Jon Williamson

We present a new framework for combining logic with probability, and demonstrate the application of this framework to breast cancer prognosis. Background knowledge concerning breast cancer prognosis is represented using logical arguments. This background knowledge and a database are used to build a Bayesian net that captures the probabilistic relationships amongst the variables. Causal hypotheses gleaned from the Bayesian net in turn generate new arguments. The Bayesian net can be queried to help decide when one argument attacks another. The Bayesian net is used to perform the prognosis, while the argumentation framework is used to provide a qualitative explanation of the prognosis.

How ought we learn causal relationships? While Popper advocated a hypothetico-deductive logic of causal discovery, inductive accounts are currently in vogue. Many inductive approaches depend on the causal Markov condition as a fundamental assumption. This condition, I maintain, is not universally valid, though it is justifiable as a default assumption. In which case the results of the inductive causal learning procedure must be tested before they can be accepted. This yields a synthesis of the hypothetico-deductive and inductive accounts, which forms the focus of this paper. I discuss the justification of this synthesis and draw an analogy between objective Bayesianism and the account of causal learning presented here.

This volume arose out of an international, interdisciplinary academic network on Probabilistic Logic and Probabilistic Networks involving four of us (Haenni, Romeijn, Wheeler and Williamson), called Progicnet and funded by the Leverhulme Trust from 2006–8. Many of the papers in this volume were presented at an associated conference, the Third Workshop on Combining Probability and Logic (Progic 2007), held at the University of Kent on 5–7 September 2007. The papers in this volume concern either the special focus on the connection between probabilistic logic and probabilistic networks or the more general question of the links between probability and logic. Here we introduce probabilistic logic, probabilistic networks, current and future directions of research and also the themes of the papers that follow.

According to objective Bayesianism, an agent’s degrees of belief should be determined by a probability function, out of all those that satisfy constraints imposed by background knowledge, that maximises entropy. A Bayesian net offers a way of efficiently representing a probability function and efficiently drawing inferences from that function. An objective Bayesian net is a Bayesian net representation of the maximum entropy probability function. In this paper we apply the machinery of objective Bayesian nets to breast cancer prognosis. Background knowledge is diverse and comes from several different sources: a database of clinical data, a database of molecular data, and quantitative data from the literature. We show how an objective Bayesian net can be constructed from this background knowledge and how it can be applied to yield prognoses and aid translation of clinical knowledge to genomics research

Inductive logic admits a variety of semantics (Haenni et al., 2011, Part 1). This paper develops semantics based on the norms of Bayesian epistemology (Williamson, 2010, Chapter 7). §1 introduces the semantics and then, in §2, the paper explores methods for drawing inferences in the resulting logic and compares the methods of this paper with the methods of Barnett and Paris (2008). §3 then evaluates this Bayesian inductive logic in the light of four traditional critiques of inductive logic, arguing (i) that it is language independent in a key sense, (ii) that it admits connections with the Principle of Indifference but these connections do not lead to paradox, (iii) that it can capture the phenomenon of learning from experience, and (iv) that while the logic advocates scepticism with regard to some universal hypotheses, such scepticism is not problematic from the point of view of scientific theorising

In this paper, we examine what is to be said in defence of Machamer, Darden and Craver’s (MDC) controversial dualism about activities and entities (Machamer, Darden and Craver’s in Philos Sci 67:1–25, 2000). We explain why we believe the notion of an activity to be a novel, valuable one, and set about clearing away some initial objections that can lead to its being brushed aside unexamined. We argue that substantive debate about ontology can only be effective when desiderata for an ontology are explicitly articulated. We distinguish three such desiderata. The first is a more permissive descriptive ontology of science, the second a more reductive ontology prioritising understanding, and the third a more reductive ontology prioritising minimalism. We compare MDC’s entities-activities ontology to its closest rival, the entities-capacities ontology, and argue that the entities-activities ontology does better on all three desiderata

It is tempting to analyse causality in terms of just one of the indicators of causal relationships, e.g., mechanisms, probabilistic dependencies or independencies, counterfactual conditionals or agency considerations. While such an analysis will surely shed light on some aspect of our concept of cause, it will fail to capture the whole, rather multifarious, notion. So one might instead plump for pluralism: a different analysis for a different occasion. But we do not seem to have lots of different concepts of cause – just one eclectic notion. The resolution of this conundrum, I think, requires us to accept that our causal beliefs are generated by a wide variety of indicators, but to deny that this variety of indicators yields a variety of concepts of cause. This focus on the relation between evidence and causal beliefs leads to what I call epistemic causality. Under this view, certain causal beliefs are appropriate or rational on the basis of observed evidence; our notion of cause can be understood purely in terms of these rational beliefs. Causality, then, is a feature of our epistemic representation of the world, rather than of the world itself. This yields one, multifaceted notion of cause

