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

We propose a method to evaluate the complexity of probabil- ity measures from data that is based on a reproducing kernel Hilbert space seminorm of the logarithm of conditional probability densities. The mo- tivation 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 un- derstanding 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.