When an agent learns of an expert's credence in a proposition about which they are an expert, the agent should defer to the expert and adopt that credence as their own. This is a popular thought about how agents ought to respond to (ideal) experts. In a Bayesian framework, it is often modelled by endowing the agent with a set of priors that achieves this result. But this model faces a number of challenges, especially when applied to non-ideal agents (who (...) nevertheless interact with ideal experts). I outline these problems, and use them as desiderata for the development of a new model. Taking inspiration from Richard Jeffrey's development of Jeffrey conditioning, I develop a model in which expert reports are taken as exogenous constraints on the agent's posterior probabilities. I show how this model can handle a much wider class of expert reports (for example reports of conditional probabilities), and can be naturally extended to cover propositions for which the agent has no prior. (shrink)

Bayesians often appeal to “merging of opinions” to rebut charges of excessive subjectivity. But what happens in the short run is often of greater interest than what happens in the limit. Seidenfeld and coauthors use this observation as motivation for investigating the counterintuitive short run phenomenon of dilation, since, they allege, dilation is “the opposite” of asymptotic merging of opinions. The measure of uncertainty relevant for dilation, however, is not the one relevant for merging of opinions. We explicitly investigate the (...) short run behavior of the metric relevant for merging, and show that dilation is independent of the opposite of merging. (shrink)

As a general rule, whenever a hearer is justified in forming the belief that p on the basis of a speaker’s testimony, she will also be justified in assuming that the speaker has formed her belief appropriately in light of a relevantly large and representative sample of the evidence that bears on p. In simpler terms, a justification for taking someone’s testimony entails a justification for trusting her assessment of the evidence. This introduces the possibility of what I will call (...) “evidential preemption.” Evidential preemption occurs when a speaker, in addition to offering testimony that p, also warns the hearer of the likelihood that she will subsequently be confronted with apparently contrary evidence: this is done, however, not so as to encourage the hearer to temper her confidence in p in anticipation of that evidence, but rather to suggest that the (apparently) contrary evidence is in fact misleading evidence or evidence that has already been taken into account. Either way, the speaker is signalling to the hearer that the subsequent disclosure of this evidence will not require her to significantly revise her belief that p. Such preemption can effectively inoculate an audience against future contrary evidence, and thereby creates an opening for a form of exploitative manipulation that I will call “epistemic grooming.” Nonetheless, I argue, not all uses of evidential preemption are nefarious; it can also serve as an important tool for guiding epistemically limited agents though complex evidential scenarios. (shrink)

Our senses provide us with information about the world, but what exactly do they tell us? I argue that in order to optimally respond to sensory stimulations, an agent’s doxastic space may have an extra, “imaginary” dimension of possibility; perceptual experiences confer certainty on propositions in this dimension. To some extent, the resulting picture vindicates the old-fashioned empiricist idea that all empirical knowledge is based on a solid foundation of sense-datum propositions, but it avoids most of the problems traditionally associated (...) with that idea. The proposal might also explain why experiences appear to have a non-physical phenomenal character, even if the world is entirely physical. (shrink)

General epistemic polarization arises when the beliefs of a population grow further apart, in particular when all agents update on the same evidence. Epistemic factionalization arises when the beliefs grow further apart, but different beliefs also become correlated across the population. I present a model of how factionalization can emerge in a population of ideally rational agents. This kind of factionalization is driven by probabilistic relations between beliefs, with background beliefs shaping how the agents’ beliefs evolve in the light of (...) new evidence. Moreover, I show that in such a model, the only possible outcomes from updating on identical evidence are general convergence or factionalization. Beliefs cannot spread out in all directions: if the beliefs overall polarize, then it must result in factionalization. (shrink)

How should a group with different opinions (but the same values) make decisions? In a Bayesian setting, the natural question is how to aggregate credences: how to use a single credence function to naturally represent a collection of different credence functions. An extension of the standard Dutch-book arguments that apply to individual decision-makers recommends that group credences should be updated by conditionalization. This imposes a constraint on what aggregation rules can be like. Taking conditionalization as a basic constraint, we gather (...) lessons from the established work on credence aggregation, and extend this work with two new impossibility results. We then explore contrasting features of two kinds of rules that satisfy the constraints we articulate: one kind uses fixed prior credences, and the other uses geometric averaging, as opposed to arithmetic averaging. We also prove a new characterisation result for geometric averaging. Finally we consider applications to neighboring philosophical issues, including the epistemology of disagreement. (shrink)

