Consistent Kernel Mean Estimation for Functions of Random Variables

By Carl-Johann Simon-Gabriel, Adam Scibior, Ilya O. Tolstikhin, Bernhard Schölkopf. 2016

15 citations recorded by the source corpus.

We provide a theoretical foundation for non-parametric estimation of functions of random variables using kernel mean embeddings. We show that for any continuous function f, consistent estimators of the mean embedding of a random variable X lead to consistent estimators of the mean embedding of f(X). For Matern kernels and sufficiently smooth functions we also provide rates of convergence. Our results extend to functions of multiple random variables. If the variables are dependent, we require an estimator of the mean embedding of their joint distribution as a starting point; if they are independent, it is sufficient to have separate estimators of the mean embeddings of their marginal distributions. In either case, our results cover both mean embeddings based on i.i.d. samples as well as "reduced set" expansions in terms of dependent expansion points. The latter serves as a justification for using such expansions to limit memory resources when applying the approach as a basis for probabilistic programming.

Subfields: Artificial Intelligence, Statistics and Probability, Statistical Learning Theory

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “Simon-Gabriel et al. (2016) provides consistency results and finite sample guarantees for the mean embedding of f(Z), where f is a continuous function and Z may consist of multiple random variables.”