Cumulative Distribution Networks and the Derivative-Sum-Product Algorithm

by Jim C. Huang and Brendan J. Frey

The paper first proposed novel probabilistic graphical model, i.e. CDN. The model differs with MRF and Bayesian belief network in several aspects. The basic idea of the network resembles the factor graph of MRF. The factor graph of an MRF consists a vertex set of two parts: one of the original vertices of the MRF, the other of each cliques. The representer theorem of MRF actually tells us each clique is a nonnegative potential function (unnormalized PDF, hence the vertex is often referred to by function vertex). In CDN, however, each function vertex corresponds to a CDF-like function (i.e. cumulative function as in the paper). As long as these cumulative functions satisfy several simple properties (which might be easily proved as in the paper), their product is a decent CDF.

To do inference in CDF, the authors proposed an algorithm that resembles sum-product algorithm for factor graphs (i.e. belief propagation for polytrees). We know the derivative of the CDF is PDF, therefore it looks natural their inference is called Derivative-Sum-Product (DSP) algorithm. I will read the details about the algorithm later.

This framework is applied to a ranking problem.

Here we can see, we have three teams, (X₁, X₂), (X₃, X₄) and (X₅, X₆, X₇). The function s_i(x_i) is the skill of the player. The performance of each team is the sum of all players, a function of x_i. And we get R₁, R₂ and R₃. We do have the rank of these teams, which comes as an ordered graph whose relationships are represented with h. The prediction is carried out with skill functions learned with previous results. In this model every function vertex is a CDF of a Gaussian and DSP could be computed easily. I think the details are worth trying to see.

In the end, several applications are pointed out, webpage ranking, collaborative filtering and etc. They also intended to focus on the learning part of the model in later research and the DSP algorithm, as the belief propagation, is only applicable to trees.

Several related topic:
convolutional factor graph: Convolutional factor graphs as probabilistic models, UAI 2004.