The presumably main goal of any empirical investigation is to reveal the association structure among the variables of concern and to interpret the associations in terms of causality. The latter, however, has been subject of numerous controversial debates, since there was and still is great doubt about the legitimacy of drawing causal conclusions from empirical results only.

The book by Judea Pearl provides an overdue comprehensive account of this problem embedded in a fully mathematical framework. The author is an expert in the field of Bayesian networks, artificial intelligence, and causal reasoning. With his new book, he demonstrates how and under which conditions causal inference from empirical data can be given a clear mathematical formulation and solution, yet emphasizing that substantive background information on the underlying mechanisms is a necessary ingredient. Besides presenting a unifying approach to the various facets of the concept of causality, one of the main merits of this book is the illustration by numerous fictitious and real-data examples coming from such different fields as epidemiology, sociology, and legal reasoning.

At first reading, however, the argumentation might not be easy to follow for readers being not familiar with the mathematical background. It is therefore recommended to start with the Epilogue which contains the material of a public lecture given by the author on "The Art and Science of Cause and Effect". This Epilogue gives an easy-to-read introduction to the historical and more
nonmathematical aspects of causation, but nevertheless already raises several of the main issues covered in detail by the main body of the book.

Preceding the Epilogue are ten chapters which address the problem of causal inference from different perspectives and relate it to concepts such as confounding, structural models, exogeneity, or instrumental variables. This is completed by an exhaustive bibliography. The mathematical background is developed throughout the book.

To understand Pearl's approach to causal modelling and analysis it is crucial to be familiar with some basic concepts of probability theory and in particular graphical models. These prerequisites are therefore briefly introduced in Chapter 1 with focus on conditional probability, graphical models, and Markov properties without dwelling on the technical details. They are supplemented by a first basic outline of the idea underlying causal models which is revised and deepened in the following chapters. In contrast to models describing pure statistical associations, causal models, as conceived by Pearl, aim at reflecting the behaviour of a system under interventions, i.e. subject to some external manipulation. The notion of intervention and the corresponding calculus is thus payed specific attention also in Chapter 3. As to the causal models, it becomes clear that the author favours the structural approach requiring quite extensive assumptions about the underlying data generating process. In fact, this is not of so much importance for the first part of the book, but gains weight in the later chapters.

Causal inference is challenged by the problem that typically only the non-interventional associations are observable except for the specific situation of randomized experiments. Different questions might then be of interest and are successively addressed in the subsequent chapters: (1) How and under which conditions can a causal model be deduced from raw data? (2) Given a causal model, how and under which conditions is the effect of an intervention identifyable? (3) Given a causal model and empirical evidence, how and under which conditions can we trace back the cause of a specific effect?

The first question is addressed in Chapter 2. Inferring causation from observational data has to be justified by several reasons and conditions also pervading the argumentation in the remaining chapters. One of the basic principles is that any observed association between variables *X* and *Y* has a causal explanation: Either *X* is the cause of *Y* or vice versa, or they have a common cause. In addition, it is argued that specific patterns of statistical associations only make sense when being interpreted in terms of causality. Finally, the search for causal patterns requires the restrictive assumption of no unobserved common causes. The application of some general model selection principles such as Occam's razor then yields general algorithms to uncover causal models consistent with the data.

While the above method of model search gets by without any structural assumptions w.r.t. the underlying data generating process, it becomes evident in Chapter 3 and the following that causality cannot be conjured up without substantive background information. Either, Chapter 3 and Chapter 4, are concerned with the identifyability of the effect of interventions based on a given causal graph but at different levels of complexity. To this end, the new calculus of intervention is formally introduced. The main idea is to extend the concept of conditional probabilities that an event A occurs given that we *see B* to the calculation of probabilities that an event *A* occurs given that we *do* *B*. Next, it is derived how to compute the effects of interventions by reducing it to observable quantities. For controlling a possible bias due to confounding, two well-known graphical criteria originally introduced by the author are presented: the back-door and front-door criterion. These can be applied to check if a set of variables is sufficient for identifying the postintervention distribution. This newly developped probablistic framework of preintervention and postintervention distributions, its implications, and possible extensions are further related to other approaches as e.g. Robins' *G*-estimation. The question of identifyability can be extended to conditional interventions and stochastic policies as shown in Chapter 4. One of the main results consists in providing an algorithm to find an admissible sequence of intervention variables such that the identifiability and existence of a closed form expression for the causal effect is ensured.

