MAP inference algorithms without approximation for collective graphical models on path graphs via discrete difference of convex algorithm

Abstract

Collective graphical model (CGM) is a probabilistic model that provides a framework for analyzing aggregated count data. Maximum a posteriori (MAP) inference of unobserved variables under given observations is one of the essential operations in CGM. Because the MAP inference problem is known to be NP-hard in general, the current mainstream approach is to solve an alternative problem obtained by approximating the objective function and applying continuous relaxation. However, this approach has two significant drawbacks. First, the quality of the solution deteriorates when the values in the count data are negligible due to the inaccuracy of Stirling’s approximation. Second, the application of continuous relaxation causes the violation of integrality constraints. This paper proposes novel algorithms for MAP inference in CGMs on path graphs to overcome these problems. Our method is based on the discrete difference of convex algorithm (DCA); DCA is a general framework to minimize the sum of a convex function and a concave function by repeatedly minimizing surrogate functions. Utilizing the particular structure of path graphs, we efficiently solve the surrogate function minimization by minimum convex cost flow algorithms. Furthermore, our approach also leads to a new method of solving another important task; MAP inference of the sample size in CGM on path graphs. Our method is naturally applicable to this task, allowing us to design very efficient algorithms. Experimental results on synthetic and real-world datasets show the effectiveness of the proposed algorithms.