y-ness has been attained. The primary reason for this is that the weight of a piece of evidence (...) varies with the means by which it is provided, so knowledge of Y-ness does not have uniform impact on the probability of X. A better updating of the probability of X is provided by P(X|Y*), where Y* is not an outcome of the chance process being observed, but the sentence 'the outcome Y has been observed', an 'outcome' of the subsequent observation process. This alternative formula is widely endorsed in practice, but not well recognized in theory, where the oversight has generated some unsatisfactory consequences. (shrink)

Nebel argues for the Repugnant Conclusion via the “Intrapersonal Repugnant Conclusion,” on which certainty of a mediocre life is better for individuals than a sufficiently small chance of an excellent life. In this article, I deny that acceptance of the Intrapersonal Repugnant Conclusion leads us to the Repugnant Conclusion. I point out that on many views which avoid the Repugnant Conclusion we should discount very small probabilities down to zero. If we do, then Nebel’s crucial premise of Ex Ante (...) Pareto fails, because discounting at the individual level can fail to match up with discounting at the population level. (shrink)

Bayesianism is the position that scientific reasoning is probabilistic and that probabilities are adequately interpreted as an agent's actual subjective degrees of belief, measured by her betting behaviour. Confirmation is one important aspect of scientific reasoning. The thesis of this paper is the following: if scientific reasoning is at all probabilistic, the subjective interpretation has to be given up in order to get right confirmation—and thus scientific reasoning in general. The Bayesian approach to scientific reasoning Bayesian confirmation theory The (...) example The less reliable the source of information, the higher the degree of Bayesian confirmation Measure sensitivity A more general version of the problem of old evidence Conditioning on the entailment relation The counterfactual strategy Generalizing the counterfactual strategy The desired result, and a necessary and sufficient condition for it Actual degrees of belief The common knock-down feature, or ‘anything goes’ The problem of prior probabilities. (shrink)

The usage of imprecise probabilities has been advocated in many domains: A number of philosophers have argued that our belief states should be “imprecise” in response to certain sorts of evidence, and imprecise probabilities have been thought to play an important role in disciplines such as artificial intelligence, climate science, and engineering. In this paper I’m interested in the question of whether the usage of imprecise probabilities can be given a practical motivation (a motivation based on practical (...) rather than epistemic, or alethic concerns). My aim is to challenge the central motivation for using imprecise probabilities in decision-making that has been offered in the literature: the idea that, in at least some contexts, it’s desirable to be ambiguity averse. If I succeed, this will show that we need to reconsider whether there are good reasons to use imprecise probabilities in contexts in which making good decisions is what's of primary concern. (shrink)

A number of philosophers have attempted to solve the problem of null-probability possible events in Bayesian epistemology by proposing that there are infinitesimal probabilities. Hájek and Easwaran have argued that because there is no way to specify a particular hyperreal extension of the real numbers, solutions to the regularity problem involving infinitesimals, or at least hyperreal infinitesimals, involve an unsatisfactory ineffability or arbitrariness. The arguments depend on the alleged impossibility of picking out a particular hyperreal extension of the real (...) numbers and/or of a particular value within such an extension due to the use of the Axiom of Choice. However, it is false that the Axiom of Choice precludes a specification of a hyperreal extension—such an extension can indeed be specified. Moreover, for all we know, it is possible to explicitly specify particular infinitesimals within such an extension. Nonetheless, I prove that because any regular probability measure that has infinitesimal values can be replaced by one that has all the same intuitive features but other infinitesimal values, the heart of the arbitrariness objection remains. (shrink)

Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences φ and θ, we consider the structures with domain {1,..., N} that satisfy θ, and compute the fraction of them in which φ is true. We then consider what happens to this fraction as N gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, by considering asymptotic (...) conditional probabilities. As shown by Liogon'kii [24], if there is a non-unary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kii also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is well-defined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable. (shrink)

