AMP Chain Graphs: Minimal Separators and Structure Learning Algorithms
This work addresses structure learning challenges in probabilistic graphical models, specifically for AMP CGs, which are incremental improvements over existing methods for researchers in machine learning and statistics.
The paper tackles the problem of finding minimal separators in Andersson-Madigan-Perlman chain graphs (AMP CGs) and develops polynomial-time algorithms for various versions, while also addressing order-dependence in structure learning by proposing modifications to the PC-like algorithm and extending a decomposition-based method to AMP CGs. The results show that the LCD-AMP algorithm usually outperforms the PC-like algorithm, with modifications improving accuracy and stability, especially in high-dimensional and sparse graph settings.
We address the problem of finding a minimal separator in an Andersson-Madigan-Perlman chain graph (AMP CG), namely, finding a set Z of nodes that separates a given nonadjacent pair of nodes such that no proper subset of Z separates that pair. We analyze several versions of this problem and offer polynomial-time algorithms for each. These include finding a minimal separator from a restricted set of nodes, finding a minimal separator for two given disjoint sets, and testing whether a given separator is minimal. To address the problem of learning the structure of AMP CGs from data, we show that the PC-like algorithm (Pena, 2012) is order-dependent, in the sense that the output can depend on the order in which the variables are given. We propose several modifications of the PC-like algorithm that remove part or all of this order-dependence. We also extend the decomposition-based approach for learning Bayesian networks (BNs) proposed by (Xie et al., 2006) to learn AMP CGs, which include BNs as a special case, under the faithfulness assumption. We prove the correctness of our extension using the minimal separator results. Using standard benchmarks and synthetically generated models and data in our experiments demonstrate the competitive performance of our decomposition-based method, called LCD-AMP, in comparison with the (modified versions of) PC-like algorithm. The LCD-AMP algorithm usually outperforms the PC-like algorithm, and our modifications of the PC-like algorithm learn structures that are more similar to the underlying ground truth graphs than the original PC-like algorithm, especially in high-dimensional settings. In particular, we empirically show that the results of both algorithms are more accurate and stabler when the sample size is reasonably large and the underlying graph is sparse.