We propose an independence criterion based on the eigen-spectrum of covariance operators in reproducing kernel Hilbert spaces (RKHSs), consisting of an empirical estimate of the Hilbert-Schmidt norm of the cross-covariance operator (we term this a Hilbert-Schmidt Independence Criterion, or HSIC). This approach has several advantages, compared with previous kernel-based independence criteria. First, the empirical estimate is simpler than any other kernel dependence test, and requires no user-defined regularisation. Second, there is a clearly defined population quantity which the empirical estimate approaches in the large sample limit, with exponential convergence guaranteed between the two: this ensures that independence tests based on HSIC do not suffer from slow learning rates. Finally, we show in the context of independent component analysis (ICA) that the performance of HSIC is competitive with that of previously published kernel-based criteria, and of other recently published ICA methods.

Measuring statistical dependence with Hilbert-Schmidt norms

What is the difference of a prediction that is made with a causal model and a non-causal model? Suppose we intervene on the predictor variables or change the whole environment. The predictions from a causal model will in general work as well under interventions as for observational data. In contrast, predictions from a non-causal model can potentially be very wrong if we actively intervene on variables. Here, we propose to exploit this invariance of a prediction under a causal model for causal inference: given different experimental settings (for example various interventions) we collect all models that do show invariance in their predictive accuracy across settings and interventions. The causal model will be a member of this set of models with high probability. This approach yields valid confidence intervals for the causal relationships in quite general scenarios. We examine the example of structural equation models in more detail and provide sufficient assumptions under which the set of causal predictors becomes identifiable. We further investigate robustness properties of our approach under model misspecification and discuss possible extensions. The empirical properties are studied for various data sets, including large-scale gene perturbation experiments.

Causal inference using invariant prediction: identification and confidence intervals

A fundamental task in various disciplines of science, including biology, is to find underlying causal relations and make use of them. Causal relations can be seen if interventions are properly applied; however, in many cases they are difficult or even impossible to conduct. It is then necessary to discover causal relations by analyzing statistical properties of purely observational data, which is known as causal discovery or causal structure search. This paper aims to give a introduction to and a brief review of the computational methods for causal discovery that were developed in the past three decades, including constraint-based and score-based methods and those based on functional causal models, supplemented by some illustrations and applications.

/pdf/review-of-causal-discovery-methods-based-on-graphical-models-3doov8liuv.pdf

Review of Causal Discovery Methods Based on Graphical Models.

The discovery of causal relationships from purely observational data is a fundamental problem in science. The most elementary form of such a causal discovery problem is to decide whether X causes Y or, alternatively, Y causes X, given joint observations of two variables X,Y. An example is to decide whether altitude causes temperature, or vice versa, given only joint measurements of both variables. Even under the simplifying assumptions of no confounding, no feedback loops, and no selection bias, such bivariate causal discovery problems are challenging. Nevertheless, several approaches for addressing those problems have been proposed in recent years. We review two families of such methods: methods based on Additive Noise Models (ANMs) and Information Geometric Causal Inference (IGCI). We present the benchmark CAUSEEFFECTPAIRS that consists of data for 100 different causee ffect pairs selected from 37 data sets from various domains (e.g., meteorology, biology, medicine, engineering, economy, etc.) and motivate our decisions regarding the "ground truth" causal directions of all pairs. We evaluate the performance of several bivariate causal discovery methods on these real-world benchmark data and in addition on artificially simulated data. Our empirical results on real-world data indicate that certain methods are indeed able to distinguish cause from effect using only purely observational data, although more benchmark data would be needed to obtain statistically significant conclusions. One of the best performing methods overall is the method based on Additive Noise Models that has originally been proposed by Hoyer et al. (2009), which obtains an accuracy of 63 ± 10 % and an AUC of 0.74 ± 0.05 on the real-world benchmark. As the main theoretical contribution of this work we prove the consistency of that method.

/pdf/distinguishing-cause-from-effect-using-observational-data-39ffkai7u5.pdf

Distinguishing cause from effect using observational data: methods and benchmarks

We consider the problem of learning causal directed acyclic graphs from an observational joint distribution. One can use these graphs to predict the outcome of interventional experiments, from which data are often not available. We show that if the observational distribution follows a structural equation model with an additive noise structure, the directed acyclic graph becomes identifiable from the distribution under mild conditions. This constitutes an interesting alternative to traditional methods that assume faithfulness and identify only the Markov equivalence class of the graph, thus leaving some edges undirected. We provide practical algorithms for finitely many samples, RESIT (regression with subsequent independence test) and two methods based on an independence score. We prove that RESIT is correct in the population setting and provide an empirical evaluation.