Conciliatory views of peer disagreement hold that when an agent encounters peer disagreement she should conciliate by adjusting her doxastic attitude towards that of her peer. In this paper I distinguish different ways conciliation can be understood and argue that the way conciliationism is typically understood violates the principle of commutativity of evidence. Commutativity of evidence holds that the order in which evidence is acquired should not influence what it is reasonable to believe based on that evidence. I argue that (...) when an agent encounters more than one peer, and applies the process of conciliation serially, the order she encounters the peers influences the resulting credence. I argue this is a problem for conciliatory views of disagreement, and suggest some responses available to advocates of conciliation. (shrink)

The applicability of Bayesian conditionalization in setting one’s posterior probability for a proposition, α, is limited to cases where the value of a corresponding prior probability, PPRI(α|∧E), is available, where ∧E represents one’s complete body of evidence. In order to extend probability updating to cases where the prior probabilities needed for Bayesian conditionalization are unavailable, I introduce an inference schema, defeasible conditionalization, which allows one to update one’s personal probability in a proposition by conditioning on a proposition that represents a (...) proper subset of one’s complete body of evidence. While defeasible conditionalization has wider applicability than standard Bayesian conditionalization (since it may be used when the value of a relevant prior probability, PPRI(α|∧E), is unavailable), there are circumstances under which some instances of defeasible conditionalization are unreasonable. To address this difficulty, I outline the conditions under which instances of defeasible conditionalization are defeated. To conclude the article, I suggest that the prescriptions of direct inference and statistical induction can be encoded within the proposed system of probability updating, by the selection of intuitively reasonable prior probabilities. (shrink)

One of the fundamental problems of epistemology is to say when the evidence in an agent’s possession justifies the beliefs she holds. In this paper and its prequel, we defend the Bayesian solution to this problem by appealing to the following fundamental norm: Accuracy An epistemic agent ought to minimize the inaccuracy of her partial beliefs. In the prequel, we made this norm mathematically precise; in this paper, we derive its consequences. We show that the two core tenets of Bayesianism (...) follow from the norm, while the characteristic claim of the Objectivist Bayesian follows from the norm along with an extra assumption. Finally, we consider Richard Jeffrey’s proposed generalization of conditionalization. We show not only that his rule cannot be derived from the norm, unless the requirement of Rigidity is imposed from the start, but further that the norm reveals it to be illegitimate. We end by deriving an alternative updating rule for those cases in which Jeffrey’s is usually supposed to apply. (shrink)

Suppose that several individuals who have separately assessed prior probability distributions over a set of possible states of the world wish to pool their individual distributions into a single group distribution, while taking into account jointly perceived new evidence. They have the option of first updating their individual priors and then pooling the resulting posteriors or first pooling their priors and then updating the resulting group prior. If the pooling method that they employ is such that they arrive at the (...) same final distribution in both cases, the method is said to be externally Bayesian, a property first studied by Madansky . We show that a pooling method for discrete distributions is externally Bayesian if and only if it commutes with Jeffrey conditioning, parameterized in terms of certain ratios of new to old odds, as in Wagner , rather than in terms of the posterior probabilities of members of the disjoint family of events on which such conditioning originates. (shrink)

Traditional procedures for rational updating fail when it comes to self-locating opinions, such as your credences about where you are and what time it is. This paper develops an updating procedure for rational agents with self-locating beliefs. In short, I argue that rational updating can be factored into two steps. The first step uses information you recall from your previous self to form a hypothetical credence distribution, and the second step changes this hypothetical distribution to reflect information you have genuinely (...) learned as time has passed. While the second step resembles traditional procedures of updating by conditionalization, the first is best understood by analogy to traditional models of how agents transmit self-locating opinions through ordinary interpersonal communication. (shrink)

Jeffrey conditioning allows updating in Bayesian style when the evidence is uncertain. A weighted average, essentially, over classically updating on the alternatives. Unlike classical Bayesian conditioning, this allows learning to be unlearned.

