Consistent Kernel Mean Estimation for Functions of Random Variables
This provides a theoretical foundation for probabilistic programming and kernel methods, but it is incremental as it extends existing consistency results to functions of random variables.
The paper tackles the problem of non-parametric estimation of functions of random variables using kernel mean embeddings, showing that consistent estimators for a random variable's mean embedding lead to consistent estimators for functions of that variable, with convergence rates provided for specific kernels and smooth functions.
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 Matérn 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.