/pdf/causal-discovery-with-continuous-additive-noise-models-46arfkekjw.pdf

Causal discovery with continuous additive noise models

Structural equation models and Bayesian networks have been widely used to analyze causal relations between continuous variables. In such frameworks, linear acyclic models are typically used to model the data-generating process of variables. Recently, it was shown that use of non-Gaussianity identifies the full structure of a linear acyclic model, i.e., a causal ordering of variables and their connection strengths, without using any prior knowledge on the network structure, which is not the case with conventional methods. However, existing estimation methods are based on iterative search algorithms and may not converge to a correct solution in a finite number of steps. In this paper, we propose a new direct method to estimate a causal ordering and connection strengths based on non-Gaussianity. 
In contrast to the previous methods, our algorithm requires no algorithmic parameters and is guaranteed to converge to the right solution within a small fixed number of steps if the data strictly follows the model.

DirectLiNGAM: A direct method for learning a linear non-Gaussian structural equation model

Structural equation models and Bayesian networks have been widely used to analyze causal relations between continuous variables. In such frameworks, linear acyclic models are typically used to model the data-generating process of variables. Recently, it was shown that use of non-Gaussianity identifies the full structure of a linear acyclic model, that is, a causal ordering of variables and their connection strengths, without using any prior knowledge on the network structure, which is not the case with conventional methods. However, existing estimation methods are based on iterative search algorithms and may not converge to a correct solution in a finite number of steps. In this paper, we propose a new direct method to estimate a causal ordering and connection strengths based on non-Gaussianity. In contrast to the previous methods, our algorithm requires no algorithmic parameters and is guaranteed to converge to the right solution within a small fixed number of steps if the data strictly follows the model, that is, if all the model assumptions are met and the sample size is infinite.

/pdf/directlingam-a-direct-method-for-learning-a-linear-non-3i31u3ry6s.pdf

DirectLiNGAM: A Direct Method for Learning a Linear Non-Gaussian Structural Equation Model

Discovering causal relations among observed variables in a given data set is a main topic in studies of statistics and artificial intelligence. Recently, some techniques to discover an identifiable causal structure have been explored based on non-Gaussianity of the observed data distribution. However, most of these are limited to continuous data. In this paper, we present a novel causal model for binary data and propose a new approach to derive an identifiable causal structure governing the data based on skew Bernoulli distributions of external noise. Experimental evaluation shows excellent performance for both artificial and real world data sets.

/pdf/discovering-causal-structures-in-binary-exclusive-or-skew-1pi97b2ouc.pdf

Discovering causal structures in binary exclusive-or skew acyclic models

We discuss causal structure learning based on linear structural equation models. Conventional learning methods most often assume Gaussianity and create many indistinguishable models. Therefore, in many cases it is difficult to obtain much information on the structure. Recently, a non-Gaussian learning method called LiNGAM has been proposed to identify the model structure without using prior knowledge on the structure. However, more efficient learning can be achieved if some prior knowledge on a part of the structure is available. In this paper, we propose to use prior knowledge to improve the performance of a state-of-art non-Gaussian method. Experiments on artificial data show that the accuracy and computational time are significantly improved even if the amount of prior knowledge is not so large.

http://www.ar.sanken.osaka-u.ac.jp/~sshimizu/papers/Inazumi10LVA.pdf

Use of prior knowledge in a non-Gaussian method for learning linear structural equation models

Discovering causal relations among observed variables in a given data set is a major objective in studies of statistics and artificial intelligence. Recently, some techniques to discover a unique causal model have been explored based on non-Gaussianity of the observed data distribution. However, most of these are limited to continuous data. In this paper, we present a novel causal model for binary data and propose an efficient new approach to deriving the unique causal model governing a given binary data set under skew distributions of external binary noises. Experimental evaluation shows excellent performance for both artificial and real world data sets.

Takanori Inazumi

Papers

DirectLiNGAM: A direct method for learning a linear non-Gaussian structural equation model

DirectLiNGAM: A Direct Method for Learning a Linear Non-Gaussian Structural Equation Model

Discovering causal structures in binary exclusive-or skew acyclic models

Use of prior knowledge in a non-Gaussian method for learning linear structural equation models

Causal Discovery in a Binary Exclusive-or Skew Acyclic Model: BExSAM