## Multidimensional Conditional Probability Estimation Using the V-Matrix Method

**Dr. Rauf Izmailov**

*USA, Vencore Labs*

**Prof. Vladimir Vapnik**

*USA, Columbia University, Facebook*

We present direct setting and rigorous solution of one of the main prob- lems of statistical inference – estimation of conditional probability. We show that its rigorous solution requires solving multidimensional Fredholm integral equations of the first kind in the situation where not only the right- hand side of the equation is an approximation, but the operator in the equation is also defined approximately. Using the Stefanyuk-Vapnik the- ory for solving such ill-posed operator equations, a constructive method of conditional probability estimation is described. This method is based on a recently introduced concept called V-matrix, which captures geometric properties of the observation data that are ignored by classical statistical methods.

We also consider the application of the developed estimation method to the problem of combining monotonic pattern recognition classifiers which can be viewed as an additional mechanism of learning in “deep learning” methods.