Given the semantics of causality as provided by causal graphs and the intervention calculus, Pearl next explores the relation to structural equation models (SEMs). In Chapter 5, he strongly advocates that the causal meaning originally attached to SEMs (and obliterated in the last decades) should be restored. This implies that the model specificatioil should not be read as assumptions about the association structure among the variables but instead as assumptions about the underlying mechanisms which are supposed to be invariant under intervention. The author shows that most of the preceding results and graphical criteria can be applied to SEMs yielding important insights into concepts such as total or direct effects, instrumental variables, and exogeneity. Moreover, the limitations of SEMs are emphasized, e.g. that they provide no method for testing causal assumptions. This chapter should be read by any researcher using SEMs as it raises several important points challenging their conventional application and interpretation.

Further basic concepts that have been discussed in connection with causality, such as Simpson's paradox and confounding, are revised in Chapter 6. Pearl demonstrates in a very illustrative manner that they are difficult to conceive only in terms of statistical association without causal interpretation. Here, again, the importance of graphical tools and the intervention calculus become apparent when (re)defining confounding and for mulating conditions for its testability. The reasoning in this chapter often refers to and discusses in detail approaches proposed by other authors whereby it tends to be hard to understand when not familiar with the literature.

While the first half of the book almost avoids the-topic of counterfactuals, Chapters 7 to 10 are heavily based on this controversial approach to causality. Clearly, most of the intuitive reasoning about causality is based on such statements, but the possibility of a precise mathematical formulation could be disputed. However, Pearl gives a very lucid introduction to counterfactuals, their mathematical representation and handling, and their empirical content. The clarity is mainly due to several simple examples and a distinct endeavor to relate the formalization to intuitive human reasoning. One of the motivations for the use of counterfactuals is that they provide a formalization of the claim that all relevant circumstances can be held constant as required e.g. in statistical prediction. The counterfactual approach is further contrasted with others such as the Neyman-Rubin framework and the probabilistic approach. More than in the previous chapters, this and the following are based on the assumption of a structural model underlying the data generating process.

The practical use of counterfactuals is made clear in the subsequent chapters. In some situations it might for instance not be possible to identify the causal effect of an inter vention even under restrictive assumptions. A typical example is given by the partial compliance situation in clinical studies discussed in detail in Chapter 8. Instead, Pearl proposes the derivation of bounds which is heavily based on counterfactual reasons, un fortunately without mentioning that other ways of derivation are possible, too (Dawid, 2000). Somewhat hidden in this chapter is also a brief discussion of causal inference from finite samples. While most of the results presented in this book discard sampling variation, Pearl proposes here to use the method of Gibbs sampling in order to compute the posterior distribution of the average causal effect. Unfortunately, this important topic is not paid enough attention although illustrated by some small real data examples.

The final two chapters, Chapter 9 and 10, apply the counterfactual approach to different facets of question (3) raised above. Again, it seems that this kind of problem can only be tackled by referring to counterfactuals. Its relevance for epidemiology or legal in quiries - providing the main examples in these two chapters - is obvious. The notions of sufficient, necessary, and actual causes are formally introduced and their limitations discussed. Despite some promising results, this last chapter leaves no doubt that the "quest for the actual cause" cannot be satisfactorily solved at the present stage and that more research on this topic is needed.

In conclusion, this book constitutes an important contribution to the ongoing debate about the possibility of causal inference. It demonstrates that empirical data can reveal much more about causal relations than admitted by "main stream" statistical analysis, albeit sometimes tied to restrictive assumptions. Even if some of these conditions are highly questionable and typically not testable, they are at least made explicit and their plausibility can thus be subject to a critical scrutiny. Many of the issues raised in this book are crucial for the thorough interpretation of empirical findings. It therefore deserves to be read by any empirical researcher.

*Vanessa Didelez, Department of Statistical Science, University College London, Gower Street, UK-London WC1E 6BT
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Iris Pigeot, University of Munich, Department of Statistics, Ludwigstr. 33, D-80539 MiInchen

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