Jeffrey updating is a natural extension of Bayesian updating to cases where the evidence is uncertain. But, the resulting degrees of belief appear to be sensitive to the order in which the uncertain evidence is acquired, a rather un-Bayesian looking effect. This order dependence results from the way in which basic Jeffrey updating is usually extended to sequences of updates. The usual extension seems very natural, but there are other plausible ways to extend Bayesian updating that maintain order-independence. I will (...) explore three models of sequential updating, the usual extension and two alternatives. I will show that the alternative updating schemes derive from extensions of the usual rigidity requirement, which is at the heart of Jeffrey updating. Finally, I will establish necessary and sufficient conditions for order-independent updating, and show that extended rigidity is closely related to these conditions. (shrink)

In this paper, we investigate various possible (Bayesian) precisifications of the (somewhat vague) statements of “the equal weight view” (EWV) that have appeared in the recent literature on disagreement. We will show that the renditions of (EWV) that immediately suggest themselves are untenable from a Bayesian point of view. In the end, we will propose some tenable (but not necessarily desirable) interpretations of (EWV). Our aim here will not be to defend any particular Bayesian precisification of (EWV), but rather to (...) raise awareness about some of the difficulties inherent in formulating such precisifications. (shrink)

Bayesianism can be characterized as the following twofold position: (i) rational credences obey the probability calculus; (ii) rational learning, i.e., the updating of credences, is regulated by some form of conditionalization. While the formal aspect of various forms of conditionalization has been explored in detail, the philosophical application to learning from experience is still deeply problematic. Some philosophers have proposed to revise the epistemology of perception; others have provided new formal accounts of conditionalization that are more in line with how (...) we learn from perceptual experience. The current investigation argues that Bayesian epistemology is still incomplete; the epistemology of perception and the epistemology of rational reasoning have not been reconciled. (shrink)

We explore which types of probabilistic updating commute with convex IP pooling. Positive results are stated for Bayesian conditionalization, imaging, and a certain parameterization of Jeffrey conditioning. This last observation is obtained with the help of a slight generalization of a characterization of externally Bayesian pooling operators due to Wagner :336–345, 2009). These results strengthen the case that pooling should go by imprecise probabilities since no precise pooling method is as versatile.

Bayesian Epistemology is a general framework for thinking about agents who have beliefs that come in degrees. Theories in this framework give accounts of rational belief and rational belief change, which share two key features: (i) rational belief states are represented with probability functions, and (ii) rational belief change results from the acquisition of evidence. This dissertation focuses specifically on the second feature. I pose the Evidence Question: What is it to have evidence? Before addressing this question we must have (...) an understanding of Bayesian Epistemology. The first chapter argues that we should understand Bayesian Epistemology as giving us theories that are evaluative and not action-guiding. I reach this verdict after considering the popular ‘ought’-implies-‘can’ objection to Bayesian Epistemology. The second chapter argues that it is important for theories in Bayesian Epistemology to answer the Evidence Question, and distinguishes between internalist and externalist answers. The third and fourth chapters present and defend a specific answer to the Evidence Question. The account is inspired by reliabilist accounts of justification, and attempts to understand what it is to have evidence by appealing solely to considerations of reliability. Chapter 3 explains how to understand reliability, and how the account fits with Bayesian Epistemology, in particular, the requirement that an agent’s evidence receive probability 1. Chapter 4 responds to objections, which maintain that the account gives the wrong verdict in a variety of situations including skeptical scenarios, lottery cases, scientific cases, and cases involving inference. After slight modifications, I argue that my account has the resources to answer the objections. The fifth chapter considers the possibility of losing evidence. I show how my account can model these cases. To do so, however, we require a modification to Conditionalization, the orthodox principle governing belief change. I present such a modification. The sixth and seventh chapters propose a new understanding of Dutch Book Arguments, historically important arguments for Bayesian principles. The proposal shows that the Dutch Book Arguments for implausible principles are defective, while the ones for plausible principles are not. The final chapter is a conclusion. (shrink)

Handbook of the History of Logic, vol. 10, eds. Dov Gabbay, Stephan Hartmann, and John Woods, forthcoming.

