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How is the kernel embedding of distributions used?
In machine learning, the kernel embedding of distributions (also called the kernel mean or mean map) comprises a class of nonparametric methods in which a probability distribution is represented as an element of a reproducing kernel Hilbert space (RKHS).
How to measure the statistical ” distance ” between two?
Smirnov-Kolmogorov test: a test to determine whether two cumulative distribution functions for continuous random variables come from the same sample. Chi-squared test: a goodness-of-fit test to decide how well a frequency distribution differs from an expected frequency distribution.
How is the MMD used to estimate distance between distributions?
While most distance-measures between distributions such as the widely used Kullback–Leibler divergence either require density estimation (either parametrically or nonparametrically) or space partitioning/bias correction strategies, the MMD is easily estimated as an empirical mean which is concentrated around the true value of the MMD.
Are there dimensionality-independent rates of convergence for the empirical kernel mean?
Dimensionality-independent rates of convergence for the empirical kernel mean (estimated using samples from the distribution) to the kernel embedding of the true underlying distribution can be proven.
How is the embedding of a conditional distribution expressed?
The analog of this rule in the kernel embedding framework expresses the kernel embedding of the conditional distribution in terms of conditional embedding operators which are modified by the prior distribution
How is the maximum mean discrepancy ( MMD ) defined?
The maximum mean discrepancy (MMD) is a distance-measure between distributions () and () which is defined as the squared distance between their embeddings in the RKHS (,) = ‖ − ‖
Which is a generalization of the kernel Hilbert space?
The basic idea behind this framework is to map distributions into a reproducing kernel Hilbert space (RKHS) in which the whole arsenal of kernel methods can be extended to probability measures. It can be viewed as a generalization of the original “feature map” common to support vector machines (SVMs) and other kernel methods.
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