Statistical Theory and Algorithms for Machine Learning of Conditional Distribution Functions

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We propose an algorithm for machine learning of conditional distribution functions for a dependent variable (Y ) with continuous support. The algorithm produces a complete description of the conditional distribution function at all observed points in the covariate (X) space, and provides a similar estimate for other possible covariate values. The descriptions it provides are quite general and are globally valid conditional densities. The algorithm is multi-layered and feed-forward. Each layer has the same statistical interpretation: Layer k takes a vector e(k-1)that is nearly perfectly marginally Gaussian and makes it more marginally Gaussian and more independent of X. It does this by applying a continuous monotonic transformation that varies depending on an observation’s X value. Each layer is estimated by an elastic net regularization of maximum likelihood. We demonstrate Wilks’ phenomenon for the composite algorithm and show how to calculate the algorithm’s effective dimension.

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