A main message from the causal modelling literature in the last several decades is that under some plausible assumptions, there can be statistically consistent procedures for inferring (features of) the causal structure of a set of random variables from observational data. But whether we can control the error probabilities with a finite sample size depends on the kind of consistency the procedures can achieve. It has been shown that in general, under the standard causal Markov and Faithfulness assumptions, the procedures (...) can only be pointwise but not uniformly consistent without substantial background knowledge. This implies the impossibility of choosing a finite sample size to control the worst case error probabilities. In this paper, I consider the simpler task of inferring causal directions when the skeleton of the causal structure is known, and establish a similarly negative result concerning the possibility of controlling error probabilities. Although the result is negative in form, it has an interesting positive implication for causal discovery methods. (shrink)

The framework of causal Bayes nets, currently influential in several scientific disciplines, provides a rich formalism to study the connection between causality and probability from an epistemological perspective. This article compares three assumptions in the literature that seem to constrain the connection between causality and probability in the style of Occam's razor. The trio includes two minimality assumptions—one formulated by Spirtes, Glymour, and Scheines (SGS) and the other due to Pearl—and the more well-known faithfulness or stability assumption. In terms of (...) logical strength, it is fairly obvious that the three form a sequence of increasingly stronger assumptions. The focus of this article, however, is to investigate the nature of their relative strength. The comparative analysis reveals an important sense in which Pearl's minimality assumption is as strong as the faithfulness assumption and identifies a useful condition under which it is as safe as SGS's relatively secure minimality assumption. Both findings have notable implications for the theory and practice of causal inference. 1 Introduction2 Background: Inference of Causal Structure in Markovian Causal Models3 Three Assumptions of Simplicity4 A Comparison of P-minimality and Faithfulness5 A Comparison of P-minimality and SGS-minimality6 Methodological Formulations and Prior Knowledge of Causal Order7 Conclusion. (shrink)

Much of the recent work on the epistemology of causation has centered on two assumptions, known as the Causal Markov Condition and the Causal Faithfulness Condition. Philosophical discussions of the latter condition have exhibited situations in which it is likely to fail. This paper studies the Causal Faithfulness Condition as a conjunction of weaker conditions. We show that some of the weaker conjuncts can be empirically tested, and hence do not have to be assumed a priori. Our results lead to (...) two methodologically significant observations: (1) some common types of counterexamples to the Faithfulness condition constitute objections only to the empirically testable part of the condition; and (2) some common defenses of the Faithfulness condition do not provide justification or evidence for the testable parts of the condition. It is thus worthwhile to study the possibility of reliable causal inference under weaker Faithfulness conditions. As it turns out, the modification needed to make standard procedures work under a weaker version of the Faithfulness condition also has the practical effect of making them more robust when the standard Faithfulness condition actually holds. This, we argue, is related to the possibility of controlling error probabilities with finite sample size (“uniform consistency”) in causal inference. (shrink)

The first part of this paper reveals a conflict between the core principles of deterministic causation and the standard method of difference, which is widely seen as a correct method of causally analyzing deterministic structures. We show that applying the method of difference to deterministic structures can give rise to causal inferences that contradict the principles of deterministic causation. The second part then locates the source of this conflict in an inference rule implemented in the method of difference according to (...) which factors that can make a difference to investigated effects relative to one particular test setup are to be identified as causes, provided the causal background of the corresponding setup is homogeneous. The paper ends by modifying the method of difference in a way that renders it compatible with the principles of deterministic causation. (shrink)

Spirtes, Glymour and Scheines [Causation, Prediction, and Search Springer] described a pointwise consistent estimator of the Markov equivalence class of any causal structure that can be represented by a directed acyclic graph for any parametric family with a uniformly consistent test of conditional independence, under the Causal Markov and Causal Faithfulness assumptions. Robins et al. [Biometrika 90 491–515], however, proved that there are no uniformly consistent estimators of Markov equivalence classes of causal structures under those assumptions. Subsequently, Kalisch and B¨uhlmann (...) [J. Mach. Learn. Res. 8 613–636] described a uniformly consistent estimator of the Markov equivalence class of a linear Gaussian causal structure under the Causal Markov and Strong Causal Faithfulness assumptions. However, the Strong Faithfulness assumption may be false with high probability in many domains. We describe a uniformly consistent estimator of both the Markov equivalence class of a linear Gaussian causal structure and the identifiable structural coefficients in the Markov equivalence class under the Causal Markov assumption and the considerably weaker k-Triangle-Faithfulness assumption. (shrink)

