Bayesian methods have become among the most popular methods in phylogenetics, but theoretical opposition to this methodology remains. After providing an introduction to Bayesian theory in this context, I attempt to tackle the problem mentioned most often in the literature: the “problem of the priors”—how to assign prior probabilities to tree hypotheses. I first argue that a recent objection—that an appropriate assignment of priors is impossible—is based on a misunderstanding of what ignorance and bias are. I then consider (...) different methods of assigning prior probabilities to trees. I argue that priors need to be derived from an understanding of how distinct taxa have evolved and that the appropriate evolutionary model is captured by the Yule birth–death process. This process leads to a well-known statistical distribution over trees. Though further modifications may be necessary to model more complex aspects of the branching process, they must be modifications to parameters in an underlying Yule model. Ignoring these Yule priors commits a fallacy leading to mistaken inferences both about the trees themselves and about macroevolutionary processes more generally. (shrink)

Although Bayesian methods are widely used in phylogenetic systematics today, the foundations of this methodology are still debated among both biologists and philosophers. The Bayesian approach to phylogenetic inference requires the assignment of prior probabilities to phylogenetic trees. As in other applications of Bayesian epistemology, the question of whether there is an objective way to assign these prior probabilities is a contested issue. This paper discusses the strategy of constraining the prior probabilities of phylogenetic (...) trees by means of the Principal Principle. In particular, I discuss a proposal due to Velasco (Biol Philos 23:455–473, 2008) of assigning prior probabilities to tree topologies based on the Yule process. By invoking the Principal Principle I argue that prior probabilities of tree topologies should rather be assigned a weighted mixture of probability distributions based on Pinelis’ (P Roy Soc Lond B Bio 270:1425–1431, 2003) multi-rate branching process including both the Yule distribution and the uniform distribution. However, I argue that this solves the problem of the priors of phylogenetic trees only in a weak form. (shrink)

Meehl's statement "in most psychological research, Improved power of a statistical design leads to a prior probability approaching 1/2 of finding a significant difference in the theoretically predicted direction" (philosophy of science, Volume 34, Pages 103-115), Is without foundation. The computation of prior probabilities of accepting or rejecting a hypothesis presupposes knowledge of the prior probabilities that this hypothesis or any of its conceivable alternatives are true. As we do not have such knowledge, We cannot (...) give any numerical values of prior probabilities of accepting or rejecting hypotheses in any statistical test procedure. Only topological statements are possible, As for example: when the region of acceptance of a hypothesis narrows, The prior probability that it will be accepted remains constant or increases. (shrink)

The theoretical construction and practical use of prior probabilities, in particular for systems having many degrees of freedom, are investigated. It becomes clear that it is operationally unsound to use mutually consistent priors if one wishes to draw sensible conclusions from practical experiments. The prior cannot usefully be identified with a state of knowledge, and indeed it is not so identified in common scientific practice. Rather, it can be identified with the question one asks. Accordingly, priors are (...) free constructions. Their informal, ill-defined and subjective characteristics must carry over into the conclusions one chooses to draw from experiments or observations. (shrink)

Theism is a far simpler hypothesis, and so a priori more probably true, than naturalism, understood as the hypothesis that the existence of this law-governeduniverse has no explanation. Theism postulates only one entity (God) with very simple properties, whereas naturalism has to postulate either innumerableentities all having the same properties, or one very complicated entity with the power to produce the former. If theism is true, it is moderately probable that God would create humanoid beings and so humanoid bodies; but (...) laws of nature would have to have very special properties if they are to bring about the existencne of humaoid bodies. Given laws of the present form (quantum theory with the Pauli principle and the four forces), the constants of the laws and variables of the boundary conditions of the universe would need to be extremely fine-tuned; and no simpler set of laws would allow the existence of humanoid bodies at all. So the evidence of the existence of humanoid bodies adds further to the probability of theismas against naturalism. (shrink)

A consideration of some basic problems that arise in the attempt to provide an adequate characterization of statistical explanation is taken to show that an understanding of the nature of scientific explanation requires us to deal with the philosophical problems connected with the nature of prior probabilities.

