Easy Conditioning far beyond Gaussian

TL;DR

This paper explores classes of multivariate distributions that allow analytical conditioning beyond Gaussian, introduces a generative method using copulas for conditional density estimation, and demonstrates its effectiveness in data imputation tasks.

Contribution

It extends the concept of easy conditioning to broader distribution families via mixtures and transformations, and proposes a copula-based generative approach for conditional estimation.

Findings

Effective in conditional density estimation and data imputation

Broader class of distributions inherit easy conditioning properties

Demonstrated advantages over traditional methods

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 motivatedby probabilistic imputation problems, we review and discuss families of multivariate distributions that do enjoy analytical conditioning, also providing a few counter-examples. Proving that trans-dimensional stability under conditioning extends to mixtures and 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 showcase the efficacy of our method in tasks involving conditional density estimation and data imputation. We also touch upon links to Gaussian process modelling and how stability by mixtures and transformations and mixtures carries over towards easy conditioning of non-Gaussian processes.