# Maximum-likelihood learning of cumulative distribution functions on graphs

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Jim Huang,
Nebojsa Jojic
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Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, PMLR 9:342-349, 2010.

#### Abstract

For many applications, a probability model can be easily expressed as a cumulative distribution functions (CDF) as compared to the use of probability density or mass functions (PDF/PMFs). Cumulative distribution networks (CDNs) have recently been proposed as a class of graphical models for CDFs. 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 data may follow extreme value statistics and exhibit spatial correlations so that dependencies between model variables must be accounted for. The problem of learning from data in such settings may nevertheless consist 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. 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 will demonstrate the GDP algorithm on real-world rainfall and H1N1 mortality data and we will show that CDNs provide a natural choice of parameterizations for the heavy-tailed multivariate distributions that arise in these problems.

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