Bayesian Conditionalization Research Articles

Conditionalization and not Knowing that One Knows

Bayesian Conditionalization is a widely used proposal for how to update one’s beliefs upon the receipt of new evidence. This is in part because of its attention to the totality of one’s evidence, which often includes facts about what one’s new evidence is and how one has come to have it. However, an increasingly popular position in epistemology holds that one may gain new evidence, construed as knowledge, without being in a position to know that one has gained this evidence. These are cases of KK-Failure, cases where one knows p but is not in a position to know that one knows p. This paper assumes that certain KK-Failures are possible and argues that Conditionalization goes wrong in those cases.

Objective Bayesianism and the Maximum Entropy Principle

Objective Bayesian epistemology invokes three norms: the strengths of our beliefs should be probabilities; they should be calibrated to our evidence of physical probabilities; and they should otherwise equivocate sufficiently between the basic propositions that we can express. The three norms are sometimes explicated by appealing to the maximum entropy principle, which says that a belief function should be a probability function, from all those that are calibrated to evidence, that has maximum entropy. However, the three norms of objective Bayesianism are usually justified in different ways. In this paper, we show that the three norms can all be subsumed under a single justification in terms of minimising worst-case expected loss. This, in turn, is equivalent to maximising a generalised notion of entropy. We suggest that requiring language invariance, in addition to minimising worst-case expected loss, motivates maximisation of standard entropy as opposed to maximisation of other instances of generalised entropy. Our argument also provides a qualified justification for updating degrees of belief by Bayesian conditionalisation. However, conditional probabilities play a less central part in the objective Bayesian account than they do under the subjective view of Bayesianism, leading to a reduced role for Bayes’ Theorem.

Defeasible Conditionalization

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.

Updating as Communication*

On many traditional theories of belief, your belief state is represented by an assignment of credences to propositions, or sets of possible worlds. If you are rational, your credence distribution will be a probability measure. Traditional theories of belief fit with a standard Bayesian theory of rational belief change: on learning a proposition, you must update your belief state by conditionalizing your credence distribution on the proposition you learn. That is, you must update by assigning 0 credence to those worlds incompatible with what you learn, and re-normalizing your credence distribution over the remaining worlds. Following Quine 1969, Lewis 1979 argues that we should instead represent your belief state by an assignment of credences to sets of centered worlds: world-timeindividual triples. For instance, if you have .5 credence that it is 3:00pm, your belief state should be represented by a measure that assigns .5 to the set of centered worlds with that time coordinate. Unlike traditional theories of belief, Lewis’s theory does not fit with a standard Bayesian theory of rational belief change. For instance, Bayesian conditionalization preserves certainties. If you update by conditionalizing on the set of centered worlds you learn, it follows that if you are ever certain that it is 3:00pm, you must always remain certain that it is 3:00pm. But clearly this is not what rationality requires. If we agree with Lewis about how to represent belief states, we must develop another set of principles governing rational belief change.

Symmetry and the Brown-Freiling Refutation of the Continuum Hypothesis

Freiling [1] and Brown [2] have put forward a probabilistic reductio argument intended to refute the Continuum Hypothesis. The argument relies heavily upon intuitions about symmetry in a particular scenario. This paper argues that the argument fails, but is still of interest for two reasons. First, the failure is unusual in that the symmetry intuitions are demonstrably coherent, even though other constraints make it impossible to find a probability model for the scenario. Second, the best probability models have properties analogous to non-conglomerability, motivating a proposed extension of that concept (and corresponding limits on Bayesian conditionalization).

Defining uncertainty with maximum entropy method.

The maximum entropy (ME) method was strongly defended and advocated by E. T. Jaynes as a means to define uncertainty. Here, the ME method is applied to the estimation of ocean waveguide parameter probability distributions from measured acoustic data. An ME analysis produces a canonical distribution, well known from equilibrium statistical mechanics, which is the distribution that maximizes the entropy subject to constraints that reflect selected features of the measured data and a model. The ME method gives the most conservative distribution based only on the measured data and observed features. A Bayesian approach also has the goal of defining uncertainty, but starts from the specification of the likelihood function and the model priors. Data noise is naturally handled in the specification of the likelihood function. The discussion introduces simple examples, showing basic relationships between the constraints in ME and the maximum likelihood estimation. In special cases the form of the likelihood function used in Bayesian conditionalization can be derived from the ME approach. The form of the cost function is an important consideration in comparing ME and Bayesian methods of inferences. In general ME and Bayesian inferences lead to different results. [Work supported by ONR Code 321 OA.]

