Optimal sub-graphical models

The complexity of inference in graphical models is typically exponential in some parame- ter of the graph, such as the size of the largest clique. Therefore, it is often required to find a subgraphical model that has lower complexity (smaller clique size) without introducing a large error in inference results. The KL-divergence between the original probability dis- tribution and the probability distribution on the simplified graphical model is often used to measure the impact on inference. Existing techniques for reducing the complexity of graph- ical models including annihilation and edge-removal [4] are greedy in nature and cannot make any guarantees regarding the optimality of the solution. This problem is NP-complete [9] and so, in general, one cannot expect a polynomial time algorithm to find the optimal solution. However, we show that when we restrict the problem to some sets of subgraphs, the optimal solution can be found quite quickly using a dynamic programming algorithm in time polynomial in the tree-width of the graph.