What if gravity satisfied the Klein-Gordon equation? Both particle physics from the 1920s-30s and the 1890s Neumann-Seeliger modification of Newtonian gravity with exponential decay suggest considering a "graviton mass term" for gravity, which is _algebraic_ in the potential. Unlike Nordström's "massless" theory, massive scalar gravity is strictly special relativistic in the sense of being invariant under the Poincaré group but not the 15-parameter Bateman-Cunningham conformal group. It therefore exhibits the whole of Minkowski space-time structure, albeit only indirectly concerning volumes. Massive (...) scalar gravity is plausible in terms of relativistic field theory, while violating most interesting versions of Einstein's principles of general covariance, general relativity, equivalence, and Mach. Geometry is a poor guide to understanding massive scalar gravity: matter sees a conformally flat metric due to universal coupling, but gravity also sees the rest of the flat metric in the mass term. What is the 'true' geometry, one might wonder, in line with Poincaré's modal conventionality argument? Infinitely many theories exhibit this bimetric 'geometry,' all with the total stress-energy's trace as source; thus geometry does not explain the field equations. The irrelevance of the Ehlers-Pirani-Schild construction to a critique of conventionalism becomes evident when multi-geometry theories are contemplated. Much as Seeliger envisaged, the smooth massless limit indicates underdetermination of theories by data between massless and massive scalar gravities---indeed an unconceived alternative. At least one version easily could have been developed before General Relativity; it then would have motivated thinking of Einstein's equations along the lines of Einstein's newly re-appreciated "physical strategy" and particle physics and would have suggested a rivalry from massive spin 2 variants of General Relativity. The Putnam-Grünbaum debate on conventionality is revisited with an emphasis on the broad modal scope of conventionalist views. Massive scalar gravity thus contributes to a historically plausible rational reconstruction of much of 20th-21st century space-time philosophy in the light of particle physics. An appendix reconsiders the Malament-Weatherall-Manchak conformal restriction of conventionality and constructs the 'universal force' influencing the causal structure. Subsequent works will discuss how massive gravity could have provided a template for a more Kant-friendly space-time theory that would have blocked Moritz Schlick's supposed refutation of synthetic _a priori_ knowledge, and how Einstein's false analogy between the Neumann-Seeliger-Einstein modification of Newtonian gravity and the cosmological constant \Lambda generated lasting confusion that obscured massive gravity as a conceptual possibility. (shrink)

There are some candidates that have been thought to measure the degree to which evidence incrementally confirms a hypothesis. This paper provides an argument for one candidate—the log-likelihood ratio measure. For this purpose, I will suggest a plausible requirement that I call the Requirement of Collaboration. And then, it will be shown that, of various candidates, only the log-likelihood ratio measure \(l\) satisfies this requirement. Using this result, Jeffrey conditionalization will be reformulated so as to disclose explicitly what determines new (...) credences after experience. (shrink)

This paper discusses simultaneous belief updates. I argue here that modeling such belief updates using the Principle of Minimum Information can be regarded as applying Jeffrey conditionalization successively, and so that, contrary to what many probabilists have thought, the simultaneous belief updates can be successfully modeled by means of Jeffrey conditionalization.

In this article, I suggest an argument that seems to show a conflict between the reflection principle and conditionalization. In particular, I show that when the reflection principle is formulated in a standard way, the principle conflicts with Jeffrey conditionalization. And it is argued that the source of the conflict resides in an ambiguity of the standard formulation. Furthermore, I attempt to rescue the principle using Bayes factors. That is, I suggest a new formulation of the principle so as to (...) avoid the conflict. (shrink)

Bayesianism is a collection of positions in several related fields, centered on the interpretation of probability as something like degree of belief, as contrasted with relative frequency, or objective chance. However, Bayesianism is far from a unified movement. Bayesians are divided about the nature of the probability functions they discuss; about the normative force of this probability function for ordinary and scientific reasoning and decision making; and about what relation (if any) holds between Bayesian and non-Bayesian concepts.

Lange (2000) famously argues that although Jeffrey Conditionalization is non-commutative over evidence, it’s not defective in virtue of this feature. Since reversing the order of the evidence in a sequence of updates that don’t commute does not reverse the order of the experiences that underwrite these revisions, the conditions required to generate commutativity failure at the level of experience will fail to hold in cases where we get commutativity failure at the level of evidence. If our interest in commutativity is, (...) fundamentally, an interest in the order-invariance of information, an updating sequence that does not violate such a principle at the more fundamental level of experiential information should not be deemed defective. This paper claims that Lange’s argument fails as a general defense of the Jeffrey framework. Lange’s argument entails that the inputs to the Jeffrey framework differ from those of classical Bayesian Conditionalization in a way that makes them defective. Therefore, either the Jeffrey framework is defective in virtue of not commuting its inputs, or else it is defective in virtue of commuting the wrong kinds of ones. (shrink)

