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Abstract:
A fundamental problem in several applications in bioinformatics, computer vision, statistics and machine learning is that of accounting for uncertainty in observed data as well as the presence of latent explanatory variables. This requires a structured approach to designing probability models and marginalizing over many variables under some joint density model. An approach to probabilistic modelling I am investigating is the cumulative distribution network (CDNs), a graphical model that describes the joint /cumulative distribution function/ instead of the joint density function. The conditional independence properties are quite different from other graphical models and marginalization involves tractable operations such as driving variables to \infty and computing derivatives of local functions. I derive relevant theorems and lemmas for CDNs and describe a message-passing algorithm called the derivative-product algorithm for performing inference in such models. I compare the performance of learnt CDNs to that obtained from an undirected density model on data sets generated such that they have conditional independence properties that are well-suited to CDNs. To demonstrate the potential for practical application, I apply the proposed framework to the problem of filling-in missing image pixels. The results indicate that CDNs can model data in which long-range conditional dependencies between model variables exist and must be accounted for.