The Dempster–Shafer approach to expressing beliefabout a parameter in a statistical model is notconsistent with the likelihood principle. Thisinconsistency has been recognized for some time, andmanifests itself as a non-commutativity, in which theorder of operations (combining belief, combininglikelihood) makes a difference. It is proposed herethat requiring the expression of belief to be committed to the model (and to certain of itssubmodels) makes likelihood inference very nearly aspecial case of the Dempster–Shafer theory.

If we receive information from multiple independent and partially reliable information sources, then whether we are justified to believe these information items is affected by how reliable the sources are, by how well the information coheres with our background beliefs and by how internally coherent the information is. We consider the following question. Is coherence a separable determinant of our degree of belief, i.e. is it the case that the more coherent the new information is, the more justified we are (...) in believing the new information, ceteris paribus? We show that if we consider sets of information items of any size (Holism), and if we assume that there exists a coherence Ordering over such sets and that coherence is a function of the probability distribution over the propositions in such sets (Probabilism), then Separability fails to hold. (shrink)

I outline four competing probabilistic accounts of contrastive evidential support and consider various considerations that might help arbitrate between these. The upshot of the discussion is that the so-called ‘Law of Likelihood’ is to be preferred to any of the alternatives considered.

This paper is a response to Forster and Sober's [1994] solution to the curve-fitting problem. If their solution is correct, it will provide us with a solution to the New Riddle of Induction as well as provide a basis for choosing realism over conventionalism. Examining this solution is also important as Forster and Sober incorporate it in much of their other philosophical work (see Forster [1995a, b, 1994] and Sober [1996, 1995, 1993]). I argue that Forster and Sober's solution is (...) subject precisely to the problem they seek to solve. They provide a method of choosing among hypotheses but only at the cost of requiring that we have a method of choosing between different ways of conceptualizing the world. Thus the solution raises a new problem—the world-fitting problem. (shrink)

In his "From classical to constructive probability," Weatherson offers a generalization of Kolmogorov's axioms of classical probability that is neutral regarding the logic for the object-language. Weatherson's generalized notion of probability can hardly be regarded as adequate, as the example of supervaluationist logic shows. At least, if we model credences as betting rates, the Dutch-Book argument strategy does not support Weatherson's notion of supervaluationist probability, but various alternatives. Depending on whether supervaluationist bets are specified as (a) conditional bets (Cantwell), (b) (...) unconditional bets with graded payoffs (Milne), or (c) unconditional bets with ungraded payoffs(Dietz), supervaluationist probability amounts to (a) conditional probability of truth given a truth-value, (b) the expected truth-value, or (c) the probability of truth, respectively. It is suggested that for supervaluationist logic, the third option is the most attractive one, for (unlike the other options) it preserves respect for single-premise entailment. (shrink)

Van Fraassen (1989) argues that Inference to the Best Explanation is incoherent in the sense that adopting it as a rule for belief change will make one susceptible to a dynamic Dutch book. The present paper argues against this. A strategy is described that allows us to infer to the best explanation free of charge.

In the first paper, I discussed the basic claims of Bayesianism (that degrees of belief are important, that they obey the axioms of probability theory, and that they are rationally updated by either standard or Jeffrey conditionalization) and the arguments that are often used to support them. In this paper, I will discuss some applications these ideas have had in confirmation theory, epistemol- ogy, and statistics, and criticisms of these applications.

The “puzzle of the unmarked clock” derives from a conflict between the following: (1) a plausible principle of epistemic modesty, and (2) “Rational Reflection”, a principle saying how one’s beliefs about what it is rational to believe constrain the rest of one’s beliefs. An independently motivated improvement to Rational Reflection preserves its spirit while resolving the conflict.