Shoot an arrow at a wall, and then paint a target around it so that the arrow sticks squarely in the bull's eye. Alternatively, paint a fixed target on a wall, and then shoot an arrow so that it sticks squarely in the bull's eye. How do these situations differ? In both instances the precise place where the arrow lands is highly improbable. Yet in the one, one can do no better than attribute the arrow's landing to chance, whereas in (...) the other one rightly attributes the arrow's landing to the archer's skill. ;Highly improbable events occur all the time, and by themselves may legitimately be attributed to chance. Yet when an event also conforms to a pattern given independently of that event , one rightly refuses to attribute the event to chance. Patterns given independently of events are called specifications. This dissertation explicates the relation of independence between specification and event, and sets precise bounds on what may count as a probability small enough to eliminate chance. Although this dissertation connects with how statistics eliminates chance, it considerably extends and clarifies the range of application for chance-elimination arguments based on small probabilities. ;The dual notions of specification and small probability together help justify a regulative principle of probability called the Law of Small Probability, which asserts that specified events of small probability do not occur by chance. This principle engenders a form of inference called the design inference, and a mode of explanation called design. To infer design in explaining an event is to eliminate decisively explanations of that event which appeal to law-like regularities or chance. (shrink)

The propensity interpretation of probability, bred by Popper in 1957(K. R. Popper, in Observation and Interpretation in the Philosophy of Physics,S. Körner, ed. (Butterworth, London, 1957, and Dover, New York, 1962), p. 65; reprinted in Popper Selections,D. W. Miller, ed. (Princeton University Press, Princeton, 1985), p. 199) from pure frequency stock, is the only extant objectivist account that provides any proper understanding of single-case probabilities as well as of probabilities in ensembles and in the long run. In Sec. (...) 1 of this paper I recall salient points of the frequency interpretations of von Mises and of Popper himself, and in Sec. 2 I filter out from Popper's numerous expositions of the propensity interpretation its most interesting and fertile strain. I then go on to assess it. First I defend it, in Sec. 3, against recent criticisms(P. Humphreys, Philos. Rev.94,557 (1985); P. Milne, Erkenntnis25,129 (1986)) to the effect that conditional [or relative] probabilities, unlike absolute probabilities, can only rarely be made sense of as propensities. I then challenge its predominance, in Sec. 4, by outlining a rival theory: an irreproachably objectivist theory of probability, fully applicable to the single case, that interprets physical probabilities asinstantaneous frequencies. (shrink)

In this note we demonstrate that the results of observations in the EPR–Bohm–Bell experiment can be described within the classical probabilistic framework. However, the “quantum probabilities” have to be interpreted as conditional probabilities, where conditioning is with respect to fixed experimental settings. Our approach is based on the complete account of randomness involved in the experiment. The crucial point is that randomness of selections of experimental settings has to be taken into account within one consistent framework covering all (...) events related to the experiment. This approach can be applied to any complex experiment in which statistical data are collected for various experimental settings. (shrink)

We consider a setting where a decision maker’s uncertainty is represented by a set of probability measures, rather than a single measure. Measure-by-measure updating of such a set of measures upon acquiring new information is well known to suffer from problems. To deal with these problems, we propose using weighted sets of probabilities: a representation where each measure is associated with a weight, which denotes its significance. We describe a natural approach to updating in such a situation and a (...) natural approach to determining the weights. We then show how this representation can be used in decision making, by modifying a standard approach to decision making—minimizing expected regret—to obtain minimax weighted expected regret. We provide an axiomatization that characterizes preferences induced by MWER both in the static and dynamic case. (shrink)

