Distinguishing between cause and effect via kernel-based complexity measures for conditional distributions

We propose a method to evaluate the complexity of probability measures from data that is based on a reproducing kernel Hilbert space seminorm of the logarithm of conditional probability densities. The motivation is to provide a tool for a causal inference method which assumes that conditional probabilities for effects given their causes are typically simpler and smoother than vice-versa. We present experiments with toy data where the quantitative results are consistent with our intuitive understanding of complexity and smoothness. Also in some examples with real-world data the probability measure corresponding to the true causal direction turned out to be less complex than those of the reversed order.