We argue that David Lewis’s principal principle implies a version of the principle of indifference. The same is true for similar principles that need to appeal to the concept of admissibility. Such principles are thus in accord with objective Bayesianism, but in tension with subjective Bayesianism. 1 The Argument2 Some Objections Met.

We argue that David Lewis’s principal principle implies a version of the principle of indifference. The same is true for similar principles that need to appeal to the concept of admissibility. Such principles are thus in accord with objective Bayesianism, but in tension with subjective Bayesianism. 1 The Argument2 Some Objections Met.

Varied evidence confirms more strongly than less varied evidence, ceteris paribus. This epistemological Variety of Evidence Thesis enjoys widespread intuitive support. We put forward a novel explication of one notion of varied evidence and the Variety of Evidence Thesis within Bayesian models of scientific inference by appealing to measures of entropy. Our explication of the Variety of Evidence Thesis holds in many of our models which also pronounce on disconfirmatory and discordant evidence. We argue that our models pronounce rightly. Against (...) a backdrop of failures of the Variety of Evidence Thesis, the intuitive case for the Variety of Evidence Thesis emerges strengthened. Our models do however not support the general case for the thesis since our explication of it fails to hold in certain cases. The parameter space of this failure is explored and an explanation for the failure is offered. (shrink)

This paper highlights the role of Lewis’ Principal Principle and certain auxiliary conditions on admissibility as serving to explicate normal informal standards of what is reasonable. These considerations motivate the presuppositions of the argument that the Principal Principle implies the Principle of Indifference, put forward by Hawthorne et al.. They also suggest a line of response to recent criticisms of that argument, due to Pettigrew and Titelbaum and Hart, 621–632, 2020). The paper also shows that related concerns of Hart and (...) Titelbaum, 252–262, 2015) do not undermine the argument of Hawthorne et al.. (shrink)

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. (shrink)

The intuitive Variety of Evidence Thesis states that, ceteris paribus, more varied evidence for a hypothesis confirms it more strongly than less varied evidence. Recent Bayesian analyses have raised serious doubts in its validity. Claveau suggests the existence of a novel type of counter-example to this thesis: a gradual increase in source independence can lead to a decrease in hypothesis confirmation. I show that Claveau’s measure of gradual source independence suffers from two unsuspected types of inconsistencies. I hence put forward (...) a more natural measure of gradual source independence which is not plagued by inconsistencies. Claveau’s counter-examples to the variety of evidence thesis disappear with the measure I suggest. I hence argue that my measure is preferable and that this thesis does at least not seem to be troubled by Claveau’s arguments. (shrink)

According to the variety of evidence thesis items of evidence from independent lines of investigation are more confirmatory, ceteris paribus, than, for example, replications of analogous studies. This thesis is known to fail (Bovens and Hartmann; Claveau). However, the results obtained by Bovens and Hartmann only concern instruments whose evidence is either fully random or perfectly reliable; instead, for Claveau, unreliability is modelled as deterministic bias. In both cases, the unreliable instrument delivers totally irrelevant information. We present a model that (...) formalizes both reliability and unreliability differently. Our instruments either are reliable, but affected by random error, or are biased but not deterministically so.Bovens and Hartmann’s results are counter-intuitive in that in their model a long series of consistent reports from the same instrument does not raise suspicion of ‘too-good-to-be-true’ evidence. This happens precisely because they contemplate neither the role of systematic bias, nor unavoidable random error of reliable instruments. In our model, the variety of evidence thesis fails as well, but the area of failure is considerably smaller than for Bovens and Hartmann and Claveau, and holds for (the majority of) realistic cases (that is, where biased instruments are very biased). The essential mechanism that triggers variety of evidence thesis failure is the rate of false to true positives for the two kinds of instruments. Our emphasis is on modelling beliefs about sources of knowledge and their role in hypothesis confirmation in interaction with dimensions of evidence, such as variety and consistency. (shrink)

Amalgamating evidence from heterogeneous sources and across levels of inquiry is becoming increasingly important in many pure and applied sciences. This special issue provides a forum for researchers from diverse scientific and philosophical perspectives to discuss evidence amalgamation, its methodologies, its history, its pitfalls, and its potential. We situate the contributions therein within six themes from the broad literature on this subject: the variety-of-evidence thesis, the philosophy of meta-analysis, the role of robustness/sensitivity analysis for evidence amalgamation, its bearing on questions (...) of extrapolation and external validity of experiments, its connection with theory development, and its interface with causal inference, especially regarding causal theories of cancer. (shrink)

