Jürgen Landes

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: (i) applying it to finite sublanguages and taking the pointwise limit of the resulting probabilities as the size n of the sublanguage increases; (ii) 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.

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.

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.

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.

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.

Models developed by the knowledge representation and reasoning community permit us to study defeasible inference based on argumentation and data. Scientific reasoning progresses by evaluating scientific hypotheses based on data and meta-evidence. Meta-evidence can be understood as arguments for discounting or even ignoring data or other meta-evidence. Non-monotonic reasoning is underpinned by non-monotonic logic. We here develop a method for modelling scientific inferences within formal argumentation models. We show how these models capture hypothesis testing, meta-analysis, strong inference and non-monotonic consequence relations.

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: (i) applying it to finite sublanguages and taking a limit; (ii) 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 Σ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _2$$\end{document}-premiss sentences. I discuss implications of this failure for the first strategy and consequences for the entropy-limit conjecture.

Explanatory relationships between data and hypotheses have been suggested to play a role in the formation of posterior probabilities. This suggestion was tested in a toy environment and supported by simulations by David H. Glass. We here put forward a variety of inference to the best explanation approaches for determining posterior probabilities by intertwining Bayesian and inference to the best explanation approaches. We then simulate their performances for the estimation of parameters in the Brock and Hommes agent-based model for asset pricing in finance. We find that performances depend on circumstances and also on the evaluation metric. However, most of the time our suggested approaches outperform the Bayesian approach.

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.

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.

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.

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.

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.

Philosophical discussions on causal inference in medicine are stuck in dyadic camps, each defending one kind of evidence or method rather than another as best support for causal hypotheses. Whereas Evidence Based Medicine advocates the use of Randomised Controlled Trials and systematic reviews of RCTs as gold standard, philosophers of science emphasise the importance of mechanisms and their distinctive informational contribution to causal inference and assessment. Some have suggested the adoption of a pluralistic approach to causal inference, and an inductive rather than hypothetico-deductive inferential paradigm. However, these proposals deliver no clear guidelines about how such plurality of evidence sources should jointly justify hypotheses of causal associations. We here develop such guidelines by first giving a philosophical analysis of the underpinnings of Hill’s viewpoints on causality. We then put forward an evidence-amalgamation framework adopting a Bayesian net approach to model causal inference in pharmacology for the assessment of harms. Our framework accommodates a number of intuitions already expressed in the literature concerning the EBM vs. pluralist debate on causal inference, evidence hierarchies, causal holism, relevance, and reliability.

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.

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.