Un argumento trascendental para la inducción

El problema de la justificación de la inducción requiere, en mi opinión, la distinción entre justificación persuasiva y justificación explicativa. Manifiesto mi acuerdo con David Hume en que no hay justificación persuasiva posible de la inducción y encuentro una justificación explicativa para el razonamiento inductivo en la filosofía de Donald Davidson.

Does Roush show that evidence should be probable?

This paper critically analyzes Sherrilyn Roush’s (Tracking truth: knowledge, evidence and science, 2005) definition of evidence and especially her powerful defence that in the ideal, a claim should be probable to be evidence for anything. We suggest that Roush treats not one sense of ‘evidence’ but three: relevance, leveraging and grounds for knowledge; and that different parts of her argument fare differently with respect to different senses. For relevance, we argue that probable evidence is sufficient but not necessary for Roush’s own two criteria of evidence to be met. With respect to grounds for knowledge, we agree that high probability evidence is indeed ideal for the central reason Roush gives: When believing a hypothesis on the basis of e it is desirable that e be probable. But we maintain that her further argument that Bayesians need probable evidence to warrant the method they recommend for belief revision rests on a mistaken interpretation of Bayesian conditionalization. Moreover, we argue that attempts to reconcile Roush’s arguments with Bayesianism fail. For leveraging, which we agree is a matter of great importance, the requirement that evidence be probable suffices for leveraging to the probability of the hypothesis if either one of Roush’s two criteria for evidence are met. Insisting on both then seems excessive. To finish, we show how evidence, as Roush defines it, can fail to track the hypothesis. This can remedied by adding a requirement that evidence be probable, suggesting another rationale for taking probable evidence as ideal—but only for a grounds-for-knowledge sense of evidence.

Objective Bayesianism, Bayesian conditionalisation and voluntarism

Objective Bayesianism has been criticised on the grounds that objective Bayesian updating, which on a finite outcome space appeals to the maximum entropy principle, differs from Bayesian conditionalisation. The main task of this paper is to show that this objection backfires: the difference between the two forms of updating reflects negatively on Bayesian conditionalisation rather than on objective Bayesian updating. The paper also reviews some existing criticisms and justifications of conditionalisation, arguing in particular that the diachronic Dutch book justification fails because diachronic Dutch book arguments are subject to a reductio: in certain circumstances one can Dutch book an agent however she changes her degrees of belief. One may also criticise objective Bayesianism on the grounds that its norms are not compulsory but voluntary, the result of a stance. It is argued that this second objection also misses the mark, since objective Bayesian norms are tied up in the very notion of degrees of belief.

Sleeping Beauty and Self-location: A Hybrid Model

The Sleeping Beauty problem is test stone for theories about self- locating belief, i.e. theories about how we should reason when data or theories contain indexical information. Opinion on this problem is split between two camps, those who defend the “1/2 view” and those who advocate the “1/3 view”. I argue that both these positions are mistaken. Instead, I propose a new “hybrid” model, which avoids the faults of the standard views while retaining their attractive properties. This model appears to violate Bayesian conditionalization, but I argue that this is not the case. By paying close attention to the details of conditionalization in contexts where indexical information is relevant, we discover that the hybrid model is in fact consistent with Bayesian kinematics. If the proposed model is correct, there are important lessons for the study of self-location, observation selection theory, and anthropic reasoning.

Assimilation of fuzzy data by the BME method

Modern spatiotemporal geostatistics provides a powerful framework for generation of predictive maps over a spatiotemporal domain by accounting for general knowledge to define a space of plausible events and then restricting this space of plausible events to be consistent with available site-specific knowledge. The Bayesian maximum entropy (BME) method is one of the most widely used modern geostatistics methods. BME results from assigning probabilities of plausible events based on general knowledge through information maximization and then applying operational Bayesian conditionalization that can explicitly assimilate stochastic representations of various uncertain (soft) data bases. The paper demonstrates that fuzzy data sets can be indirectly assimilated by BME through a two-step process: (a) reinterpretation of the fuzzy data as probabilistic through a generalized defuzzification procedure, and (b) efficient assimilation of the probabilistic results of generalized defuzzification by the BME method. A numerical demonstration involves site-specific probabilistic results obtained from the generalized defuzzification of a simulated fuzzy data set and general knowledge that includes the spatial mean trend and correlation structure models. The parameters of these models can be inferred from the hard data equivalent values of the probabilistic results. Accordingly, details of inference based on probabilistic soft data are also considered.