Likelihoodists and Bayesians seem to have a fundamental disagreement about the proper probabilistic explication of relational (or contrastive) conceptions of evidential support (or confirmation). In this paper, I will survey some recent arguments and results in this area, with an eye toward pinpointing the nexus of the dispute. This will lead, first, to an important shift in the way the debate has been couched, and, second, to an alternative explication of relational support, which is in some sense a "middle way" (...) between Likelihoodism and Bayesianism. In the process, I will propose some new work for an old probability puzzle: the "Monty Hall" problem. (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)

Recent solutions to the curve-fitting problem, described in Forster and Sober ([1995]), trade off the simplicity and fit of hypotheses by defining simplicity as the paucity of adjustable parameters. Scott De Vito ([1997]) charges that these solutions are 'conventional' because he thinks that the number of adjustable parameters may change when the hypotheses are described differently. This he believes is exactly what is illustrated in Goodman's new riddle of induction, otherwise known as the grue problem. However, the 'number of adjustable (...) parameters' is actually a loose way of referring to a quantity that is not language dependent. The quantity arises out of Akaike's theorem in a way that ensures its language invariance. (shrink)

If we receive information from multiple independent and partially reliable information sources, then whether we are justified to believe these information items is affected by how reliable the sources are, by how well the information coheres with our background beliefs and by how internally coherent the information is. We consider the following question. Is coherence a separable determinant of our degree of belief, i.e. is it the case that the more coherent the new information is, the more justified we are (...) in believing the new information, ceteris paribus? We show that if we consider sets of information items of any size (Holism), and if we assume that there exists a coherence Ordering over such sets and that coherence is a function of the probability distribution over the propositions in such sets (Probabilism), then Separability fails to hold. (shrink)

Fine-tuning arguments attempt to infer God’s existence from the empirical fact that life would not be possible if any of approximately two-dozen fundamental laws and properties of the universe had been even slightly different. In this essay, I consider a version that relies on the following principle: if an observation O is more likely to occur under hypothesis H1 than under hypothesis H2, then O supports accepting H1 over H2. I argue that this particular application of this principle is vulnerable (...) to straightforward counterexamples and attempt to explain the proper application conditions for this principle. (shrink)

Many people believe that there is a Dutch Book argument establishing that the principle of countable additivity is a condition of coherence. De Finetti himself did not, but for reasons that are at first sight perplexing. I show that he rejected countable additivity, and hence the Dutch Book argument for it, because countable additivity conflicted with intuitive principles about the scope of authentic consistency constraints. These he often claimed were logical in nature, but he never attempted to relate this idea (...) to deductive logic and its own concept of consistency. This I do, showing that at one level the definitions of deductive and probabilistic consistency are identical, differing only in the nature of the constraints imposed. In the probabilistic case I believe that R.T. Cox's ‘scale-free’ axioms for subjective probability are the most suitable candidates. 1 Introduction 2 Coherence and Consistency 3 The Infinite Fair Lottery 4 The Puzzle Resolved—But Replaced by Another 5 Countable Additivity, Conglomerability and Dutch Books 6 The Probability Axioms and Cox's Theorem 7 Truth and Probability 8 Conclusion: ‘Logical Omniscience’ CiteULike Connotea Del.icio.us What's this? (shrink)

A central problem facing a probabilistic approach to the problem of induction is the difficulty of sufficiently constraining prior probabilities so as to yield the conclusion that induction is cogent. The Principle of Indifference, according to which alternatives are equiprobable when one has no grounds for preferring one over another, represents one way of addressing this problem; however, the Principle faces the well-known problem that multiple interpretations of it are possible, leading to incompatible conclusions. I propose a partial solution to (...) the latter problem, drawing on the notion of explanatory priority. The resulting synthesis of Bayesian and inference-to-best-explanation approaches affords a principled defense of prior probability distributions that support induction. (shrink)

I discuss the philosophical significance of the statistical model selection criteria, in particular their relevance for philosophical problems of underdetermination. I present an easily comprehensible account of their simplest possible application and contrast it with their application to curve-fitting problems. I embed philosophers' earlier discussion concerning the situations in which the criteria yield implausible results into a more general framework. Among other things, I discuss a difficulty which is related to the so-called subfamily problem, and I show that it has (...) analogies in all legitimate applications of the model selection criteria, and that an analogy of Goodman's new riddle of induction can be formulated in only some of their applications. (shrink)

