Jon Williamson

Subjective Probability: The Real Thing is the last book written by the late Richard Jeffrey, a key proponent of the Bayesian interpretation of probability.Bayesians hold that probability is a mental notion: saying that the probability of rain is 0.7 is just saying that you believe it will rain to degree 0.7. Degrees of belief are themselves cashed out in terms of bets—in this case you consider 7:3 to be fair odds for a bet on rain. There are two extreme Bayesian positions. Strict subjectivists think that an agent can adopt whatever degrees of belief she likes, as long as they satisfy the axioms of probability. Thus your degree of belief in rain and degree of belief in no rain must sum to one but are otherwise unconstrained. At the other extreme, objectivists claim that an agent's background knowledge considerably narrows down the choice of appropriate degrees of belief. In particular, if you know only that the frequency of rain is 0.7 then you should believe it will rain to degree 0.7; if you know absolutely nothing about the weather then you should set your degree of belief in rain to be 0.5; in neither of these cases is there room for subjective choice of degree of belief. In this book, Jeffrey advocates what is sometimes called empirically-based subjectivism, a position that lies between the two extremes of strict subjectivism and objectivism. According to this position, knowledge of frequencies constrains degree of belief, but lack of knowledge does not impose any constraints, so that if you know nothing about the weather you may adopt any degree of belief in rain you like.1The aim of the book is not so much to justify this point of view as to provide a comprehensive exposition of probability theory from the …

Cancer treatment decisions should be based on all available evidence. But this evidence is complex and varied: it includes not only the patient’s symptoms and expert knowledge of the relevant causal processes, but also clinical databases relating to past patients, databases of observations made at the molecular level, and evidence encapsulated in scientific papers and medical informatics systems. Objective Bayesian nets offer a principled path to knowledge integration, and we show in this chapter how they can be applied to integrate various kinds of evidence in the cancer domain. This is important from the systems biology perspective, which needs to integrate data that concern different levels of analysis, and is also important from the point of view of medical informatics.

:A normative Bayesian theory of deliberation and judgement requires a procedure for merging the evidence of a collection of agents. In order to provide such a procedure, one needs to ask what the evidence is that grounds Bayesian probabilities. After finding fault with several views on the nature of evidence, it is argued that evidence is whatever is rationally taken for granted. This view is shown to have consequences for an account of merging evidence, and it is argued that standard axioms for merging need to be altered somewhat.

The Recursive Bayesian Net (RBN) formalism was originally developed for modelling nested causal relationships. In this paper we argue that the formalism can also be applied to modelling the hierarchical structure of mechanisms. The resulting network contains quantitative information about probabilities, as well as qualitative information about mechanistic structure and causal relations. Since information about probabilities, mechanisms and causal relations is vital for prediction, explanation and control respectively, an RBN can be applied to all these tasks. We show in particular how a simple two-level RBN can be used to model a mechanism in cancer science. The higher level of our model contains variables at the clinical level, while the lower level maps the structure of the cell's mechanism for apoptosis

Expert probability forecasts can be useful for decision making . But levels of uncertainty escalate: however the forecaster expresses the uncertainty that attaches to a forecast, there are good reasons for her to express a further level of uncertainty, in the shape of either imprecision or higher order uncertainty . Bayesian epistemology provides the means to halt this escalator, by tying expressions of uncertainty to the propositions expressible in an agent’s language . But Bayesian epistemology comes in three main varieties. Strictly subjective Bayesianism and empirically-based subjective Bayesianism have difficulty in justifying the use of a forecaster’s probabilities for decision making . On the other hand, objective Bayesianism can justify the use of these probabilities, at least when the probabilities are consistent with the agent’s evidence . Hence objective Bayesianism offers the most promise overall for explaining how testimony of uncertainty can be useful for decision making. Interestingly, the objective Bayesian analysis provided in Sect. 5 can also be used to justify a version of the Principle of Reflection.

According to Russo and Williamson (Int Stud Philos Sci 21(2):157–170, 2007, Hist Philos Life Sci 33:389–396, 2011a, Philos Sci 1(1):47–69, 2011b ), in order to establish a causal claim of the form, ‘_C_ is a cause of _E_’, one typically needs evidence that there is an underlying mechanism between _C_ and _E_ as well as evidence that _C_ makes a difference to _E_. This thesis has been used to argue that hierarchies of evidence, as championed by evidence-based movements, tend to give primacy to evidence of difference making over evidence of mechanisms and are flawed because the two sorts of evidence are required and they should be treated on a par. An alternative approach gives primacy to evidence of mechanism over evidence of difference making. In this paper, we argue that this alternative approach is equally flawed, again because both sorts of evidence need to be treated on a par. As an illustration of this parity, we explain how scientists working in the ‘EnviroGenomarkers’ project constantly make use of the two evidential components in a dynamic and intertwined way. We argue that such an interplay is needed not only for causal assessment but also for policy purposes.