Paul Draper has recently developed an account of intrinsic probability according to which a theory's intrinsic probability is determined by its modesty and coherence. He employs this account in an argument that Source Physicalism (SP) and Source Idealism (SI) are equally intrinsically probable. Since SP and SI are not exhaustive, and Theism is one very specific version of SI, it follows that the intrinsic probability of Theism is very low. I argue here that considerations of fundamentality show that more work (...) needs to be done to defend the claim that P(SP) = P(SI). (shrink)

Wittgenstein did not write very much on the topic of probability. The little we have comes from a few short pages of the Tractatus, some 'remarks' from the 1930s, and the informal conversations which went on during that decade with the Vienna Circle. Nevertheless, Wittgenstein's views were highly influential in the later development of the logical theory of probability. This paper will attempt to clarify and defend Wittgenstein's conception of probability against some oft-cited criticisms that stem from a misunderstanding of (...) his views. Max Black, for instance, criticises Wittgenstein for formulating a theory of probability that is capable of being used only against the backdrop of the ideal language of the Tractatus. I argue that on the contrary, by appealing to the 'hypothetical laws of nature', Wittgenstein is able to make sense of probability statements involving propositions that have not been completely analysed. G.H. von Wright criticises Wittgenstein's characterisation of these very hypothetical laws. He argues that by introducing them Wittgenstein makes what is distinctive about his theory superfluous, for the hypothetical laws are directly inspired by statistical observations and hence these observations indirectly determine the mechanism by which the logical theory of probability operates. I argue that this is not the case at all, and that while statistical observations play a part in the formation of the hypothetical laws, these observations are only necessary, but not sufficient conditions for the introduction of these hypotheses. (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)

Consistent with Koehler's position, we propose a generalization of the base rate fallacy and earlier conservatism literatures. In studies using both traditional tasks and new tasks based on ecologically valid base rates, our subjects typically underweight individuating information at least as much as they underweight base rates. The implications of cue consistency for averaging heuristics are discussed.

Hume’s celebrated argument concerning miracles, and an 18th century criticism of it put forward by Richard Price, is here interpreted in terms of the modern controversy over the base-rate fallacy. When considering to what degree we should trust a witness, should we or should we not take into account the prior probability of the event reported? The reliability of the witness (’Pr’(says e/e)) is distinguished from the credibility of the testimony (’Pr’(e/says e)), and it is argued that Hume, as (...) a good proto-Bayesian, argued that the credibility of the testimony should be calculated in terms of both the reliability of the witness and the prior probability of the event reported. (shrink)

Do participants bring their own priors to an experiment? If so, do they share the same priors as the researchers who design the experiment? In this article, we examine the extent to which self-generated priors conform to experimenters’ expectations by explicitly asking participants to indicate their own priors in estimating the probability of a variety of events. We find in Study 1 that despite being instructed to follow a uniform distribution, participants appear to have used their own priors, which deviated (...) from the given instructions. Using subjects’ own priors allows us to account better for their responses rather than merely to test the accuracy of their estimates. Implications for the study of judgment and decision making are discussed. (shrink)

The current debate about peer disagreement has so far mainly focused on the question of whether peer disagreements provide genuine counterevidence to which we should respond by revising our credences. By contrast, comparatively little attention has been devoted to the question by which process, if any, such revision should be brought about. The standard assumption is that we update our credences by conditionalizing on the evidence that peer disagreements provide. In this paper, we argue that non-dogmatist views have good reasons (...) to reject conditionalization. Instead, peer disagreements should be understood to call for a revision of our prior conditional probabilities: rather than merely adding to our original evidence, they pose a challenge to the thought that we have properly assessed the probative force of our original evidence. (shrink)

I argue that information is a goal-relative concept for Bayesians. More precisely, I argue that how much information is provided by a piece of evidence depends on whether the goal is to learn the truth or to rank actions by their expected utility, and that different confirmation measures should therefore be used in different contexts. I then show how information measures may reasonably be derived from confirmation measures, and I show how to derive goal-relative non-informative and informative priors given background (...) information. Finally, I argue that my arguments have important implications for both objective and subjective Bayesianism. In particular, the Uniqueness Thesis is either false or must be modified. Moreover, objective Bayesians must concede that pragmatic factors systematically influence which priors are rational, and subjective Bayesians must concede that pragmatic factors sometimes partly determine which prior distribution most accurately represents an agent’s epistemic state. (shrink)

