Towards a Learning Theory of Cause-Effect Inference

We pose causal inference as the problem of learning to classify probability distributions. In particular, we assume access to a collection $\{(S_i,l_i)\}_{i=1}^n$, where each $S_i$ is a sample drawn from the probability distribution of $X_i \times Y_i$, and $l_i$ is a binary label indicating whether "$X_i \to Y_i$" or "$X_i \leftarrow Y_i$". Given these data, we build a causal inference rule in two steps. First, we featurize each $S_i$ using the kernel mean embedding associated with some characteristic kernel. Second, we train a binary classifier on such embeddings to distinguish between causal directions. We present generalization bounds showing the statistical consistency and learning rates of the proposed approach, and provide a simple implementation that achieves state-of-the-art cause-effect inference. Furthermore, we extend our ideas to infer causal relationships between more than two variables

Structural Agnostic Modeling: Adversarial Learning of Causal Graphs

A new causal discovery method, Structural Agnostic Modeling (SAM), is presented in this paper. Leveraging both conditional independencies and distributional asymmetries in the data, SAM aims at recovering full causal models from continuous observational data along a multivariate non-parametric setting. The approach is based on a game between $d$ players estimating each variable distribution conditionally to the others as a neural net, and an adversary aimed at discriminating the overall joint conditional distribution, and that of the original data. An original learning criterion combining distribution estimation, sparsity and acyclicity constraints is used to enforce the end-to-end optimization of the graph structure and parameters through stochastic gradient descent. Besides the theoretical analysis of the approach in the large sample limit, SAM is extensively experimentally validated on synthetic and real data

Non-linear Causal Inference using Gaussianity Measures

We provide theoretical and empirical evidence for a type of asymmetry between causes and effects that is present when these are related via linear models contaminated with additive non-Gaussian noise. Assuming that the causes and the effects have the same distribution, we show that the distribution of the residuals of a linear fit in the anti-causal direction is closer to a Gaussian than the distribution of the residuals in the causal direction. This Gaussianization effect is characterized by reduction of the magnitude of the high-order cumulants and by an increment of the differential entropy of the residuals. The problem of non-linear causal inference is addressed by performing an embedding in an expanded feature space, in which the relation between causes and effects can be assumed to be linear. The effectiveness of a method to discriminate between causes and effects based on this type of asymmetry is illustrated in a variety of experiments using different measures of Gaussianity. The proposed method is shown to be competitive with state-of-the-art techniques for causal inference.Comment: 35 pages, 9 figure