On the assimilation of uncertain physical knowledge bases: Bayesian and non-Bayesian techniques

We study the assimilation of uncertain physical knowledge across space and time, and we discuss the resulting issues of environmental inference and prediction. The term “assimilation” refers to a general framework that includes physical knowledge integration and processing activities in a variety of scientific disciplines (atmospheric data analysis and weather forecasting, environmental pollution mapping, exposure assessment, etc.). The choice of an adequate assimilation approach is primarily a conceptual modelling affair that is supported by the physical and logical features of the environmental situation under consideration, rather than merely relying on pure inductive schemes and statistical arguments. In many cases, the assimilation approach involves conditionalization techniques which enable it to merge various forms of site-specific data with modelling frameworks. We distinguish between interpretive conditionalization (which refers to the physical and epistemic characteristics of assimilation modelling) and formal conditionalization (which is a purely mathematical construction), and emphasize the importance of an adequate choice of the latter to represent the former. We point out the considerable improvements of the operational Bayesian conditionalization approach over the standard Bayesian rule. Statistical learning techniques are useful at the descriptive/correlation level of scientific inquiry, whereas the level of explanation/prediction is the domain of advanced operational techniques. An operational approach is based on an epistemic view in which the relevant natural processes are represented in terms of spatiotemporal random fields and the probability laws are expressed in terms of physically meaningful operators. The spirit of this view is to keep the field solutions consistent with the site-specific information (hard data, uncertain observations, etc.) while exactly satisfying the constraints arising from general knowledge sources (physical laws, primitive equations, scientific theories, etc.). Natural field probabilities in space and time are derived for non-Bayesian operational conditionals based on deductively sound inference, which are valid for a wide variety of knowledge bases. Analytical and numerical comparisons are made between Bayesian and non-Bayesian conditionals, and insight is gained in terms of examples and applications which cut across various earth science disciplines. Operational Bayesian conditionalization is a powerful and versatile component of assimilation modelling with many applications in environmental sciences, although in certain cases a non-Bayesian conditional (based on deductive logic and the characterization of physical connection) may provide a meaningful description of the data assimilation framework suggested by the laws of nature.

Calibration and the Epistemological Role of Bayesian Conditionalization

Evaluation de la pertinence de la conditionnalisation bayesienne (BC) dans le domaine de la theorie de la confirmation en science. Mesurant le role epistemologique de (BC) au regard de la valeur et de la decouverte d'une nouvelle evidence, d'une part, et distinguant l'approche diachronique et le moment particulier de l'opinion, d'autre part, l'A. defend la permissibilite rationnelle d'une mise a jour pour violer (BC) et utilise la notion de calibrage afin de montrer en quel sens un argument justificatif ne peut violer (BC)

Combinatorial Semantics: Semantics for Frequent Validity

In ordinary first–order logic, a valid inference in a language L is one in which the conclusion is true in every model of the language in which the premises are true. To accommodate inductive/uncertain/probabilistic/nonmonotonic inference, we weaken that demand to the demand that the conclusion be true in a large proportion of the models in which the relevant premises are true. More generally, we say that an inference is [p,q] valid if its conclusion is true in a proportion lying between p and q of those models in which the relevant premises are true. If we include a statistical variable binding operator “%” in our language, there are many quite general (and useful) things we can say about uncertain validity. A surprising result is that some of these things may conflict with Bayesian conditionalization.

The constraint rule of the maximum entropy principle

The principle of maximum entropy is a method for assigning values to probability distributions on the basis of partial information. In usual formulations of this and related methods of inference one assumes that this partial information takes the form of a constraint on allowed probability distributions. In practical applications, however, the information consists of empirical data. A constraint rule is then employed to construct constraints on probability distributions out of these data. Usually one adopts the rule that equates the expectation values of certain functions with their empirical averages. There are, however, various other ways in which one can construct constraints from empirical data, which makes the maximum entropy principle lead to very different probability assignments. This paper shows that an argument by Jaynes to justify the usual constraint rule is unsatisfactory and investigates several alternative choices. The choice of a constraint rule is also shown to be of crucial importance to the debate on the question whether there is a conflict between the methods of inference based on maximum entropy and Bayesian conditionalization.

