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

Distinguishing cause from effect using observational data: methods and benchmarks

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: Additive Noise Methods (ANM) and Information Geometric Causal Inference (IGCI). We present the benchmark CauseEffectPairs that consists of data for 100 different cause-effect pairs selected from 37 datasets 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 additive-noise method originally 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.Comment: 101 pages, second revision submitted to Journal of Machine Learning Researc

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