Jon Williamson, Bayesian Nets and Causality: Philosophical and Computational Foundations (Oxford: Oxford University, 2005).
Jon Williamson is currently located at the Department of Philosophy, Logic, and Scientific Method, London School of Economics, London. Within this volume, Williamson seeks to address the age-old question of how humanity should reason with causal relationships. From a philosophical perspective, this book explores the ontology and epistemology of probability and causality, whereas from a computational point of view, this book investigates the relationship between Bayesian nets and maximum entropy methods. Williamson’s positions argued within this book, including objective Bayesianism and epistemic causality, are part of a coherent scientific framework in which entities are neither physical, mind-independent features of the world, nor arbitrary and subjective entities varying according to the individual.
The aim of this book is to present coherent foundations to the recent work that hypothesizes that Bayesian nets provide a calculus for causal reasoning, and that one can learn actual causal relationships by the automated learning of Bayesian nets from observational data (i.e. inductively). The predictive features of Bayesian Systems’ products are based on a fundamental principal of logic known as Bayes’ theorem. This principle was discovered in 1761 by the Englishman Thomas Bayes, and brought into its modern form shortly thereafter by the great French mathematician Pierre Simon de Laplace. Properly understood, the theorem is the fundamental mathematical law governing the process of logical inference – determining what degree of confidence we may have in various possible conclusions, based on the body of evidence available.
Chapter one of this text gives a coherent and concise overview of the entirety of the book. Chapter two gives one a foundational understanding of probability and its interpretations. Chapter three introduces the reader to Bayesian nets, and thereafter gives an erudite summary of them as well. Chapter four relates the problems that are accompanied by the attempt to apply Bayesian nets to causal reasoning, which leads Williamson to offer new foundations for Bayesian nets based upon an objective Bayesian interpretation of probability, as presented in chapter five. Within the fifth chapter, Williamson argues cogently for a particular interpretation of probability, i.e., objective Bayesianism, which holds that probabilities are an agent’s rational degrees of belief and these degrees of belief are fixed as a function of the agent’s background knowledge. Williamson then offers a two-staged methodology for constructing Bayesian nets in chapter six.
Chapters seven through nine all discuss either the nature of causality, or investigate the possibility of discovering causal structures via Bayesian networks. In my humble opinion, chapter seven is worth the price of the book, for Williamson therein introduces the vast array of current causality models. He introduces three varieties of position upon causality; one can argue that a). the concept of causality is of heuristic value only, b). that it is a fundamental feature of the world, or c). that it is reducible to other concepts. Williamson notes that the latter concept is predominant in current philosophical literature, and that four approaches to it are recognized: 1). The mechanistic theory, 2). The probabilistic account, 3). The counterfactual account, and 4). The agent-oriented account. In an important note, Williamson’s position does not correspond or compare to any of the current positions that he outlines in chapter seven.
In chapter eight, Williamson presents proposals for discovering causal relationships, only to then develop an account of epistemic causality in chapter nine. Williamson asserts that his account of epistemic causality, wherein causal relations (though objective) are part of an agent’s epistemic state, comports well with the objective Bayesian interpretation of probability, and thus forms the basis of new approach to discovering causal relations using Bayesian nets. Thus, probability and causality are treated as mental notions. However, probability and causality are also objective, insomuch as different agents with the same background knowledge ought to adopt similar probabilistic and causal beliefs.
Chapters ten through twelve for a coherent unit in addressing various extensions and applications of the framework that Williamson develops in the first nine chapters of this text. Indeed, chapter ten examines the extension of Bayesian nets to the possibility of recursive causality, whereas chapter eleven explores how Bayesian nets can be employed in reasoning about logical relations. Within chapter eleven, moreover, attempts to show that objective Bayesianism can be used to provide practical semantics for probabilistic logic. Finally, chapter twelve discusses how his proposed Bayesian framework might accommodate itself to changes in the domain variables.
In sum, this book develops a systematic account of causal reasoning and shows how Bayesian nets can be coherently employed to automate the reasoning processes of an artificial agent. The resulting framework for causal reasoning involves new algorithms and new conceptual foundations. This book is intentioned for researchers and graduate students in computer science, mathematics and philosophy, and it provides a general introduction to the philosophical views related to Bayesian causality. As a result of recognizing its intended audience, I would caution the reader not to come to this text lightheartedly, for it necessitates much concentration and circumspect reading.
Bradford McCall
Regent University, Virginia Beach, VA.