The discovery of causal relationships between a set of observed variables is a fundamental problem in science. For continuous-valued data linear acyclic causal models with additive noise are often used because these models are well understood and there are well-known methods to fit them to data. In reality, of course, many causal relationships are more or less nonlinear, raising some doubts as to the applicability and usefulness of purely linear methods. In this contribution we show that in fact the basic linear framework can be generalized to nonlinear models. In this extended framework, nonlinearities in the data-generating process are in fact a blessing rather than a curse, as they typically provide information on the underlying causal system and allow more aspects of the true data-generating mechanisms to be identified. In addition to theoretical results we show simulations and some simple real data experiments illustrating the identification power provided by nonlinearities. 1

Causal reasoning is primarily concerned with what would happen to a system under external interventions. In particular, we are often interested in predicting the probability distribution of some random variables that would result if some other variables were forced to take certain values. One prominent approach to tackling this problem is based on causal Bayesian networks, using directed acyclic graphs as causal diagrams to relate post-intervention probabilities to pre-intervention probabilities that are estimable from observational data. However, such causal diagrams are seldom fully testable given observational data. In consequence, many causal discovery algorithms based on data-mining can only output an equivalence class of causal diagrams (rather than a single one). This paper is concerned with causal reasoning given an equivalence class of causal diagrams, represented by a (partial) ancestral graph. We present two main results. The first result extends Pearl (1995)’s celebrated do-calculus to the context of ancestral graphs. In the second result, we focus on a key component of Pearl’s calculus—the property of invariance under interventions, and give stronger graphical conditions for this property than those implied by the first result. The second result also improves the earlier, similar results due to Spirtes et al. (1993).

Causal analysis of continuous-valued variables typically uses either autoregressive models or linear Gaussian Bayesian networks with instantaneous effects. Estimation of Gaussian Bayesian networks poses serious identifiability problems, which is why it was recently proposed to use non-Gaussian models. Here, we show how to combine the non-Gaussian instantaneous model with autoregressive models. We show that such a non-Gaussian model is identifiable without prior knowledge of network structure, and we propose an estimation method shown to be consistent. This approach also points out how neglecting instantaneous effects can lead to completely wrong estimates of the autoregressive coefficients. 1.

We propose a method that detects the true direction of time series, by fitting an autoregressive moving average model to the data. Whenever the noise is independent of the previous samples for one ordering of the observations, but dependent for the opposite ordering, we infer the former direction to be the true one. We prove that our method works in the population case as long as the noise of the process is not normally distributed (for the latter case, the direction is not identifiable). A new and important implication of our result is that it confirms a fundamental conjecture in causal reasoning — if after regression the noise is independent of signal for one direction and dependent for the other, then the former represents the true causal direction — in the case of time series. We test our approach on two types of data: simulated data sets conforming to our modeling assumptions, and real world EEG time series. Our method makes a decision for a significant fraction of both data sets, and these decisions are mostly correct. For real world data, our approach outperforms alternative solutions to the problem of time direction recovery. 1.

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Summary. We propose two kernel based methods for detecting the time direction in empirical time series. First we apply a Support Vector Machine on the finitedimensional distributions of the time series (classification method) by embedding these distributions into a Reproducing Kernel Hilbert Space. For the ARMA method we fit the observed data with an autoregressive moving average process and test whether the regression residuals are statistically independent of the past values. Whenever the dependence in one direction is significantly weaker than in the other we infer the former to be the true one. Both approaches were able to detect the direction of the true generating model for simulated data sets. We also applied our tests to a large number of real world time series. The ARMA method made a decision for a significant fraction of them, in which it was mostly correct, while the classification method did not perform as well, but still exceeded chance level.

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Analysis of causal effects between continuous-valued variables typically uses either autoregressive models or structural equation models with instantaneous effects. Estimation of Gaussian, linear structural equation models poses serious identifiability problems, which is why it was recently proposed to use non-Gaussian models. Here, we show how to combine the non-Gaussian instantaneous model with autoregressive models. This is effectively what is called a structural vector autoregression (SVAR) model, and thus our work contributes to the long-standing problem of how to estimate SVAR’s. We show that such a non-Gaussian model is identifiable without prior knowledge of network structure. We propose computationally efficient methods for estimating the model, as well as methods to assess the significance of the causal influences. The model is successfully applied on financial and brain imaging data.

observations

In science, one faces the problem of selecting the true theory from a range of alternative theories. The typical response is to select the simplest theory compatible with available evidence, on the authority of “Ockham’s Razor”. But how can a fixed bias toward simplicity help one find possibly complex truths? A short survey of standard answers to this question reveals them to be either wishful, circular, or irrelevant. A new explanation is presented, based on minimizing the reversals of opinion prior to convergence to the truth. According to this alternative approach, Ockham’s razor does not inform one which theory is true but is, nonetheless, the uniquely most efficient strategy for arriving at the true theory, where efficiency is a matter of minimizing reversals of opinion prior to finding the true theory. 1