# Detecting confounding in multivariate linear models via spectral analysis

**Dominik Janzing**(Max Planck Institute for Intelligent Systems)

January 31, 2017 — 10:30 — "Salle du conseil du L2S"

###### Abstract

We study a model where one target variable Y is correlated with a vector X:=(X_1,…,X_d) of predictor variables being potential causes of Y. We describe a method that infers to what extent the statistical dependences between X and Y are due to the influence of X on Y and to what extent due to a hidden common cause (confounder) of X and Y. The method is based on an independence assumption stating that, in the absence of confounding, the vector of regression coefficients describing the influence of each X on Y has ‘generic orientation’ relative to the eigenspaces of the covariance matrix of X. For the special case of a scalar confounder we show that confounding typically spoils this generic orientation in a characteristic way that can be used to quantitatively estimate the amount of confounding. I also show some encouraging experiments with real data, but the method is work in progress and critical comments are highly appreciated. Postulating ‘generic orientation’ is inspired by a more general postulate stating that P(cause) and P(effect|cause) are independent objects of Nature and therefore don’t contain information about each other [1,2,3], an idea that inspired several causal inference methods already, e.g. [4,5]. [1] Janzing, Schoelkopf: Causal inference using the algorithmic Markov condition, IEEE TIT 2010. [2] Lemeire, Janzing: Replacing causal faithfulness with the algorithmic independence of conditionals, Minds and Machines, 2012. [3] Schoelkopf et al: On causal and anticausal learning, ICML 2012. [4] Janzing et al: Telling cause frome effect based on high-dimensional observations, ICML 2010. [5] Shajarisales et al: Telling cause from effect in deterministic linear dynamical systems, ICML 2015.