We investigate the general properties of general Bayesian learning, where “general Bayesian learning” means inferring a state from another that is regarded as evidence, and where the inference is conditionalizing the evidence using the conditional expectation determined by a reference probability measure representing the background subjective degrees of belief of a Bayesian Agent performing the inference. States are linear functionals that encode probability measures by assigning expectation values to random variables via integrating them with respect to the probability measure. If (...) a state can be learned from another this way, then it is said to be Bayes accessible from the evidence. It is shown that the Bayes accessibility relation is reflexive, antisymmetric and non-transitive. If every state is Bayes accessible from some other defined on the same set of random variables, then the set of states is called weakly Bayes connected. It is shown that the set of states is not weakly Bayes connected if the probability space is standard. The set of states is called weakly Bayes connectable if, given any state, the probability space can be extended in such a way that the given state becomes Bayes accessible from some other state in the extended space. It is shown that probability spaces are weakly Bayes connectable. Since conditioning using the theory of conditional expectations includes both Bayes’ rule and Jeffrey conditionalization as special cases, the results presented generalize substantially some results obtained earlier for Jeffrey conditionalization. (shrink)

Conditionalization and Jeffrey Conditionalization cannot simultaneously satisfy two widely held desiderata on rules for empirical learning. The first desideratum is confirmational holism, which says that the evidential import of an experience is always sensitive to our background assumptions. The second desideratum is commutativity, which says that the order in which one acquires evidence shouldn't affect what conclusions one draws, provided the same total evidence is gathered in the end. (Jeffrey) Conditionalization cannot satisfy either of these desiderata without violating the other. (...) This is a surprising problem, and I offer a diagnosis of its source. I argue that (Jeffrey) Conditionalization is inherently anti-holistic in a way that is just exacerbated by the requirement of commutativity. The dilemma is thus a superficial manifestation of (Jeffrey) Conditionalization's fundamentally anti-holistic nature. (shrink)

The foundations of probability are viewed through the lens of the subjectivist interpretation. This article surveys conditional probability, arguments for probabilism, probability dynamics, and the evidential and subjective interpretations of probability.

Consider: -/- The Evidence Question: When, and under what conditions does an agent have proposition E as evidence (at t)? -/- Timothy Williamson's (2000) answer to this question is the well-known E = K thesis: -/- E = K: E is a member of S's evidence set at t iff S knows E at t. -/- I will argue that this answer is inconsistent with the version of Bayesianism that Williamson advocates. This is because E = K allows an agent (...) to garner evidence via inductive inference whereas standard Bayesian views disallow such a thing. Since Williamson's version of Bayesianism shares the key features with the standard Bayesian view, there is an inconsistency. (shrink)

There are narrowest bounds for P(h) when P(e) = y and P(h/e) = x, which bounds collapse to x as y goes to 1. A theorem for these bounds -- bounds for probable modus ponens -- entails a principle for updating on possibly uncertain evidence subject to these bounds that is a generalization of the principle for updating by conditioning on certain evidence. This way of updating on possibly uncertain evidence is appropriate when updating by ’probability kinematics’ or ’Jeffrey-conditioning’ is, (...) and apparently in countless other cases as well. A more complicated theorem due to Karl Wagner -- bounds for probable modus tollens -- registers narrowest bounds for P(not h) when P(not e) = y and P(e/h) = x. This theorem serves another principle for updating on possibly uncertain evidence that might be termed ’contraditioning’, though it is for a way of updating that seems in practice to be frequently not appropriate. It is definitely not a way of putting down a theory -- for example, a random-chanc. (shrink)

This paper considers a special case of belief updating—when an agent learns testimonial data, or in other words, the beliefs of others on some issue. The interest in this case is twofold: (1) the linear averaging method for updating on testimony is somewhat popular in epistemology circles, and it is important to assess its normative acceptability, and (2) this facilitates a more general investigation of what it means/requires for an updating method to have a suitable Bayesian representation (taken here as (...) the normative standard). The paper initially defends linear averaging against Bayesian-compatibility concerns raised by Bradley (Soc Choice Welf 29:609-632, 2007), as well as problems associated with multiple testimony updates. The resolution of these issues, however, requires an extremely nuanced interpretation of the parameters of the linear averaging model—the so-called weights of respect. We go on to propose a role that the parameters of any 'shortcut' updating function should play, by way of minimal interpretation of these parameters. The class of updating functions that is consistent with this role, however, excludes linear averaging, at least in its standard form. (shrink)

