  We consider the problem of inferring causal relationships between two or more
passively observed variables. While the problem of such causal discovery has
been extensively studied especially in the bivariate setting, the majority of
current methods assume a linear causal relationship, and the few methods which
consider non-linear dependencies usually make the assumption of additive noise.
Here, we propose a framework through which we can perform causal discovery in
the presence of general non-linear relationships. The proposed method is based
on recent progress in non-linear independent component analysis and exploits
the non-stationarity of observations in order to recover the underlying sources
or latent disturbances. We show rigorously that in the case of bivariate causal
discovery, such non-linear ICA can be used to infer the causal direction via a
series of independence tests. We further propose an alternative measure of
causal direction based on asymptotic approximations to the likelihood ratio, as
well as an extension to multivariate causal discovery. We demonstrate the
capabilities of the proposed method via a series of simulation studies and
conclude with an application to neuroimaging data.
