Hierarchical correlation reconstruction with missing data

Machine learning often needs to estimate density from a multidimensional data sample, where we would also like to model correlations between coordinates. Additionally, we often have missing data case: that data points have only partial information - can miss information about some coordinates. This paper adapts rapid parametric density estimation technique for this purpose: modelling density as a linear combination, for which $L^2$ optimization says that estimated coefficient for a given function is just average over the sample of this function. Hierarchical correlation reconstruction first models probability density for each separate coordinate using all its appearances in data sample, then adds corrections from independently modelled pairwise correlations using all samples having both coordinates, and so on independently adding correlations for growing numbers of variables using decreasing evidence in our data sample. A basic application of such modelled multidimensional density can be imputation of missing coordinates: by inserting known coordinates to the density, and taking expected values for the missing coordinates, and maybe also variance to estimate their uncertainty.