I argue that information is a goal-relative concept for Bayesians. More precisely, I argue that how much information is provided by a piece of evidence depends on whether the goal is to learn the truth or to rank actions by their expected utility, and that different confirmation measures should therefore be used in different contexts. I then show how information measures may reasonably be derived from confirmation measures, and I show how to derive goal-relative non-informative and informative priors given background (...) information. Finally, I argue that my arguments have important implications for both objective and subjective Bayesianism. In particular, the Uniqueness Thesis is either false or must be modified. Moreover, objective Bayesians must concede that pragmatic factors systematically influence which priors are rational, and subjective Bayesians must concede that pragmatic factors sometimes partly determine which prior distribution most accurately represents an agent's epistemic state. (shrink)

The original development of the formalism of quantum mechanics involved the study of isolated quantum systems in pure states. Such systems fail to capture important aspects of the warm, wet, and noisy physical world which can better be modelled by quantum statistical mechanics and local quantum field theory using mixed states of continuous systems. In this context, we need to be able to compute quantum probabilities given only partial information. Specifically, suppose that B is a set of operators. This (...) set need not be a von Neumann algebra. Simple axioms are proposed which allow us to identify a function which can be interpreted as the probability, per unit trial of the information specified by B, of observing the (mixed) state of the world restricted to B to be σ when we are given ρ – the restriction to B of a prior state. This probability generalizes the idea of a mixed state (ρ) as being a sum of terms (σ) weighted by probabilities. The unique function satisfying the axioms can be defined in terms of the relative entropy. The analogous inference problem in classical probability would be a situation where we have some information about the prior distribution, but not enough to determine it uniquely. In such a situation in quantum theory, because only what we observe should be taken to be specified, it is not appropriate to assume the existence of a fixed, definite, unknown prior state, beyond the set B about which we have information. The theory was developed for the purposes of a fairly radical attack on the interpretation of quantum theory, involving many-worlds ideas and the abstract characterization of observers as finite information-processing structures, but deals with quantum inference problems of broad generality. (shrink)

The idea that perceptual and cognitive systems must incorporate knowledge about the structure of the environment has become a central dogma of cognitive theory. In a Bayesian context, this idea is often realized in terms of “tuning the prior”—widely assumed to mean adjusting prior probabilities so that they match the frequencies of events in the world. This kind of “ecological” tuning has often been held up as an ideal of inference, in fact defining an “ideal observer.” But (...) widespread as this viewpoint is, it directly contradicts Bayesian philosophy of probability, which views probabilities as degrees of belief rather than relative frequencies, and explicitly denies that they are objective characteristics of the world. Moreover, tuning the prior to observed environmental frequencies is subject to overfitting, meaning in this context overtuning to the environment, which leads (ironically) to poor performance in future encounters with the same environment. Whenever there is uncertainty about the environment—which there almost always is—an agent's prior should be biased away from ecological relative frequencies and toward simpler and more entropic priors. (shrink)

This paper articulates a way to ground a relatively high prior probability for grand explanatory theories apart from an appeal to simplicity. I explore the possibility of enumerating the space of plausible grand theories of the universe by using the explanatory properties of possible views to limit the number of plausible theories. I motivate this alternative grounding by showing that Swinburne’s appeal to simplicity is problematic along several dimensions. I then argue that there are three plausible grand views—theism, atheism, (...) and axiarchism—which satisfy explanatory requirements for plausibility. Other possible views lack the explanatory virtue of these three theories. Consequently, this explanatory grounding provides a way of securing a nontrivial prior probability for theism, atheism, and axiarchism. An important upshot of my approach is that a modest amount of empirical evidence can bear significantly on the posterior probability of grand theories of the universe. (shrink)

Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of (...) being true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter) examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic. (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)

Stalnaker's Thesis about indicative conditionals is, roughly, that the probability one ought to assign to an indicative conditional equals the probability that one ought to assign to its consequent conditional on its antecedent. The thesis seems right. If you draw a card from a standard 52-card deck, how confident are you that the card is a diamond if it's a red card? To answer this, you calculate the proportion of red cards that are diamonds -- that is, you calculate the (...) probability of drawing a diamond conditional on drawing a red card. Skyrms' Thesis about counterfactual conditionals is, roughly, that the probability that one ought to assign to a counterfactual equals one's rational expectation of the chance, at a relevant past time, of its consequent conditional on its antecedent. This thesis also seems right. If you decide not to enter a 100-ticket lottery, how confident are you that you would have won had you bought a ticket? To answer this, you calculate the prior chance--that is, the chance just before your decision not to buy a ticket---of winning conditional on entering the lottery. The central project of this article is to develop a new uniform theory of conditionals that allows us to derive a version of Skyrms' Thesis from a version of Stalnaker's Thesis, together with a chance-deference norm relating rational credence to beliefs about objective chance. (shrink)

