Computing functions of random variables via reproducing kernel Hilbert space representations

By Bernhard Schölkopf, Krikamol Muandet, Kenji Fukumizu, Stefan Harmeling, Jan Peters. 2015

39 citations recorded by the source corpus.

We describe a method to perform functional operations on probability distributions of random variables. The method uses reproducing kernel Hilbert space representations of probability distributions, and it is applicable to all operations which can be applied to points drawn from the respective distributions. We refer to our approach as kernel probabilistic programming. We illustrate it on synthetic data and show how it can be used for nonparametric structural equation models, with an application to causal inference.

Subfields: Artificial Intelligence, Probabilistic & Bayesian Methods, Statistical Learning Theory

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “Schölkopf et al. (2015) proposed to estimate µ[f(X,Y )] by the following estimator µ̂[f(X,Y )] := 1∑m i=1 αi ∑n j=1 βj m∑ i=1 n∑ j=1 αiβjφz(f(xi,yj)).”
Consistent Kernel Mean Estimation for Functions of Random Variables (2016, methodology): “For this reason Schölkopf et al. (2015) propose to use a reduced set method both on X̂ ′ and Ŷ ′ to get new samples X̂ = {xi, wi}ni=1 and Ŷ = {yj , uj}nj=1 of size n N , and then estimate µf(X,Y ) using µ̂3 := ∑n i,j=1 wiujkx(f(xi, yj), .).”