Weighted averaging is a method for aggregating the totality of information, both regimented and unregimented, possessed by an individual or group of individuals. The application of such a method may be warranted by a theorem of the calculus of probability, simple conditionalization, or Jeffrey's formula for probability kinematics, all of which average in terms of the prior probability of evidence statements. Weighted averaging may, however, be applied as a method of rational aggregation of the probabilities of diverse perspectives or persons (...) in cases in which the weights cannot be articulated as the prior probabilities of statements of evidence. The method is justified by Wagner's Theorem exhibiting that any method satisfying the conditions of the Irrelevance of Alternatives and Zero Unanimity must, when applied to three or more alternatives, be weighted averaging. (shrink)

We defend a set of acceptance rules that avoids the lottery paradox, that is closed under classical entailment, and that accepts uncertain propositions without ad hoc restrictions. We show that the rules we recommend provide a semantics that validates exactly Adams’ conditional logic and are exactly the rules that preserve a natural, logical structure over probabilistic credal states that we call probalogic . To motivate probalogic, we first expand classical logic to geo-logic , which fills the entire unit cube, and (...) then we project the upper surfaces of the geo-logical cube onto the plane of probabilistic credal states by means of standard, linear perspective, which may be interpreted as an extension of the classical principle of indifference. Finally, we apply the geometrical/logical methods developed in the paper to prove a series of trivialization theorems against question-invariance as a constraint on acceptance rules and against rational monotonicity as an axiom of conditional logic in situations of uncertainty. (shrink)

Probabilistic belief contraction has been a much neglected topic in the field of probabilistic reasoning. This is due to the difficulty in establishing a reasonable reversal of the effect of Bayesian conditionalization on a probabilistic distribution. We show that indifferent contraction, a solution proposed by Ramer to this problem through a judicious use of the principle of maximum entropy, is a probabilistic version of a full meet contraction. We then propose variations of indifferent contraction, using both the Shannon entropy measure (...) as well as the Hartley entropy measure, with an aim to avoid excessive loss of beliefs that full meet contraction entails. (shrink)

Probabilistic support is not transitive. There are cases in which x probabilistically supports y , i.e., Pr( y | x ) > Pr( y ), y , in turn, probabilistically supports z , and yet it is not the case that x probabilistically supports z . Tomoji Shogenji, though, establishes a condition for transitivity in probabilistic support, that is, a condition such that, for any x , y , and z , if Pr( y | x ) > Pr( y (...) ), Pr( z | y ) > Pr( z ), and the condition in question is satisfied, then Pr( z | x ) > Pr( z ). I argue for a second and weaker condition for transitivity in probabilistic support. This condition, or the principle involving it, makes it easier (than does the condition Shogenji provides) to establish claims of probabilistic support, and has the potential to play an important role in at least some areas of philosophy. (shrink)

Igor Douven establishes several new intransitivity results concerning evidential support. I add to Douven’s very instructive discussion by establishing two further intransitivity results and a transitivity result.

There is considerable confusion about the role of p-values in statistical model checking. To clarify that point, I introduce the distinction between measures of surprise and measures of evidence which come with different epistemological functions. I argue that p-values, often understood as measures of evidence against a null model, do not count as proper measures of evidence and are closer to measures of surprise. Finally, I sketch how the problem of old evidence may be tackled by acknowledging the epistemic role (...) of surprise indices. (shrink)

“Double-halfers” think that throughout the Sleeping Beauty Problem, Beauty should keep her credence that a fair coin flip came up heads equal to 1/2. I introduce a new wrinkle to the problem that shows even double-halfers can't keep Beauty's credences equal to the objective chances for all coin-flip propositions. This leaves no way to deny that self-locating information generates an unexpected kind of inadmissible evidence.