A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings
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We present a new operator-free, measure-theoretic definition of the conditional mean embedding as a random variable taking values in a reproducing kernel Hilbert space. While the kernel mean embedding of marginal distributions has been defined rigorously, the existing operator-based approach of the conditional version lacks a rigorous definition, and depends on strong assumptions that hinder its analysis. Our definition does not impose any of the assumptions that the operator-based counterpart requires. We derive a natural regression interpretation to obtain empirical estimates, and provide a thorough analysis of its properties, including universal consistency. As natural by-products, we obtain the conditional analogues of the Maximum Mean Discrepancy and Hilbert-Schmidt Independence Criterion, and demonstrate their behaviour via simulations.
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