Bayesian prior probabilities have an important place in probabilistic and statistical methods. In spite of this fact, the analysis of where these priors come from and how they are formed has received little attention. It is reasonable to excuse the lack, in the foundational literature, of detailed psychological theory of what are the mechanisms by which prior probabilities are formed. But it is less excusable that there is an almost total absence of a detailed discussion of (...) the highly differentiating nature of past experience in forming a prior. The focus here is on what kind of account, even if necessarily schematic, can be given about the psychological mechanisms back of the formation of Bayesian priors. The last section examines a detailed experiment relevant to how priors are learned. (shrink)

Arthur Prior was one of the logicians who participated in the invention of the possible worlds’ semantics. The ontology, which is connected with his systems of modal logic, is unique. Prior tried to reduce the number of abstract entities as much as possible. Hence he did not elect to introduce possible worlds and possibilia into his ontology. In addition, he held a reductionist view, which is called modal actualism by Fine or modalism by Melia. Prior was inspired (...) by various authors but this paper mainly discusses the influence of two prominent sources, Wittgenstein and logicians from Warsaw School. Prior acknowledged Wittgenstein’s influence, when he discuss the view that possible worlds consist of propositions. The second important feature of Prior’s concept is the denial of the real existence of possible worlds and the denial of the existence of entities which are in them. It was probably enabled by Prior’s adoption of some systems of logic invented by logicians from the Warsaw school, especially Lesniewski’s and Łukasiewicz’s systems. (shrink)

In the Bayesian approach to quantum mechanics, probabilities—and thus quantum states—represent an agent’s degrees of belief, rather than corresponding to objective properties of physical systems. In this paper we investigate the concept of certainty in quantum mechanics. Particularly, we show how the probability-1 predictions derived from pure quantum states highlight a fundamental difference between our Bayesian approach, on the one hand, and Copenhagen and similar interpretations on the other. We first review the main arguments for the general claim that (...) probabilities always represent degrees of belief. We then argue that a quantum state prepared by some physical device always depends on an agent’s prior beliefs, implying that the probability-1 predictions derived from that state also depend on the agent’s prior beliefs. Quantum certainty is therefore always some agent’s certainty. Conversely, if facts about an experimental setup could imply agent-independent certainty for a measurement outcome, as in many Copenhagen-like interpretations, that outcome would effectively correspond to a preexisting system property. The idea that measurement outcomes occurring with certainty correspond to preexisting system properties is, however, in conflict with locality. We emphasize this by giving a version of an argument of Stairs [(1983). Quantum logic, realism, and value-definiteness. Philosophy of Science, 50, 578], which applies the Kochen–Specker theorem to an entangled bipartite system. (shrink)

John Maynard Keynes’s A Treatise on Probability is the seminal text for the logical interpretation of probability. According to his analysis, probabilities are evidential relations between a hypothesis and some evidence, just like the relations of deductive logic. While some philosophers had suggested similar ideas prior to Keynes, it was not until his Treatise that the logical interpretation of probability was advocated in a clear, systematic and rigorous way. I trace Keynes’s influence in the philosophy of probability through (...) a heterogeneous sample of thinkers who adopted his interpretation. This sample consists of Frederick C. Benenson, Roy Harrod, Donald C. Williams, Henry E. Kyburg and David Stove. The ideas of Keynes prove to be adaptable to their diverse theories of probability. My discussion indicates both the robustness of Keynes’s probability theory and the importance of its influence on the philosophers whom I describe. I also discuss the Problem of the Priors. I argue that none of those I discuss have obviously improved on Keynes’s theory with respect to this issue. (shrink)