The pivotal problem of comorbidity research lies in the psychometric foundation it rests on, that is, latent variable theory, in which a mental disorder is viewed as a latent variable that causes a constellation of symptoms. From this perspective, comorbidity is a (bi)directional relationship between multiple latent variables. We argue that such a latent variable perspective encounters serious problems in the study of comorbidity, and offer a radically different conceptualization in terms of a network approach, where comorbidity is hypothesized to (...) arise from direct relations between symptoms of multiple disorders. We propose a method to visualize comorbidity networks and, based on an empirical network for major depression and generalized anxiety, we argue that this approach generates realistic hypotheses about pathways to comorbidity, overlapping symptoms, and diagnostic boundaries, that are not naturally accommodated by latent variable models: Some pathways to comorbidity through the symptom space are more likely than others; those pathways generally have the same direction (i.e., from symptoms of one disorder to symptoms of the other); overlapping symptoms play an important role in comorbidity; and boundaries between diagnostic categories are necessarily fuzzy. (shrink)

Using the flexibility of recently developed methods for causal discovery based on Boolean satisfiability solvers, we encode a variety of assumptions that weaken the Faithfulness assumption. The encoding results in a number of SAT-based algorithms whose asymptotic correctness relies on weaker conditions than are standardly assumed. This implementation of a whole set of assumptions in the same platform enables us to systematically explore the effect of weakening the Faithfulness assumption on causal discovery. An important effect, suggested by simulation results, is (...) that adopting weaker assumptions greatly alleviates the problem of conflicting constraints and substantially shortens solving time. As a result, SAT-based causal discovery is potentially more scalable under weaker assumptions. (shrink)

Recent approaches to causal modelling rely upon the causal Markov condition, which specifies which probability distributions are compatible with a directed acyclic graph. Further principles are required in order to choose among the large number of DAGs compatible with a given probability distribution. Here we present a principle that we call frugality. This principle tells one to choose the DAG with the fewest causal arrows. We argue that frugality has several desirable properties compared to the other principles that have been (...) suggested, including the well-known causal faithfulness condition. _1_ Introduction _2_ The Causal Markov Condition _3_ Faithfulness _4_ Frugality _4.1_ Basic independences and frugality _4.2_ General properties of directed acyclic graphs satisfying frugality _4.3_ Connection to minimality assumptions _5_ Frugality as a Parsimony Principle _6_ Conclusion Appendix. (shrink)

The essential precondition of implementing interventionist techniques of causal reasoning is that particular variables are identified as so-called intervention variables. While the pertinent literature standardly brackets the question how this can be accomplished in concrete contexts of causal discovery, the first part of this paper shows that the interventionist nature of variables cannot, in principle, be established based only on an interventionist notion of causation. The second part then demonstrates that standard observational methods that draw on Bayesian networks identify intervention (...) variables only if they also answer the questions that can be answered by interventionist techniques—which are thus rendered dispensable. The paper concludes by suggesting a way of identifying intervention variables that allows for exploiting the whole inferential potential of interventionist techniques. (shrink)

Over the past two decades, several consistent procedures have been designed to infer causal conclusions from observational data. We prove that if the true causal network might be an arbitrary, linear Gaussian network or a discrete Bayes network, then every unambiguous causal conclusion produced by a consistent method from non-experimental data is subject to reversal as the sample size increases any finite number of times. That result, called the causal flipping theorem, extends prior results to the effect that causal discovery (...) cannot be reliable on a given sample size. We argue that since repeated flipping of causal conclusions is unavoidable in principle for consistent methods, the best possible discovery methods are consistent methods that retract their earlier conclusions no more than necessary. A series of simulations of various methods across a wide range of sample sizes illustrates concretely both the theorem and the principle of comparing methods in terms of retractions. (shrink)