Easy Conditioning far beyond Gaussian

Abstract

Multivariate Gaussian distributions enjoy Gaussian conditional distributions that makes conditioning easy: conditioning boils down to implementing analytical formulae for conditional means and covariances. For more general distributions, however, conditional distributions may not be available in analytical form and require demanding and approximate numerical approaches. Primarily motivated by probabilistic imputation problems, we review and discuss families of multivariate distributions that do enjoy analytical conditioning, also providing a few counter-examples. Proving that transdimensional stability under conditioning extends to mixtures and increasing transformations, we demonstrate that a broader class of multivariate distributions inherit easy conditioning properties. Building on this insight, we developed a generative method to estimate conditional distributions from data by first fitting a flexible joint distribution using copulas and then performing analytical conditioning in a latent space. In our applications, we specifically opt for Gaussian Mixture Copula Models (GMCM), comparing in turn various fitting strategies. Through simulations and real-world data experiments, we demonstrate that our method achieves competitive results with state-of-the-art models in conditional density estimation and data imputation, while offering several advantages due to its analytical form and generative nature. We also touch upon links to Gaussian process modelling and how stability by mixtures and transformations carries over towards easy conditioning of non-Gaussian processes.