Maximum-likelihood learning of cumulative distribution functions on graphs
Jim Huang, Nebojša Jojić · 2010
For many applications, a probability model can be more easily expressed as a cumulative distribution functions (CDF) as compared to the use of probability density or mass functions (PDF/PMFs). One advantage of CDF models is the simplicity of representing multivariate heavy-tailed distributions. Examples of fields that can benefit from the use of graphical models for CDFs include climatology and epidemiology, where datafollowheavy-taileddistributions and exhibit spatial correlations so that dependencies between model variables must be accounted for. However, in most cases the problem of learning from data consists of optimizing the log-likelihood function with respect to model parameters where we are required to optimize a log-PDF/PMF and not a log-CDF. Given a CDF defined on a graph, we present a message-passing algorithm called the gradient-derivative-product (GDP) algorithm that allows us to learn the model in terms of the log-likelihood function whereby messages correspond to local gradients of the likelihood with respect to model parameters. We demonstrate the GDP algorithm on real-world rainfall and H1N1 mortality data and weshow that the heavy-tailed multivariate distributions that arise in these problems can both be naturally parameterized and tractably estimated from data using our algorithm. 1