Objective Bayesianism says that the strengths of one’s beliefs ought to be probabilities, calibrated to physical probabilities insofar as one has evidence of them, and otherwise sufficiently equivocal. These norms of belief are often explicated using the maximum entropy principle. In this paper we investigate the extent to which one can provide a unified justification of the objective Bayesian norms in the case in which the background language is a first-order predicate language, with a view to applying the resulting formalism (...) to inductive logic. We show that the maximum entropy principle can be motivated largely in terms of minimising worst-case expected loss. (shrink)

This paper addresses a data integration problem: given several mutually consistent datasets each of which measures a subset of the variables of interest, how can one construct a probabilistic model that fits the data and gives reasonable answers to questions which are under-determined by the data? Here we show how to obtain a Bayesian network model which represents the unique probability function that agrees with the probability distributions measured by the datasets and otherwise has maximum entropy. We provide a general (...) algorithm, OBN-cDS, which offers substantial efficiency savings over the standard brute-force approach to determining the maximum entropy probability function. Furthermore, we develop modifications to the general algorithm which enable further efficiency savings but which are only applicable in particular situations. We show that there are circumstances in which one can obtain the model directly from the data; by solving algebraic problems; and by solving relatively simple independent optimisation problems. (shrink)

Some authors claim that minimal models have limited epistemic value (Fumagalli, 2016; Grüne-Yanoff, 2009a). Others defend the epistemic benefits of modelling by invoking the role of robustness analysis for hypothesis confirmation (see, e.g., Levins, 1966; Kuorikoski et al., 2010) but such arguments find much resistance (see, e.g., Odenbaugh & Alexandrova, 2011). In this paper, we offer a Bayesian rationalization and defence of the view that robustness analysis can play a confirmatory role, and thereby shed light on the potential of minimal (...) models for hypothesis confirmation. We illustrate our argument by reference to a case study from macroeconomics. At the same time, we also show that there are cases in which robustness analysis is detrimental to confirmation. We characterize these cases and link them to recent investigations on evidential variety (Landes, 2020b, 2021; Osimani and Landes, forthcoming). We conclude that robustness analysis over minimal models can confirm, but its confirmatory value depends on concrete circumstances. (shrink)

The maximum entropy principle is widely used to determine non-committal probabilities on a finite domain, subject to a set of constraints, but its application to continuous domains is notoriously problematic. This paper concerns an intermediate case, where the domain is a first-order predicate language. Two strategies have been put forward for applying the maximum entropy principle on such a domain: applying it to finite sublanguages and taking the pointwise limit of the resulting probabilities as the size n of the sublanguage (...) increases; selecting a probability function on the language as a whole whose entropy on finite sublanguages of size n is not dominated by that of any other probability function for sufficiently large n. The entropy-limit conjecture says that, where these two approaches yield determinate probabilities, the two methods yield the same probabilities. If this conjecture is found to be true, it would provide a boost to the project of seeking a single canonical inductive logic—a project which faltered when Carnap's attempts in this direction succeeded only in determining a continuum of inductive methods. The truth of the conjecture would also boost the project of providing a canonical characterisation of normal or default models of first-order theories. Hitherto, the entropy-limit conjecture has been verified for languages which contain only unary predicate symbols and also for the case in which the constraints can be captured by a categorical statement of quantifier complexity. This paper shows that the entropy-limit conjecture also holds for categorical statements of complexity, for various non-categorical constraints, and in certain other general situations. (shrink)

