Non-approximate Inference for Collective Graphical Models on Path Graphs via Discrete Difference of Convex Algorithm

The importance of aggregated count data, which is calculated from the data of multiple individuals, continues to increase. Collective Graphical Model (CGM) is a probabilistic approach to the analysis of aggregated data. One of the most important operations in CGM is maximum a posteriori (MAP) inference of unobserved variables under given observations. Because the MAP inference problem for general CGMs has been shown to be NP-hard, an approach that solves an approximate problem has been proposed. However, this approach has two major drawbacks. First, the quality of the solution deteriorates when the values in the count tables are small, because the approximation becomes inaccurate. Second, since continuous relaxation is applied, the integrality constraints of the output are violated. To resolve these problems, this paper proposes a new method for MAP inference for CGMs on path graphs. First we show that the MAP inference problem can be formulated as a (non-linear) minimum cost flow problem. Then, we apply Difference of Convex Algorithm (DCA), which is a general methodology to minimize a function represented as the sum of a convex function and a concave function. In our algorithm, important subroutines in DCA can be efficiently calculated by minimum convex cost flow algorithms. Experiments show that the proposed method outputs higher quality solutions than the conventional approach.