I argue that when we use ‘probability’ language in epistemic contexts—e.g., when we ask how probable some hypothesis is, given the evidence available to us—we are talking about degrees of support, rather than degrees of belief. The epistemic probability of A given B is the mind-independent degree to which B supports A, not the degree to which someone with B as their evidence believes A, or the degree to which someone would or should believe A if they had B as (...) their evidence. My central argument is that the degree-of-support interpretation lets us better model good reasoning in certain cases involving old evidence. Degree-of-belief interpretations make the wrong predictions not only about whether old evidence confirms new hypotheses, but about the values of the probabilities that enter into Bayes’ Theorem when we calculate the probability of hypotheses conditional on old evidence and new background information. (shrink)

Orthodox Bayesianism is a highly idealized theory of how we ought to live our epistemic lives. One of the most widely discussed idealizations is that of logical omniscience: the assumption that an agent’s degrees of belief must be probabilistically coherent to be rational. It is widely agreed that this assumption is problematic if we want to reason about bounded rationality, logical learning, or other aspects of non-ideal epistemic agency. Yet, we still lack a satisfying way to avoid logical omniscience within (...) a Bayesian framework. Some proposals merely replace logical omniscience with a different logical idealization; others sacrifice all traits of logical competence on the altar of logical non-omniscience. We think a better strategy is available: by enriching the Bayesian framework with tools that allow us to capture what agents can and cannot infer given their limited cognitive resources, we can avoid logical omniscience while retaining the idea that rational degrees of belief are in an important way constrained by the laws of probability. In this paper, we offer a formal implementation of this strategy, show how the resulting framework solves the problem of logical omniscience, and compare it to orthodox Bayesianism as we know it. (shrink)

Humean arguments from evil are some of the most powerful arguments against Theism. They take as their data what we know about good and evil. And they argue that some rival to Theism better explains, or otherwise predicts, that data than Theism. However, this paper argues that there are many problems with various methods for defending Humean arguments. I consider Philo’s original strategy; modern strategies in terms of epistemic probability; phenomenological strategies; and strategies that appeal to scientific and metaphysical explanations. (...) None of these methods have been sufficiently developed to provide a clear and distinctive defense of Humean arguments. Defenders of Humean arguments need to spend more time on the underlying methodology of their arguments. (shrink)

We present a conservative extension of a Bayesian account of confirmation that can deal with the problem of old evidence and new theories. So-called open-minded Bayesianism challenges the assumption—implicit in standard Bayesianism—that the correct empirical hypothesis is among the ones currently under consideration. It requires the inclusion of a catch-all hypothesis, which is characterized by means of sets of probability assignments. Upon the introduction of a new theory, the former catch-all is decomposed into a new empirical hypothesis and a new (...) catch-all. As will be seen, this motivates a second update rule, besides Bayes’ rule, for updating probabilities in light of a new theory. This rule conserves probability ratios among the old hypotheses. This framework allows for old evidence to confirm a new hypothesis due to a shift in the theoretical context. The result is a version of Bayesianism that, in the words of Earman, “keep[s] an open mind, but not so open that your brain falls out”. (shrink)

Why are conditional degrees of belief in an observation E, given a statistical hypothesis H, aligned with the objective probabilities expressed by H? After showing that standard replies are not satisfactory, I develop a suppositional analysis of conditional degree of belief, transferring Ramsey’s classical proposal to statistical inference. The analysis saves the alignment, explains the role of chance-credence coordination, and rebuts the charge of arbitrary assessment of evidence in Bayesian inference. Finally, I explore the implications of this analysis for Bayesian (...) reasoning with idealized models in science. (shrink)

Can computer simulation results be evidence for hypotheses about real-world systems and phenomena? If so, what sort of evidence? Can we gain genuinely new knowledge of the world via simulation? I argue that evidence from computer simulation is aptly characterized as higher-order evidence: it is evidence that other evidence regarding a hypothesis about the world has been collected. Insofar as particular epistemic agents do not have this other evidence, it is possible that they will gain genuinely new knowledge of the (...) world via simulation. I illustrate with examples inspired by uses of simulation in meteorology and astrophysics. (shrink)

Analogy has received attention as a form of inductive reasoning in the empirical sciences. Its role in mathematics has, instead, received less consideration. This paper provides a novel account of how an analogy with a more familiar mathematical domain can contribute to the confirmation of a mathematical conjecture. By reference to case-studies, we propose a distinction between an incremental and a non-incremental form of confirmation by mathematical analogy. We offer an account of the former within the popular framework of Bayesian (...) confirmation theory. As for the non-incremental notion, we defend its role in rationally informing the prior credences of mathematicians in those circumstances in which no new mathematical evidence is introduced. The resulting framework captures many important aspects of the use of analogical inference in the domain of pure mathematics. (shrink)

Derivational robustness may increase the degree to which various pieces of evidence indirectly confirm a robust result. There are two ways in which this increase may come about. First, if one can show that a result is robust, and that the various individual models used to derive it also have other confirmed results, these other results may indirectly confirm the robust result. Confirmation derives from the fact that data not known to bear on a result are shown to be relevant (...) when it is shown to be robust. Second, robustness may increase the degree to which the robust result is indirectly confirmed if it increases the weight with which existing evidence indirectly confirms it. This may happen when it strengthens the connection between the core and the robust result by showing that auxiliaries are not responsible for the result. (shrink)