Bayesian Conditionalization and Probability Kinematics

Previous articleNext article No AccessArticlesBayesian Conditionalization and Probability KinematicsCOLIN HOWSON and ALLAN FRANKLINCOLIN HOWSONLondon School of Economics, University of ColoradoBoulder Search for more articles by this author and ALLAN FRANKLINLondon School of Economics, University of ColoradoBoulder Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by The British Journal for the Philosophy of Science Volume 45, Number 2June 1994 Society: The British Society for the Philosophy of Science Views: 7Total views on this site Citations: 4Citations are reported from Crossref Views: 7Total views on this site Citations: 4Citations are reported from Crossref Article DOIhttps://doi.org/10.1093/bjps/45.2.451 Views: 7Total views on this site Citations: 4Citations are reported from Crossref © 1994 by The Authors. All rights reserved.PDF download Crossref reports the following articles citing this article:Cristian Larroulet Philippi Against Prohibition (Or, When Using Ordinal Scales to Compare Groups Is OK), The British Journal for the Philosophy of Science 0, no.jaja (Jul 2022).https://doi.org/10.1086/721759Cristian Larroulet Philippi On Measurement Scales: Neither Ordinal Nor Interval?, Philosophy of Science 88, no.55 (Dec 2021): 929–939.https://doi.org/10.1086/714873Jonathan Weisberg Updating, Undermining, and Independence, The British Journal for the Philosophy of Science 66, no.11 (Dec 2020): 121–159.https://doi.org/10.1093/bjps/axt052 Allan Franklin , and Colin Howson Comment on "The Structure of a Scientific Paper" by Frederick Suppe, Philosophy of Science 65, no.33 (Oct 2015): 411–416.https://doi.org/10.1086/392653

Realism, Convergence, and Additivity

In one version of the debate between scepticism and scientific realism, the sceptic denies and the realist affirms the ability of science to home in on the unknown truth. Germane to this debate is the familiar claim that probabilistic updating by Bayesian conditionalization will almost surely arrive at the truth so long as one does not “close the door” on a hypothesis by assigning it probability zero. In this paper, we observe, on behalf of the sceptic, that there exists a coherent agent who in fact fails on an uncountable set of possible worlds to find the truth about a simple, universal hypothesis H even though he “keeps the door open” concerning H until H is logically refuted and even though there is a trivial, logically driven method that converges to the truth about H no matter what; that nonetheless, this agent “almost surely” arrives at the truth in the usual, Bayesian sense because the uncountable set of circumstances in which he fails is assigned probability zero by the agent; and that the guarantee that the set of all circumstances in which the agent fails has probability zero hinges on the axiom of countable additivity.

Bayesian Conditionalization Resolves Positivist/Realist Disputes

Bayesian Conditionalization Resolves Positivist/Realist Disputes

Dutch Book Arguments and Consistency

I consider Dutch Book arguments for three principles of classical Bayesianism: (i) agents' belief-probabilities are consistent only if they obey the probability axioms. (ii) beliefs are updated by Bayesian conditionalisation. (iii) that the so-called Principal Principle connects statistical and belief probabilities. I argue that while there is a sound Dutch Book argument for (i), the standard ones for (ii) based on the Lewis-Teller strategy are unsound, for reasons pointed out by Christensen. I consider a type of Dutch Book argument for (iii), where the statistical probability is a von Mises one.

Conditional probability judgments: Effects of imagining vs experiencing the conditioning event

Applications of Bayesian conditionalization often involve two temporal aspects: a probability judgment is based on knowledge at a point in time and is revised over time in light of added information. Let t, t′, and t″ designate three chronological points in time. E designates a target event which occurs or not at t″, and C designates a conditioning event which occurs or not at t′. Suppose that an individual judges P( E⧹ C) at t. If C occurs, Bayesian conditionalization requires that a judgment of P( E) at t′ is equal to the earlier judgment of P( E⧹ C). However, inconsistencies may result because a judgment of P( E⧹ C) at t is based on imagining C, while a judgment of P( E) at t′ is based on experiencing C. This study examines two sources of such inconsistencies. First, C normally is an abstraction of what might happen between t and t′. What actually happens may differ, such that an individual observes extraconditional information which affects a judgment of P( E) at t′. Second, experiencing C may change an individual's affective state, leading to greater optimism or pessimism about the occurrence of E. We report an experiment which documents both effects.