On Numerical Estimation of Joint Probability Distribution from Lebesgue Integral Quadratures

An important application of Lebesgue integral quadrature arXiv:1807.06007 is developed. Given two random processes, $f(x)$ and $g(x)$, two generalized eigenvalue problems can be formulated and solved. In addition to obtaining two Lebesgue quadratures (for $f$ and $g$) from two eigenproblems, the projections of $f$- and $g$- eigenvectors on each other allow to build a joint distribution estimator, the most general form of which is a density-matrix correlation. Examples of the density-matrix correlation can be a value-correlation $V_{f^{[i]};g^{[j]}}$, similar to a regular correlation concept, and a new one, a probability-correlation $P_{f^{[i]};g^{[j]}}$. If Christoffel function average is used instead of regular average the approach can be extended to an estimation of joint probability of three and more random processes. The theory is implemented numerically; the software is available under the GPLv3 license.