I present a formalism that combines two methodologies: objective Bayesianism and Bayesian nets. According to objective Bayesianism, an agent’s degrees of belief (i) ought to satisfy the axioms of probability, (ii) ought to satisfy constraints imposed by background knowledge, and (iii) should otherwise be as non-committal as possible (i.e. have maximum entropy). Bayesian nets offer an efficient way of representing and updating probability functions. An objective Bayesian net is a Bayesian net representation of the maximum entropy probability function.

In this chapter I discuss connections between machine learning and the philosophy of science. First I consider the relationship between the two disciplines. There is a clear analogy between hypothesis choice in science and model selection in machine learning. While this analogy has been invoked to argue that the two disciplines are essentially doing the same thing and should merge, I maintain that the disciplines are distinct but related and that there is a dynamic interaction operating between the two: a series of mutually beneficial interactions that changes over time. I will introduce some particularly fruitful interactions, in particular the consequences of automated scientific discovery for the debate on inductivism versus falsificationism in the philosophy of science, and the importance of philosophical work on Bayesian epistemology and causality for contemporary machine learning. I will close by suggesting the locus of a possible future interaction: evidence integration.

1, . . . , n | ≈ ψ ? Here 1, . . . , n, ψ are premisses of some formal language, such as a propositional language or a predicate language. | ≈ is an entailment relation: the entailment holds if all models of the premisses also satisfy the conclusion, where the logic provides some suitable notion of ‘model’ and ‘satisfy’. Proof theory is normally invoked to answer a question of this form: one tries to prove the conclusion from the premisses in a finite sequence of steps, where at each step one invokes an axiom or applies a rule of inference

Russo and Williamson (Int Stud Philos Sci 21(2):157–170, 2007) put forward the thesis that, at least in the health sciences, to establish the claim that C is a cause of E, one normally needs evidence of an underlying mechanism linking C and E as well as evidence that C makes a difference to E. This epistemological thesis poses a problem for most current analyses of causality which, in virtue of analysing causality in terms of just one of mechanisms or difference making, cannot account for the need for the other kind of evidence. Weber (Int Stud Philos Sci 23(2):277–295, 2009) has suggested to the contrary that Giere’s probabilistic analysis of causality survives this criticism. In this paper, we look in detail at the case of medical imaging technology, which, we argue, supports the thesis of Russo and Williamson, and we respond to Weber’s suggestion, arguing that Giere’s account does not survive the criticism

Causal claims in biomedical contexts are ubiquitous albeit they are not always made explicit. This paper addresses the question of what causal claims mean in the context of disease. It is argued that in medical contexts causality ought to be interpreted according to the epistemic theory. The epistemic theory offers an alternative to traditional accounts that cash out causation either in terms of “difference-making” relations or in terms of mechanisms. According to the epistemic approach, causal claims tell us about which inferences (e.g., diagnoses and prognoses) are appropriate, rather than about the presence of some physical causal relation analogous to distance or gravitational attraction. It is shown that the epistemic theory has important consequences for medical practice, in particular with regard to evidence-based causal assessment

The mechanistic and causal accounts of explanation are often conflated to yield a ‘causal-mechanical’ account. This paper prizes them apart and asks: if the mechanistic account is correct, how can causal explanations be explanatory? The answer to this question varies according to how causality itself is understood. It is argued that difference-making, mechanistic, dualist and inferentialist accounts of causality all struggle to yield explanatory causal explanations, but that an epistemic account of causality is more promising in this regard.

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.

Practical reasoning requires decision—making in the face of uncertainty. Xenelda has just left to go to work when she hears a burglar alarm. She doesn’t know whether it is hers but remembers that she left a window slightly open. Should she be worried? Her house may not be being burgled, since the wind or a power cut may have set the burglar alarm off, and even if it isn’t her alarm sounding she might conceivably be being burgled. Thus Xenelda can not be certain that her house is being burgled, and the decision that she takes must be based on her degree of certainty, together with the possible outcomes of that decision.