The field of of imprecise probability has matured, in no small part because of Teddy Seidenfeld’s decades of original scholarship and essential contributions to building and sustaining the ISIPTA community. Although the basic idea behind imprecise probability is (at least) 150 years old, a mature mathematical theory has only taken full form in the last 30 years. Interest in imprecise probability during this period has also grown, but many of the ideas that the mature theory serves can be difficult to (...) apprehend to those new to the subject. Although these fundamental ideas are common knowledge in the ISIPTA community, they are expressed, when they are expressed at all, obliquely, over the course of years with students and colleagues. -/- A single essay cannot convey the store of common knowledge from any research community, let alone the ISIPTA community. But, this essay nevertheless is an attempt to guide those familiar with the basic Bayesian framework to appreciate some of the elegant and powerful ideas that underpin the contemporary theory of lower previsions, which is the theory that most people associate with the term ‘imprecise probabilities’. (shrink)

The general properties of measurements in microphysics are studied and the three types of probabilities that, according to the authors, appear in wave mechanics are set up. Such a distinction, together with the principle of the localization of the corpuscle as was laid down at the very introduction of the theory of the double solution, provides a good grasp of certain phenomena whose explanation according to the usual theory (which makes no use of permanent localization and where the three (...) types of probabilities are intermingled) leads to paradoxical or self-contradictory situations. (shrink)

We propose a new account of indicative conditionals, giving acceptability and logical closure conditions for them. We start from Adams’ Thesis: the claim that the acceptability of a simple indicative equals the corresponding conditional probability. The Thesis is widely endorsed, but arguably false and refuted by empirical research. To fix it, we submit, we need a relevance constraint: we accept a simple conditional 'If φ, then ψ' to the extent that (i) the conditional probability p(ψ|φ) is high, provided that (ii) (...) φ is relevant for ψ. How (i) should work is well-understood. It is (ii) that holds the key to improve our understanding of conditionals. Our account has (i) a probabilistic component, using Popper functions; (ii) a relevance component, given via an algebraic structure of topics or subject matters. We present a probabilistic logic for simple indicatives, and argue that its (in)validities are both theoretically desirable and in line with empirical results on how people reason with conditionals. (shrink)

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.

We consider probabilistic theories in which the most elementary system, a two-dimensional system, contains one bit of information. The bit is assumed to be contained in any complete set of mutually complementary measurements. The requirement of invariance of the information under a continuous change of the set of mutually complementary measurements uniquely singles out a measure of information, which is quadratic in probabilities. The assumption which gives the same scaling of the number of degrees of freedom with the dimension (...) as in quantum theory follows essentially from the assumption that all physical states of a higher dimensional system are those and only those from which one can post-select physical states of two-dimensional systems. The requirement that no more than one bit of information (as quantified by the quadratic measure) is contained in all possible post-selected two-dimensional systems is equivalent to the positivity of density operator in quantum theory. (shrink)

The non-random mixing of biomembrane components, especially saturated phospholipids, exhibits important consequences in molecular biology. Particularly, the distribution of lipids within natural and model membranes is strongly determined by the selective association processes. These processes of phospholipids take place due to the cooperative modes in multiparticle systems as well as the specific lipid-lipid interactions both in the hydrophobic core and in the region of the polar headgroups. We demonstrated that the investigation of the selective association processes of saturated phospholipids might (...) contribute to the insight of the lipid domains appearance inside the bilayer membranes. The association probabilities of like- pairs and cross- pairs from a binary mixture of saturated phospholipids were tested for both parallel and anti-parallel alignments of the polar headgroups. The present model confirms the experimental evidence for saturated phospholipids to have a high tendency for association in parallel configuration of the electric dipole moments of the polar headgroups whether the cross-sectional area of the polar headgroup is in an usual range of 25-55 2. There are three major lipid domains in a binary mixture of saturated phospholipids: lipid domains in non-mixed phase of the first mixture component, in parallel alignment of the polar headgroups; lipid domains in non-mixed phase of the second mixture component, in anti-parallel alignment of the polar headgroups; lipid domains in mixed phase. We think that the selective association processes of phospholipids are neither exclusively, nor only involved in promoting the lipid domains appearance through bilayer phospholipid membranes. (shrink)