Bayesian Inference in Cumulative Distribution Fields
This work addresses scalability issues in Bayesian inference for multivariate statistical models, particularly copulas and graphical models, which is incremental as it builds on existing methods to enhance computational efficiency.
The paper tackles the problem of scaling Bayesian inference for models parameterized by products of cumulative distribution functions (CDFs), such as copulas and sparse graphical models, by introducing simplified message-passing schemes; it demonstrates MCMC approaches that enable estimation and application in copula modeling, with a focus on improving scalability beyond dynamic programming limitations.
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.