Background: Evidence suggesting adverse drug reactions often emerges unsystematically and unpredictably in form of anecdotal reports, case series and survey data. Safety trials and observational studies also provide crucial information regarding the (un-)safety of drugs. Hence, integrating multiple types of pharmacovigilance evidence is key to minimising the risks of harm. Methods: In previous work, we began the development of a Bayesian framework for aggregating multiple types of evidence to assess the probability of a putative causal link between drugs and side (...) effects. This framework arose out of a philosophical analysis of the Bradford Hill Guidelines. In this article, we expand the Bayesian framework and add “evidential modulators,” which bear on the assessment of the reliability of incoming study results. The overall framework for evidence synthesis, “E-Synthesis”, is then applied to a case study. Results: Theoretically and computationally, E-Synthesis exploits coherence of partly or fully independent evidence converging towards the hypothesis of interest (or of conflicting evidence with respect to it), in order to update its posterior probability. With respect to other frameworks for evidence synthesis, our Bayesian model has the unique feature of grounding its inferential machinery on a consolidated theory of hypothesis confirmation (Bayesian epistemology), and in allowing any data from heterogeneous sources (cell-data, clinical trials, epidemiological studies), and methods (e.g., frequentist hypothesis testing, Bayesian adaptive trials, etc.) to be quantitatively integrated into the same inferential framework. Conclusions: E-Synthesis is highly flexible concerning the allowed input, while at the same time relying on a consistent computational system, that is philosophically and statistically grounded. Furthermore, by introducing evidential modulators, and thereby breaking up the different dimensions of evidence (strength, relevance, reliability), E-Synthesis allows them to be explicitly tracked in updating causal hypotheses. (shrink)

This paper addresses the problem of finding a Bayesian net representation of the probability function that agrees with the distributions of multiple consistent datasets and otherwise has maximum entropy. We give a general algorithm which is significantly more efficient than the standard brute-force approach. Furthermore, we show that in a wide range of cases such a Bayesian net can be obtained without solving any optimisation problem.

Recent work in social epistemology has shown that, in certain situations, less communication leads to better outcomes for epistemic groups. In this paper, we show that, ceteris paribus, a Bayesian agent may believe less strongly that a single agent is biased than that an entire group of independent agents is biased. We explain this initially surprising result and show that it is in fact a consequence one may conceive on the basis of commonsense reasoning.

Varied evidence for a hypothesis confirms it more strongly than less varied evidence, ceteris paribus. This epistemological Variety of Evidence Thesis enjoys long-standing widespread intuitive support. Recent literature has raised serious doubts that the correlational approach of explicating the thesis can vindicate it. By contrast, the eliminative approach due to Horwich vindicates the Variety of Evidence Thesis but only within a relatively narrow domain. I investigate the prospects of extending the eliminative approach to a larger domain by considering a larger (...) class of sensible explications of evidential variety. For a large subclass class of such explications I show how to construct cases in which the less diverse body of evidence for a hypothesis confirms more strongly. I hence argue that these prospects are dire since the eliminative approach widely fails to vindicate the thesis. (shrink)

Objective Bayesians hold that degrees of belief ought to be chosen in the set of probability functions calibrated with one’s evidence. The particular choice of degrees of belief is via some objective, i.e., not agent-dependent, inference process that, in general, selects the most equivocal probabilities from among those compatible with one’s evidence. Maximising entropy is what drives these inference processes in recent works by Williamson and Masterton though they disagree as to what should have its entropy maximised. With regard to (...) the probability function one should adopt as one’s belief function, Williamson advocates selecting the probability function with greatest entropy compatible with one’s evidence while Masterton advocates selecting the expected probability function relative to the density function with greatest entropy compatible with one’s evidence. In this paper we discuss the significant relative strengths of these two positions. In particular, Masterton’s original proposal is further developed and investigated to reveal its significant properties; including its equivalence to the centre of mass inference process and its ability to accommodate higher order evidence. (shrink)

A necessary and sufficient condition in terms of a de Finetti style representation is given for a probability function in Polyadic Inductive Logic to satisfy being part of a Language Invariant family satisfying Spectrum Exchangeability. This theorem is then considered in relation to the unary Carnap and Nix–Paris Continua.

The application of the maximum entropy principle to determine probabilities on finite domains is well-understood. Its application to infinite domains still lacks a well-studied comprehensive approach. There are two different strategies for applying the maximum entropy principle on first-order predicate languages: applying it to finite sublanguages and taking a limit; comparing finite entropies of probability functions defined on the language as a whole. The entropy-limit conjecture roughly says that these two strategies result in the same probabilities. While the conjecture is (...) known to hold for monadic languages as well as for premiss sentences containing only existential or only universal quantifiers, its status for premiss sentences of greater quantifier complexity is, in general, unknown. I here show that the first approach fails to provide a sensible answer for some \-premiss sentences. I discuss implications of this failure for the first strategy and consequences for the entropy-limit conjecture. (shrink)

