Bayesian Inference in Cumulative Distribution Fields

TL;DR

This paper introduces Bayesian inference methods for copula models constructed via products of CDFs, leveraging message-passing techniques to improve scalability and estimation in multivariate analysis.

Contribution

It develops MCMC approaches for Bayesian estimation of product-based copula models and simplifies message-passing to enhance scalability.

Findings

Demonstrated Bayesian MCMC methods for copula models

Simplified message-passing improves scalability

Applied techniques to multivariate copula modeling

Abstract

One approach for constructing copula functions is by multiplication. Given that products of cumulative distribution functions (CDFs) are also CDFs, an adjustment to this multiplication will result in a copula model, as discussed by Liebscher (J Mult Analysis, 2008). Parameterizing models via products of CDFs has some advantages, both from the copula perspective (e.g., it is well-defined for any dimensionality) and from general multivariate analysis (e.g., it provides models where small dimensional marginal distributions can be easily read-off from the parameters). Independently, Huang and Frey (J Mach Learn Res, 2011) showed the connection between certain sparse graphical models and products of CDFs, as well as message-passing (dynamic programming) schemes for computing the likelihood function of such models. Such schemes allows models to be estimated with likelihood-based methods. We discuss and demonstrate MCMC approaches for estimating such models in a Bayesian context, their application in copula modeling, and how message-passing can be strongly simplified. Importantly, our view of message-passing opens up possibilities to scaling up such methods, given that even dynamic programming is not a scalable solution for calculating likelihood functions in many models.