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)

We explore the question of whether sustained rational disagreement is possible from a broadly Bayesian perspective. The setting is one where agents update on the same information, with special consideration being given to the case of uncertain information. The classical merging of opinions theorem of Blackwell and Dubins shows when updated beliefs come and stay closer for Bayesian conditioning. We extend this result to a type of Jeffrey conditioning where agents update on evidence that is uncertain but solid (hard Jeffrey (...) shifts). However, merging of beliefs does not generally hold for Jeffrey conditioning on evidence that is fluid (soft Jeffrey shifts, Field shifts). Several theorems on the asymptotic behavior of subjective probabilities are proven. Taken together they show that while a consensus nearly always emerges in important special cases, sustained rational disagreement can be expected in many other situations. (shrink)

In his book Prejudice, Endre Begby argues that people who hold paradigmatically prejudiced beliefs – for example, the belief that women are less adept at math than men – might be fully rational in holding those beliefs. In this article, I argue that Begby fails to provide compelling examples of beliefs that are both rational and prejudiced. On Begby’s account, whether a belief is prejudiced is determined by its content: it follows that any two token beliefs with the same content (...) will either both be prejudiced, or both unprejudiced, regardless of how they differ in other respects. I sketch an alternative account, on which whether a person’s belief counts as prejudiced might be sensitive to a greater range of factors. I then turn to Bebgy’s discussion of ‘evidential preemption,’ a phenomenon by which certain speech acts seem to inoculate themselves from having their contents disconfirmed or falsified by later counterevidence. I argue for skepticism about evidential preemption. To the extent that there is a genuine normative phenomenon in the neighborhood, it is the familiar one of testimonial defeat, in which testimony from one source is neutralized by conflicting testimony from another source that one has reason to think is more reliable. (shrink)

Richard Jeffrey’s “Conditioning, Kinematics, and Exchangeability” is one of the foundational documents of probability kinematics. However, the section entitled “Successive Updating” contains a subtle error regarding the applicability of updating by so-called relevance quotients in order to ensure the commutativity of successive probability kinematical revisions. Upon becoming aware of this error, Jeffrey formulated the appropriate remedy, but he never discussed the issue in print. To head off any confusion, it seems worthwhile to alert readers of Jeffrey’s “Conditioning, Kinematics, and Exchangeability” (...) to the aforementioned error and to document his remedy, placing it in the context of both earlier and subsequent work on commuting probability kinematical revisions.1. (shrink)

We have several intuitive paradigms of defeating evidence. For example, let E be the fact that Ernie tells me that the notorious pet Precious is a bird. This supports the premise F, that Precious can fly. However, Orna gives me *opposing* evidence. She says that Precious is a dog. Alternatively, defeating evidence might not oppose Ernie's testimony in that direct way. There might be other ways for it to weaken the support that Ernie's testimony gives me for believing F, without (...) the new evidence itself intuitively constituting reasons to believe one way or the other about Precious's flight ability. An example: Ursula tells me that Ernie has no idea what Precious's species is; he's just guessing. She may not herself want to or be in a position to weigh in about Precious's real species or flight ability. I call defeating evidence of this sort *undermining evidence. (shrink)

A paper on how to adapt your probabilisitc beliefs when learning a conditional.

Richard Jeffrey espoused an antifoundationalist variant of Bayesian thinking that he termed ‘Radical Probabilism’. Radical Probabilism denies both the existence of an ideal, unbiased starting point for our attempts to learn about the world and the dogma of classical Bayesianism that the only justified change of belief is one based on the learning of certainties. Probabilistic judgment is basic and irreducible. Bayesian conditioning is appropriate when interaction with the environment yields new certainty of belief in some proposition but leaves one’s (...) conditional beliefs untouched (the ‘Rigidity’ condition). Although Richard Jeffrey denied the general applicability of this condition, one of his main contributions to probabilistic thinking is a form of belief updating—now typically called ‘Jeffrey conditioning’ or ‘probability kinematics’—that is appropriate in circumstances in which Rigidity is satisfied, but where the interaction causes one to reevaluate one’s probability judgments over some partition of the possibility space without conferring certainty on any particular element. The most familiar occasion for Jeffrey conditioning is receipt of uncertain evidence: things partially perceived or remembered. But it also serves to illuminate belief updating occasioned by a change in one’s degrees of conditional belief, a kind of belief change largely ignored by classical Bayesianism. I argue that such changes in conditional belief can also be basic (in the sense of not being analyzable as a consequence of conditioning on factual information) and offer a kinematical model for a particular kind change in conditional belief. Both are applied to changes in preference. Finally, I argue that Rigidity can fail when changes of belief give inferential grounds for changes in conditional belief (and vice versa). These failures show that conditioning methods are properly regarded, not as valid rules of inference, but as tools in the ‘art of judgment’. (shrink)

