Testing whether linear equations are causal: A free probability theory approach

By Jakob Zscheischler, Dominik Janzing, Kun Zhang. 2011

41 citations recorded by the source corpus.

We propose a method that infers whether linear relations between two high-dimensional variables X and Y are due to a causal influence from X to Y or from Y to X. The earlier proposed so-called Trace Method is extended to the regime where the dimension of the observed variables exceeds the sample size. Based on previous work, we postulate conditions that characterize a causal relation between X and Y. Moreover, we describe a statistical test and argue that both causal directions are typically rejected if there is a common cause. A full theoretical analysis is presented for the deterministic case but our approach seems to be valid for the noisy case, too, for which we additionally present an approach based on a sparsity constraint. The discussed method yields promising results for both simulated and real world data.

Subfields: Artificial Intelligence, Statistics and Probability, Causal Inference & Discovery

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Citations in the corpus

A theory of independent mechanisms for extrapolation in generative models (2020, methodology): “TR can also be interpreted using as measuring freeness in free probability theory (Zscheischler et al., 2011).”
Distinguishing Cause from Effect Using Observational Data: Methods and Benchmarks (2014, background): “Without having access to the interventional distributions, this symmetry prevents us from drawing any conclusions regarding the direction of the causal relationship between X and Y if we only have access to the observational distribution PX,Y .”
Origo: causal inference by compression (2016, methodology): “The linear trace method [21], [22] infers linear causal relations of the form Y = AX , where A is the structure matrix that maps the cause to the effect, using the linear trace condition which operates on A, and the covariance matrix of X , ΣX .”
Causal Inference by Stochastic Complexity (2017, background): “In recent years, several methods have been proposed that exploit the sophisticated properties of the joint distribution. linear trace method [6, 32] infers linear causal relations of the form Y = AX , where A is the structure matrix that maps the cause to the effect, using the linear trace…”
Causal Discovery via Reproducing Kernel Hilbert Space Embeddings (2014, methodology): “To make the causal direction identifiable, it is assumed that A and x are generated independently and thus fulfill the trace condition (Janzing et al., 2010; Zscheischler et al., 2011), where x is the covariance matrix of x.”
Distinguishing Cause from Effect Using Observational Data: Methods and Benchmarks (2014, background): “= Y −(f(X)+ν) = U−ν is independent of X. Conversely, if EY is independent of X, p(x, y) is induced by the bivariate Additive Noise Model (pX , pEY , x 7→ E(Y”
Estimating a Causal Order among Groups of Variables in Linear Models (2012, methodology): “Many techniques have recently been developed for inferring causal relationships from data over a set of random variables [1,2,3,4,5,6,7,8,9].”
Detecting Confounding in Multivariate Linear Models via Spectral Analysis (2017, methodology): “(16) almost coincide, while our method also accounts for higher order moments which the Trace Condition ignores.”