Evidence can be complex in various ways: e.g., it may exhibit structural complexity, containing information about causal, hierarchical or logical structure as well as empirical data, or it may exhibit combinatorial complexity, containing a complex combination of kinds of information. This paper examines evidential complexity from the point of view of Bayesian epistemology, asking: how should complex evidence impact on an agent’s degrees of belief? The paper presents a high-level overview of an objective Bayesian answer: it presents the objective Bayesian norms concerning the relation between evidence and degrees of belief, and goes on to show how evidence of causal, hierarchical and logical structure lead to natural constraints on degrees of belief. The objective Bayesian network formalism is presented, and it is shown how this formalism can be used to handle both kinds of evidential complexity—structural complexity and combinatorial complexity.

This paper presents the progicnet programme. It proposes a general framework for probabilistic logic that can guide inference based on both logical and probabilistic input. After an introduction to the framework as such, it is illustrated by means of a toy example from psychometrics. It is shown that the framework can accommodate a number of approaches to probabilistic reasoning: Bayesian statistical inference, evidential probability, probabilistic argumentation, and objective Bayesianism. The framework thus provides insight into the relations between these approaches, it illustrates how the results of different approaches can be combined, and it provides a basis for doing efficient inference in each of the approaches.

Mechanistic philosophy of science views a large part of scientific activity as engaged in modelling mechanisms. While science textbooks tend to offer qualitative models of mechanisms, there is increasing demand for models from which one can draw quantitative predictions and explanations. Casini et al. (Theoria 26(1):5–33, 2011) put forward the Recursive Bayesian Networks (RBN) formalism as well suited to this end. The RBN formalism is an extension of the standard Bayesian net formalism, an extension that allows for modelling the hierarchical nature of mechanisms. Like the standard Bayesian net formalism, it models causal relationships using directed acyclic graphs. Given this appeal to acyclicity, causal cycles pose a prima facie problem for the RBN approach. This paper argues that the problem is a significant one given the ubiquity of causal cycles in mechanisms, but that the problem can be solved by combining two sorts of solution strategy in a judicious way

Objective Bayesianism has been criticised for not allowing learning from experience: it is claimed that an agent must give degree of belief 12 to the next raven being black, however many other black ravens have been observed. I argue that this objection can be overcome by appealing to objective Bayesian nets, a formalism for representing objective Bayesian degrees of belief. Under this account, previous observations exert an inductive influence on the next observation. I show how this approach can be used to capture the Johnson–Carnap continuum of inductive methods, as well as the Nix–Paris continuum, and show how inductive influence can be measured. 1. Introduction2. The Problem3. Diagnosis4. Objective Bayesian Nets5. Resolution6. The Johnson–Carnap Continuum7. The Nix–Paris Continuum8. Linguistic Slack9. Frequencies and Degrees of Belief10. Conclusion.

Metabolism is a set of chemical reactions, used by living organisms to process chemical compounds in order to take energy and eliminate toxic compounds, for example. Its processes are referred as metabolic pathways. Understanding metabolism is imperative to biology, toxicology and medicine, but the number and complexity of metabolic pathways makes this a difficult task. In our paper, we investigate the use of causal Bayesian networks to model the pathways of yeast saccharomyces cerevisiae metabolism: such a network can be used to draw predictions about the levels of metabolites and enzymes in a particular specimen. We propose a two-stage methodology for causal networks, as follows. First construct a causal network from the network of metabolic pathways. The viability of this causal network depends on the validity of the causal Markov condition. If this condition fails, however, the principle of the common cause motivates the addition of a new causal arrow or a new `hidden' common cause to the network (stage 2 of the model formation process). Algorithms for adding arrows or hidden nodes have been developed separately in a number of papers, and in this paper we combine them, showing how the resulting procedure can be applied to the metabolic pathway problem. Our general approach was tested on neural cell morphology data and demonstrated noticeable improvements in both prediction and network accuracy.

this paper we argue that the formalism can also be applied to modelling the hierarchical structure of physical mechanisms. The resulting network contains quantitative information about probabilities, as well as qualitative information about mechanistic structure and causal relations. Since information about probabilities, mechanisms and causal relations are vital for prediction, explanation and control respectively, a recursive Bayesian net can be applied to all these tasks. We show how a Recursive Bayesian Net can be used to model mechanisms in cancer science. The highest level of the proposed model will contain variables at the clinical level, while a middle level will map the structure of the DNA damage response mechanism and the lowest level will contain information about gene expression.

This paper develops connections between objective Bayesian epistemology—which holds that the strengths of an agent’s beliefs should be representable by probabilities, should be calibrated with evidence of empirical probability, and should otherwise be equivocal—and probabilistic logic. After introducing objective Bayesian epistemology over propositional languages, the formalism is extended to handle predicate languages. A rather general probabilistic logic is formulated and then given a natural semantics in terms of objective Bayesian epistemology. The machinery of objective Bayesian nets and objective credal nets is introduced and this machinery is applied to provide a calculus for probabilistic logic that meshes with the objective Bayesian semantics.