Probabilistic Pairwise Markov Models: Application to Prostate Cancer Detection

Markov Random Fields (MRFs) provide a tractable means for incorporating contextual information into a Bayesian framework. This contextual information is modeled using multiple local conditional probability density functions (LCPDFs) which the MRF framework implicitly combines into a single joint probability density function (JPDF) that describes the entire system. However, only LCPDFs of certain functional forms are consistent, meaning they reconstitute a valid JPDF. These forms are specified by the Gibbs-Markov equivalence theorem which indicates that the JPDF, and hence the LCPDFs, should be representable as a product of potential functions (i.e. Gibbs distributions). Unfortunately, potential functions are mathematical abstractions that lack intuition; and consequently, constructing LCPDFs through their selection becomes an ad hoc procedure, usually resulting in generic and/or heuristic models. In this paper we demonstrate that under certain conditions the LCDPFs can be formulated in terms of quantities that are both meaningful and descriptive: probability distributions. Using probability distributions instead of potential functions enables us to construct consistent LCPDFs whose modeling capabilities are both more intuitive and expansive than typical MRF models. As an example, we compare the efficacy of our so-called probabilistic pairwise Markov models (PPMMs) to the prevalent Potts model by incorporating both into a novel computer aided diagnosis (CAD) system for detecting cancer on whole-mount histological sections of radical prostatectomies. Using the Potts model, the CAD system is able to detect prostate cancer with a specificity of 0.82 and sensitivity of 0.71; the area under its receiver operator characteristic curve (AUC) is 0.83. If instead the PPMM model is employed the sensitivity (specificity is held fixed at 0.82) and AUC increase to 0.77 and 0.87.