The major competing statistical paradigms share a common remarkable but unremarked thread: in many of their inferential applications, different probability interpretations are combined. How this plays out in different theories of inference depends on the type of question asked. We distinguish four question types: confirmation, evidence, decision, and prediction. We show that Bayesian confirmation theory mixes what are intuitively “subjective” and “objective” interpretations of probability, whereas the likelihood-based account of evidence melds three conceptions of what constitutes an “objective” probability.

We introduce a distinction, unnoticed in the literature, between four varieties of objective Bayesianism. What we call ' strong objective Bayesianism' is characterized by two claims, that all scientific inference is 'logical' and that, given the same background information two agents will ascribe a unique probability to their priors. We think that neither of these claims can be sustained; in this sense, they are 'dogmatic'. The first fails to recognize that some scientific inference, in particular that concerning evidential relations, is (...) not (in the appropriate sense) logical, the second fails to provide a non-question-begging account of 'same background information'. We urge that a suitably objective Bayesian account of scientific inference does not require either of the claims. Finally, we argue that Bayesianism needs to be fine-grained in the same way that Bayesians fine-grain their beliefs. (shrink)

Reasoning from inconclusive evidence, or ‘induction’, is central to science and any applications we make of it. For that reason alone it demands the attention of philosophers of science. This Element explores the prospects of using probability theory to provide an inductive logic, a framework for representing evidential support. Constraints on the ideal evaluation of hypotheses suggest that overall support for a hypothesis is represented by its probability in light of the total evidence, and incremental support, or confirmation, indicated by (...) an increased probability given the evidence than otherwise. This proposal is shown to have the capacity to reconstruct many canons of the scientific method and inductive inference. Along the way, significant objections are discussed, such as the challenge from inductive scepticism and the objection that the probabilistic approach makes evidential support arbitrary. (shrink)

– 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)

Does rationality require imprecise credences? Many hold that it does: imprecise evidence requires correspondingly imprecise credences. I argue that this is false. The imprecise view faces the same arbitrariness worries that were meant to motivate it in the first place. It faces these worries because it incorporates a certain idealization. But doing away with this idealization effectively collapses the imprecise view into a particular kind of precise view. On this alternative, our attitudes should reflect a kind of normative uncertainty: uncertainty (...) about what to believe. This view refutes the claim that precise credences are inappropriately informative or committal. Some argue that indeterminate evidential support requires imprecise credences; but I argue that indeterminate evidential support instead places indeterminate requirements on credences, and is compatible with the claim that rational credences may always be precise. (shrink)

Traditional accounts of justification can be characterized as trying to analyze justification in such a way that having a justified belief brings with it assurance of truth. The internalist offers a demanding requirement on justification: one's having a justified belief requires that one see what the belief has going for it. Externalists worry that the internalist's narrow conception of justification will lead to unacceptably radical and implausible skepticism. According to the externalist, one need not know what a belief has going (...) for it in order for that belief to be justified. Externalism, though, comes with its own problems. Ernest Sosa has attempted to bridge the divide between internalism and externalism by pairing the strengths of internalism with the strengths of externalism. Sosa distinguishes two kinds of knowledge: animal knowledge that is essentially externalist in character and reflective knowledge that is intended to capture our best intellectual procedure in regards to knowledge. On Sosa's view, one gains reflective knowledge by building upon animal knowledge. As a result, Sosa's view seems to illustrate a bottom-up approach to the analysis of knowledge : reflective knowledge is the result of animal knowledge and some other epistemic factor. My project, in contrast to Sosa's, is to argue that one should start with an account of ideal justification and then proceed by loosening the standards on ideal justification in an effort to develop the possibility of non- ideal kinds of justification. The view that I will develop will adopt Sosa's strategy of distinguishing kinds of knowledge, but will result in a top-down approach to the analysis of justification. Instead of starting with an undemanding standard and layer levels on top, I will start with an ideal standard and strip layers away. I will also argue that my view has some important advantages over Sosa's. Not only does Sosa's view seem to run into many of the problems that threaten externalism, but his view is incapable of offering the kind of assurance that the internalist is after. The view I develop will maintain the internalist's interest in assurance while also providing a response to some of the skeptical problems that have plagued internalists. If my project is successful, then, even if the justification that results in many of the cases I will be exploring is not ideal, we can use these conceptions of justification to help explicate how one might have justified beliefs about a great number of things. The essentially internalist account that I have offered will not only illustrate a serious approach to dealing with skepticism, but it will also capture how many of our commonsensically justified beliefs are in fact justified. (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)

This paper has two main parts. In the first part, we motivate a kind of indeterminate, suppositional credences by discussing the prospect for a subjective interpretation of a causal Bayesian network, an important tool for causal reasoning in artificial intelligence. A CBN consists of a causal graph and a collection of interventional probabilities. The subjective interpretation in question would take the causal graph in a CBN to represent the causal structure that is believed by an agent, and interventional probabilities in (...) a CBN to represent suppositional credences. We review a difficulty noted in the literature with such an interpretation, and suggest that a natural way to address the challenge is to go for a generalization of CBN that allows indeterminate credences. In the second part, we develop a decision-theoretic foundation for such indeterminate suppositional credences, by generalizing a theory of coherent choice functions to accommodate some form of act-state dependence. The upshot is a decision-theoretic framework that is not only rich enough to, so to speak, ground the probabilities in a subjectively interpreted causal network, but also interesting in its own right, in that it accommodates both act-state dependence and imprecise probabilities. (shrink)