A diagnostic judgment of a teacher can be seen as an inference from manifest observable evidence on a student’s behavior to his or her latent traits. This can be described by a Bayesian model of in-ference: The teacher starts from a set of assumptions on the student (hypotheses), with subjective probabilities for each hypothesis (priors). Subsequently, he or she uses observed evidence (stu-dents’ responses to tasks) and knowledge on conditional probabilities of this evidence (likelihoods) to revise these assumptions. (...) Many systematic deviations from this model (biases, e.g. base-rate neglect, inverse fallacy) are reported in literature on Bayesian reasoning. In a teacher’s situation the information (hypotheses, priors, likelihoods) is usually not explicitly represented numerically (as in most research on Bayesian reasoning), but only by qualitative esti-mations in the mind of the teacher. In our study, we ask to which extent individuals (approximately) apply a rational Bayesian strategy or resort on other biased strategies of processing information for their diagnostic judgments. We explicitly pose this question with respect to non-numerical settings. To investigate this question, we developed a scenario that visually displays all relevant information (hypotheses, priors, likelihoods) in a graphically displayed hypothesis space (called “hypothegon”) – without recurring to numerical representations or mathematical procedures. 42 pre-service teach-ers were asked to judge the plausibility of different misconceptions of six students based on their responses to decimal comparison tasks (e.g. 3.39 > 3.4). Applying a Bayesian classification proce-dure, we identified three updating strategies: A Bayesian update strategy (BUS, processing all probabilities), a combined evidence strategy (CES, ignoring the prior probabilities but including all likelihoods), and a single evidence strategy (SES, only using the likelihood of the most probable hypothesis). In study 1, an instruction on the relevance of using all probabilities (priors and likelihoods) only weakly increased the processing of more information. In study 2, we found strong evidence that a visual explication of the prior-likelihood interaction led to an increase of processing the interaction of all relevant information. These results show that the phenomena found in general research on Bayesian reasoning in numerical settings extend to diagnostic judgments in non-numerical settings. (shrink)

A simple rule of probability revision ensures that the final result of a sequence of probability revisions is undisturbed by an alteration in the temporal order of the learning prompting those revisions. This Uniformity Rule dictates that identical learning be reflected in identical ratios of certain new-to-old odds, and is grounded in the old Bayesian idea that such ratios represent what is learned from new experience alone, with prior probabilities factored out. The main theorem of this paper includes (...) as special cases Field's theorem on commuting probability-kinematical revisions and the equivalence of two strategies for generalizing Jeffrey 's solution to the old evidence problem to the case of uncertain old evidence and probabilistic new explanation. (shrink)

It has been alleged that Bayesian usage of prior probabilities allows one to obtain empirical statements on the basis of no evidence whatever. We examine this charge with reference to several examples from the literature, arguing, first, that the difference between probabilities based on weighty evidence and those based on little evidence can be drawn in terms of the variance of a distribution. Moreover, qua summaries of vague prior knowledge, prior distributions only transmit the empirical (...) information therein contained and, therefore, their consequences for long-run frequency behavior are "a priori" in at best a Pickwickian sense. (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)

This paper explores the relationship between concepts of probability and hermeneutics. It seeks to examine the relationship between subjective (Bayesian) views of probability and hermeneutic descriptions of understanding. It is argued that Gadamer'saccount of the prejudicial nature of understanding, derived from Heidegger'sanalysis of foreunderstanding, offers a provocative model of clinical reasoning. The implications of this model for evidence-based medicine are discussed in conclusion.

Two of the probabilistic measures of coherence discussed in this paper take probabilistic dependence into account and so depend on prior probabilities in a fundamental way. An example is given which suggests that this prior-dependence can lead to potential problems. Another coherence measure is shown to be independent of prior probabilities in a clearly defined sense and consequently is able to avoid such problems. The issue of prior-dependence is linked to the fact that the (...) first two measures can be understood as measures of coherence as striking agreement, while the third measure represents coherence as agreement. Thus, prior (in)dependence can be used to distinguish different conceptions of coherence. (shrink)