The application of the maximum entropy principle to determine probabilities on finite domains is well-understood. Its application to infinite domains still lacks a well-studied comprehensive approach. There are two different strategies for applying the maximum entropy principle on first-order predicate languages: applying it to finite sublanguages and taking a limit; comparing finite entropies of probability functions defined on the language as a whole. The entropy-limit conjecture roughly says that these two strategies result in the same probabilities. While the conjecture is (...) known to hold for monadic languages as well as for premiss sentences containing only existential or only universal quantifiers, its status for premiss sentences of greater quantifier complexity is, in general, unknown. I here show that the first approach fails to provide a sensible answer for some \-premiss sentences. I discuss implications of this failure for the first strategy and consequences for the entropy-limit conjecture. (shrink)

We give a brief account of some de Finetti style representation theorems for probability functions satisfying Spectrum Exchangeability in Polyadic Inductive Logic, together with applications to Non-splitting, Language Invariance, extensions with Equality and Instantial Relevance.

How we can reliably draw inferences from data, evidence and/or experience has been and continues to be a pressing question in everyday life, the sciences, politics and a number of branches in philosophy (traditional epistemology, social epistemology, formal epistemology, logic and philosophy of the sciences). In a world in which we can now longer fully rely on our experiences, interlocutors, measurement instruments, data collection and storage systems and even news outlets to draw reliable inferences, the issue becomes even more pressing. (...) While we were working on this question using a formal epistemology approach Landes and Osimani (2020); De Pretis et al. (2019); Osimani and Landes (2020); Osimani (2020), we realised that the width of current active interests in the notion of reliability was much broader than we initially thought. Given the breadth of approaches and angles present in philosophy (even in this journal Schubert 2012; Avigad 2021; Claveau and Grenier 2019; Kummerfeld and Danks 2014; Landes 2021; Trpin et al. 2021; Schippers 2014; Schindler 2011; Kelly et al. 2016; Mayo-Wilson 2014; Olsson and Schubert 2007; Pittard 2017), we thought that it would be beneficial to provide a forum for an open exchange of ideas, in which philosophers working in different paradigms could come together. Our call for expression of interest received a great variety of promised manuscripts, the variety is reflected in the published papers. They range from fields far away from our own interests such as quantum probabilities de Ronde et al. (2021), evolvable software systems Primiero et al. (2021), to topics closer to our own research in the philosophy of medicine Lalumera et al. (2020), psychology Dutilh et al. (2021), traditional epistemology Dunn (2021); Tolly (2021) to finally close shared interests in formal epistemology Romero and Sprenger (2021) even within our own department Merdes et al. (2021). Our job is now to reliably inform you about all the contributions in the papers in this special issue. Unfortunately, that task is beyond our capabilities. What we can and instead do is to summarise the contributed papers to inform your reliable inference to read them all in great detail. (shrink)

This paper discusses the issue of overriding the right of individual consent to participation in cluster randomised trials (CRTs). We focus on CRTs testing the efficacy of non-pharmaceutical interventions. As an example, we consider school closures during the COVID-19 pandemic. In Norway, a CRT was promoted as necessary for providing the best evidence to inform pandemic management policy. However, the proposal was rejected by the Norwegian Research Ethics Committee since it would violate the requirement for individual informed consent. This sparked (...) debate about whether ethics stand in the way of evidence-based health policy, since the Norwegian Research Ethics law’s strict requirements for individual consent make it practically impossible to carry out CRTs of public health interventions. We argue that, in the case of the school closure trial, the suggested CRT would not have eliminated an epistemic gap and thus would not have justified the violation of consent rights. First, we focus on the methodological challenges to estimating quantifiable effects of school closures in the specific case of an airborne infectious disease. Second, in line with Evidential Pluralism, we highlight the value of alternative lines of evidence for informing school closure policy in a pandemic. In general, we propose that a trial requiring the waiver of participants’ consent rights must be highly likely to eliminate an epistemic gap. We elaborate on the practical aspects of this criterion and discuss the potential advantages of adding it to the Ottawa Statement on the Ethical Design and Conduct of Cluster Randomized Trials. (shrink)

This editorial explains the scope of the special issue and provides a thematic introduction to the contributed papers.

This article connects recent work in formal epistemology to work in economics and computer science. Analysing the Dutch Book Arguments, Epistemic Utility Theory and Objective Bayesian Epistemology we discover that formal epistemologists employ the same argument structure as economists and computer scientists. Since similar approaches often have similar problems and have shared solutions, opportunities for cross-fertilisation abound.