Two open questions of inductive reasoning are solved: (1) does the principle of maximum entropy (pme) give a solution to the obverse Majerník problem; and (2) is Wagner correct when he claims that Jeffrey’s updating principle (jup) contradicts pme? Majerník shows that pme provides unique and plausible marginal probabilities, given conditional probabilities. The obverse problem posed here is whether pme also provides such conditional probabilities, given certain marginal probabilities. The theorem developed to solve the obverse Majerník problem demonstrates that in (...) the special case introduced by Wagner pme does not contradict jup, but elegantly generalizes it and offers a more integrated approach to probability updating. (shrink)

Conditional learning, where agents learn a conditional sentence ‘If A, then B,’ is difficult to incorporate into existing Bayesian models of learning. This is because conditional learning is not uniform: in some cases, learning a conditional requires decreasing the probability of the antecedent, while in other cases, the antecedent probability stays constant or increases. I argue that how one learns a conditional depends on the causal structure relating the antecedent and the consequent, leading to a causal model of conditional learning. (...) This model extends traditional Bayesian learning by incorporating causal models into agents’ epistemic states. On this theory, conditional learning proceeds in two steps. First, an agent learns a new causal model with the appropriate relationship between the antecedent and the consequent. Then, the agent narrows down the set of possible worlds to include only those which make the conditional proposition true. This model of learning can incorporate both standard cases of Bayesian learning and the non-uniform learning required to learn conditional information. (shrink)

Rédei and Gyenis recently displayed strong constraints of Bayesian learning. However, they also presented a positive result for Bayesianism. Despite the limited significance of this positive result, I find it useful to discuss its two possible strengthenings to present new results and open new questions about the limits of Bayesianism. First, I will show that one cannot strengthen the positive result by restricting the evidence to so-called “certain evidence”. Secondly, strengthening the result by restricting the partitions—as parts of one’s evidence—to (...) Jeffrey-independent partitions requires additional constraints on one’s evidence to preserve its commutativity. So, my results provide additional grounds for caution and support for the limitations of Bayesian learning. (shrink)

It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics literature often claims that spinors \emph{as such} cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov constructed (...) spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists' concept of nonlinear group representations. Roughly and locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The Ogievetsky-Polubarinov formalism, by contrast, is Ockhamized, with most of the fluff removed. As developed nonperturbatively by Bilyalov, it admits any coordinates at a point, but "time" must be listed first. Here "time" is defined in terms of an eigenvalue problem involving the metric components and the matrix $diag$, the product of which must have no negative eigenvalues in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and \emph{values} of the fields involved. Apart from coordinate order and the usual spinorial two-valuedness, Ogievetsky-Polubarinov spinors form, with the metric, a nonlinear geometric object, for which important results on Lie and covariant differentiation are recalled. Such spinors avoid a spurious absolute object in the Anderson-Friedman analysis of substantive general covariance. They also permit the gauge-invariant localization of the infinite-component gravitational energy in General Relativity. Density-weighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of an arbitrary nonsingular matrix with 16 components, 6 of which are gauged away by a new local $O$ gauge group and one of which is irrelevant due to conformal covariance, one can, and presumably should, use density-weighted Ogievetsky-Polubarinov spinors coupled to the 9-component symmetric square root of the part of the metric that fixes null cones. Thus $\frac{7}{16}$ of the orthonormal basis is eliminated as surplus structure. Greater unity between spinors and tensors and the like is achieved, such as regarding conservation laws. Regarding the conventionality of simultaneity, an unusually wide range of $\epsilon$ values is admissible, but some extreme values are inadmissible. Standard simultaneity uniquely makes the spinor transformation law linear and independent of the metric, because transformations among the standard Cartesian coordinate systems fall within the conformal group, for which the spinor transformation law is linear. The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited. (shrink)

Given someone’s fully specified posterior probability distribution q and information about the revision method that they employed to produce q, what can you infer about their prior probabilistic commitments? This question provides an entrée into a thoroughgoing discussion of a class of parameterizations of Jeffrey conditioning in which the parameters furnish information above and beyond that incorporated in \. Our analysis highlights the ubiquity of Bayes factors in the study of probability revision.