The Problem of Old Evidence is a perennial issue for Bayesian confirmation theory. Garber famously argues that the problem can be solved by conditionalizing on the proposition that a hypothesis deductively implies the existence of the old evidence. In recent work, Hartmann and Fitelson :712–717, 2015) and Sprenger :383–401, 2015) aim for similar, but more general, solutions to the Problem of Old Evidence. These solutions are more general because they allow the explanatory relationship between a new hypothesis and old evidence (...) to be inductive, rather than deductive. In this paper, I argue that these solutions are either unsound or under-motivated, depending on the case of inductive explanation that we have in mind. This lends support to the broader claim that Garber–Style Bayesian confirmation cannot capture the sense in which new hypotheses that do not deductively imply old evidence nevertheless seem to be confirmed via old evidence. (shrink)

The Problem of Old Evidence is a perennial issue for Bayesian confirmation theory. Garber (Test Sci Theor 10:99–131, 1983) famously argues that the problem can be solved by conditionalizing on the proposition that a hypothesis deductively implies the existence of the old evidence. In recent work, Hartmann and Fitelson (Philos Sci 82(4):712–717, 2015) and Sprenger (Philos Sci 82(3):383–401, 2015) aim for similar, but more general, solutions to the Problem of Old Evidence. These solutions are more general because they allow the (...) explanatory relationship between a new hypothesis and old evidence to be inductive, rather than deductive. In this paper, I argue that these solutions are either unsound or under-motivated, depending on the case of inductive explanation that we have in mind. This lends support to the broader claim that Garber–Style Bayesian confirmation cannot capture the sense in which new hypotheses that do not deductively imply old evidence nevertheless seem to be confirmed via old evidence. (shrink)

In this discussion note, we explain how to relax some of the standard assumptions made in Garber-style solutions to the Problem of Old Evidence. The result is a more general and explanatory Bayesian approach.

How good is an explanation and when is one explanation better than another? In this paper, I address these questions by exploring probabilistic measures of explanatory power in order to defend a particular Bayesian account of explanatory goodness. Critical to this discussion is a distinction between weak and strong measures of explanatory power due to Good (Br J Philos Sci 19:123–143, 1968). In particular, I argue that if one is interested in the overall goodness of an explanation, an appropriate balance (...) needs to be struck between the weak explanatory power and the complexity of a hypothesis. In light of this, I provide a new defence of a strong measure proposed by Good by providing new derivations of it, comparing it with other measures and exploring its connection with information, confirmation and explanatory virtues. Furthermore, Good really presented a family of strong measures, whereas I draw on a complexity criterion that favours a specific measure and hence provides a more precise way to quantify explanatory goodness. (shrink)

Bayesian confirmation theory is our best formal framework for describing inductive reasoning. The problem of old evidence is a particularly difficult one for confirmation theory, because it suggests that this framework fails to account for central and important cases of inductive reasoning and scientific inference. I show that we can appeal to the fragmentation of doxastic states to solve this problem for confirmation theory. This fragmentation solution is independently well-motivated because of the success of fragmentation in solving other problems. I (...) also argue that the fragmentation solution is preferable to other solutions to the problem of old evidence. These other solutions are already committed to something like fragmentation, but suffer from difficulties due to their additional commitments. If these arguments are successful, Bayesian confirmation theory is saved from the problem of old evidence, and the argument for fragmentation is bolstered by its ability to solve yet another problem. (shrink)

The problem of old evidence, first described by Glymour [1980], is still widely regarded as one of the most pressing foundational challenges to the Bayesian account of scientific reasoning. Many solutions have been proposed, but all of them have drawbacks and none is considered to be definitive. Here, we introduce and defend a new kind of solution, according to which hypotheses are confirmed when we become more confident that they provide the only way of accounting for the known evidence.

We explore the question of whether cost-free uncertain evidence is worth waiting for in advance of making a decision. A classical result in Bayesian decision theory, known as the value of evidence theorem, says that, under certain conditions, when you update your credences by conditionalizing on some cost-free and certain evidence, the subjective expected utility of obtaining this evidence is never less than the subjective expected utility of not obtaining it. We extend this result to a type of update method, (...) a variant of Judea Pearl’s virtual conditionalization, where uncertain evidence is represented as a set of likelihood ratios. Moreover, we argue that focusing on this method rather than on the widely accepted Jeffrey conditionalization enables us to show that, under a fairly plausible assumption, gathering uncertain evidence not only maximizes expected pragmatic utility, but also minimizes expected epistemic disutility. (shrink)

I discuss a contemporary solution to the dynamic problem of old evidence (POE), as proposed by Sprenger. Sprenger’s solution combines the Garber–Jeffrey–Niiniluoto (GJN) approach with Howson’s suggestion of counterfactually removing the old evidence from scientists’ belief systems. I argue that in the dynamic POE, the challenge is to explain how an insight under beliefs in which the old evidence E is known increased the credence of a scientific hypothesis. Therefore, Sprenger’s counterfactual solution, in which E has been artificially removed, does (...) not resolve the problem. I consider several potential responses. (shrink)

‘Bayesian epistemology’ became an epistemological movement in the 20th century, though its two main features can be traced back to the eponymous Reverend Thomas Bayes (c. 1701-61). Those two features are: (1) the introduction of a formal apparatus for inductive logic; (2) the introduction of a pragmatic self-defeat test (as illustrated by Dutch Book Arguments) for epistemic rationality as a way of extending the justification of the laws of deductive logic to include a justification for the laws of inductive logic. (...) The formal apparatus itself has two main elements: the use of the laws of probability as coherence constraints on rational degrees of belief (or degrees of confidence) and the introduction of a rule of probabilistic inference, a rule or principle of conditionalization. (shrink)

In Bayes or Bust? John Earman quickly dismisses a possible resolution (or avoidance) of the problem of old evidence. In this note, I argue that his dismissal is premature, and that the proposed resolution (when charitably reconstructed) is reasonable.