According to the Doomsday Argument the probability of an impending extinction of mankind is much higher than we think. The adduced reason is that in our assignment of probabilities to soon or not so soon doom we have not fully taken into account that we live in the specific year 2001. This is relevant information, because if I consider myself as an arbitrary member of the human race I have a much higher probability of finding myself living in 2001 (...) on the hypothesis of a soon extinction, Doom Soon, than on the hypothesis of Doom Late---according to the latter hypothesis there are so many more years I could have found myself living in. Accordingly, Bayesian reasoning leads to a posterior probability of the Doom Soon hypothesis, after I have taken the evidence of my birth date fully into account, that is much higher than the prior probability. I show that the Argument is nothing but a rather trivial mathematical exercise in the calculation of posterior from prior probabilities; it is only about the relation between these probabilities and is silent about the concrete values these probabilities should have. Nothing in the Argument supports the conclusion its proponents think it supports, namely that Doom Soon is much more probable than we ordinarily think. The Argument is formally valid, but ineffective. (shrink)

The epistemic probability of A given B is the degree to which B evidentially supports A, or makes A plausible. This paper is a first step in answering the question of what determines the values of epistemic probabilities. I break this question into two parts: the structural question and the substantive question. Just as an object’s weight is determined by its mass and gravitational acceleration, some probabilities are determined by other, more basic ones. The structural question asks what (...) probabilities are not determined in this way—these are the basic probabilities which determine values for all other probabilities. The substantive question asks how the values of these basic probabilities are determined. I defend an answer to the structural question on which basic probabilities are the probabilities of atomic propositions conditional on potential direct explanations. I defend this against the view, implicit in orthodox mathematical treatments of probability, that basic probabilities are the unconditional probabilities of complete worlds. I then apply my answer to the structural question to clear up common confusions in expositions of Bayesianism and shed light on the “problem of the priors.”. (shrink)

God has major reasons for intervening in human history by becoming incarnate himself—to identify with our suffering, to provide atonement for our sins, and to reveal truths. Given there is at least a significant probability that there is a God, there is at least a modest probability that he would become incarnate and live a life and provide teaching appropriate to one who sought thereby to realize these goals. Jesus lived and taught in the appropriate way. If it was God (...) incarnate who did so live and teach, he would need to show us that it was God who had done so, and so could be expected to put his signature on that life and teaching by a super-miracle, such as the Resurrection. So there is a modest prior probability in advance of considering the direct historical evidence of the Resurrection, to expect that it would happen to someone who lived and taught as Jesus did. Jesus is the only person in human history about whom there is significant evidence both that he led the appropriate kind of life, and that his life was culminated by a super-miracle. So we do not need too many witnesses to the empty tomb or too many witnesses who claimed to have talked to the risen Jesus, to make it probable that Jesus did indeed rise. We do have some such witness evidence, which it is very improbable would occur unless the Resurrection occurred. In consequence it is overall very probable that the Resurrection occurred. (shrink)

Robert Aumann presents his Agreement Theorem as the key conditional: “if two people have the same priors and their posteriors for an event A are common knowledge, then these posteriors are equal” (Aumann, 1976, p. 1236). This paper focuses on four assumptions which are used in Aumann’s proof but are not explicit in the key conditional: (1) that agents commonly know, of some prior μ, that it is the common prior; (2) that agents commonly know that each of (...) them updates on the prior by conditionalization; (3) that agents commonly know that if an agent knows a proposition, she knows that she knows that proposition (the “K K” principle); (4) that agents commonly know that they each update only on true propositions. It is shown that natural weakenings of any one of these strong assumptions can lead to countermodels to Aumann’s key conditional. Examples are given in which agents who have a common prior and commonly know what probability they each assign to a proposition nevertheless assign that proposition unequal probabilities. To alter Aumann’s famous slogan: people can “agree to disagree”, even if they share a common prior. The epistemological significance of these examples is presented in terms of their role in a defense of the Uniqueness Thesis: If an agent whose total evidence is E is fully rational in taking doxastic attitude D to P, then necessarily, any subject with total evidence E who takes a different attitude to P is less than fully rational. (shrink)

Current dynamic-epistemic logics model different types of information change in multi-agent scenarios. We generalize these logics to a probabilistic setting, obtaining a calculus for multi-agent update with three natural slots: prior probability on states, occurrence probabilities in the relevant process taking place, and observation probabilities of events. To match this update mechanism, we present a complete dynamic logic of information change with a probabilistic character. The completeness proof follows a compositional methodology that applies to a much larger (...) class of dynamic-probabilistic logics as well. Finally, we discuss how our basic update rule can be parameterized for different update policies, or learning methods. (shrink)