A simple rule of probability revision ensures that the final result ofa sequence of probability revisions is undisturbed by an alterationin the temporal order of the learning prompting those revisions.This Uniformity Rule dictates that identical learning be reflectedin identical ratios of certain new-to-old odds, and is grounded in the oldBayesian idea that such ratios represent what is learned from new experiencealone, with prior probabilities factored out. The main theorem of this paperincludes as special cases (i) Field's theorem on commuting probability-kinematical (...) revisions and (ii) the equivalence of two strategiesfor generalizing Jeffrey's solution to the old evidence problem tothe case of uncertain old evidence and probabilistic new explanation. (shrink)

We investigate the general properties of general Bayesian learning, where ``general Bayesian learning'' means inferring a state from another that is regarded as evidence, and where the inference is conditionalizing the evidence using the conditional expectation determined by a reference probability measure representing the background subjective degrees of belief of a Bayesian Agent performing the inference. States are linear functionals that encode probability measures by assigning expectation values to random variables via integrating them with respect to the probability measure. If (...) a state can be learned from another this way, then it is said to be Bayes accessible from the evidence. It is shown that the Bayes accessibility relation is reflexive, antisymmetric and non-transitive. If every state is Bayes accessible from some other defined on the same set of random variables, then the set of states is called weakly Bayes connected. It is shown that the set of states is not weakly Bayes connected if the probability space is standard. The set of states is called weakly Bayes connectable if, given any state, the probability space can be extended in such a way that the given state becomes Bayes accessible from some other state in the extended space. It is shown that probability spaces are weakly Bayes connectable. Since conditioning using the theory of conditional expectations includes both Bayes' rule and Jeffrey conditionalization as special cases, the results presented generalize substantially some results obtained earlier for Jeffrey conditionalization. (shrink)

Richard Jeffrey developed the formula for probability kinematics with the intent that it would show that strong foundations are epistemologically unnecessary. But the reasons that support strong foundationalism are considerations of dynamics rather than kinematics. The strong foundationalist is concerned with the origin of epistemic force; showing how epistemic force is propagated therefore cannot undermine his position. The weakness of personalism is evident in the difficulty the personalist has in giving a principled answer to the question of when the conditions (...) for the application of the kinematic formula—the rigidity of the posteriors—are fulfilled, a problem made intractable by the personalist commitment to treating changes in intermediate probability as unexplained surds. Because the strong foundationalist admits changes in the intermediate probability of propositions only when there is some change in the foundations, he can avail himself of ananswer to the problem of the rigidity of the posteriors which the personalist cannot regard as complete. While probability kinematics does not make certain foundations unnecessary, the possession of certain foundations also does not make the probability kinematics formula superfluous. The formula allowsus to model the indirect routes by which the foundations influence various non-foundational propositions in the probability distribution. (shrink)

Bayesians since Savage (1972) have appealed to asymptotic results to counter charges of excessive subjectivity. Their claim is that objectionable differences in prior probability judgments will vanish as agents learn from evidence, and individual agents will converge to the truth. Glymour (1980), Earman (1992) and others have voiced the complaint that the theorems used to support these claims tell us, not how probabilities updated on evidence will actually}behave in the limit, but merely how Bayesian agents believe they will behave, suggesting (...) that the theorems are too weak to underwrite notions of scientific objectivity and intersubjective agreement. I investigate, in a very general framework, the conditions under which updated probabilities actually converge to a settled opinion and the conditions under which the updated probabilities of two agents actually converge to the same settled opinion. I call this mode of convergence deterministic, and derive results that extend those found in Huttegger (2015b). The results here lead to a simple characterization of deterministic convergence for Bayesian learners and give rise to an interesting argument for what I call strong regularity, the view that probabilities of non-empty events should be bounded away from zero. (shrink)