  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.