We introduce two novel frameworks for choice under complete uncertainty. These frameworks employ intervals to represent uncertain utility attaching to outcomes. In the first framework, utility intervals arising from one act with multiple possible outcomes are aggregated via a set-based approach. In the second framework the aggregation of utility intervals employs multi-sets. On the aggregated utility intervals, we then introduce min–max decision rules and lexicographic refinements thereof. The main technical results are axiomatic characterizations of these min–max decision rules and these (...) refinements. We also briefly touch on the independence of introduced axioms. Furthermore, we show that such characterizations give rise to novel axiomatic characterizations of the well-known min–max decision rule?mnx?mnx in the classical framework of choice under complete uncertainty. (shrink)

Personalized medicine relies on two points: 1) causal knowledge about the possible effects of X in a given statistical population; 2) assignment of the given individual to a suitable reference class. Regarding point 1, standard approaches to causal inference are generally considered to be characterized by a trade-off between how confidently one can establish causality in any given study (internal validity) and extrapolating such knowledge to specific target groups (external validity). Regarding point 2, it is uncertain which reference class leads (...) to the most reliable inferences. -/- Instead, pharmacovigilance focuses on both elements of the individual prediction at the same time, that is, the establishment of the possible causal link between a given drug and an observed adverse event, and the identification of possible subgroups, where such links may arise. We develop an epistemic framework that exploits the joint contribution of different dimensions of evidence and allows one to deal with the reference class problem not only by relying on statistical data about covariances, but also by drawing on causal knowledge. That is, the probability that a given individual will face a given side effect, will probabilistically depend on his characteristics and the plausible causal models in which such features become relevant. The evaluation of the causal models is grounded on the available evidence and theory. (shrink)

Besides the usual business of solving paradoxes, there has been recent philosophical work on their essential nature. Lycan characterises a paradox as “an inconsistent set of propositions, each of which is very plausible.” Building on this definition, Paseau offers a numerical measure of paradoxicality of a set of principles: a function of the degrees to which a subject believes the principles considered individually (all typically high) and of the degree to which the subject believes the principles considered together (typically low). (...) We argue (a) that Paseau's measure fails to score certain paradoxes properly and (b) that this failure is not due to the particular measure but rather that any such function just of credences fails to adequately capture paradoxicality. Our analysis leads us to conclude that Lycan's definition also fails to capture the notion of paradox. (shrink)

This chapter addresses the problem of ranking available drugs in guideline development to support clinicians in their work. Based on a pragmatic approach to the notion of evidence and a hierarchical view on different kinds of evidence this chapter introduces a decision aid, HiDAD, which draws on the multi criteria decision making literature. This decision aid implements the wide-spread intuition that there are different kinds of evidence with varying degrees of importance by relying on a strict ordinal ordering of kinds (...) of evidence. In order to construct a ranking every pair of drugs is first compared separately on all kinds of evidence. Next, these quantitative comparisons are then aggregated into an overall comparison between drugs based on all the available evidence in a way which avoids that evidence of less importance is trumped by evidence of the higher levels. Finally, these overall comparisons are used to determine the final ranking of drugs which then informs the process of guideline writing. Properties, modifications and applicability of the decision aid HiDAD are discussed and assessed. (shrink)

This monograph is a collection of conference contributions chosen by the editors who led a three-year project on evolution, cooperation, and rationality. The collected works are held together by a six-page introduction identifying common strands and differences of positions in the different chapters. Since no two chapters have a common author, the chapters do not build on each other. Rather, they offer a variety of perspectives on a number of different aspects of rationality and evolution. The monograph thus does not (...) offer a unified account of evolution and rationality despite the possibly somewhat misleading title, nor is there a clearly identifiable overarching theme that runs through the entire book. (shrink)

We illustrate how a variety of logical methods and techniques provide useful, though currently underappreciated, tools in the foundations and applications of reasoning under uncertainty. The field is vast spanning logic, artificial intelligence, statistics, and decision theory. Rather than (hopelessly) attempting a comprehensive survey, we focus on a handful of telling examples. While most of our attention will be devoted to frameworks in which uncertainty is quantified probabilistically, we will also touch upon generalisations of probability measures of uncertainty, which have (...) attracted a significant interest in the past few decades. (shrink)