2016/05/20 by Christian Knoll, Franz Pernkopf, Knoll, Christian +5 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · #Algebraic Geometry (math.AG) #Bayesian Modeling and Causal Inference #Error Correcting Code Techniques #FOS: Computer and information sciences #FOS: Mathematics #Gene Regulatory Network Analysis #Machine Learning (stat.ML)
paper · pdf · doi:10.48550/arxiv.1605.06451
openalex publication_date 2016/05/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/01
Belief propagation (BP) is an iterative method to perform approximate inference on arbitrary graphical models. Whether BP converges and if the solution is a unique fixed point depends on both the structure and the parametrization of the model. To understand this dependence it is interesting to find all fixed points. In this work, we formulate a set of polynomial equations, the solutions of which correspond to BP fixed points. To solve such a nonlinear system we present the numerical polynomial-homotopy-continuation (NPHC) method. Experiments on binary Ising models and on error-correcting codes show how our method is capable of obtaining all BP fixed points. On Ising models with fixed parameters we show how the structure influences both the number of fixed points and the convergence properties. We further asses the accuracy of the marginals and weighted combinations thereof. Weighting marginals with their respective partition function increases the accuracy in all experiments. Contrary to the conjecture that uniqueness of BP fixed points implies convergence, we find graphs for which BP fails to converge, even though a unique fixed point exists. Moreover, we show that this fixed point gives a good approximation, and the NPHC method is able to obtain this fixed point.