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 separate a given non-adjacent 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. We provide an extension of the decomposition approach for learning Bayesian networks (BNs) proposed by (Xie et. al.) to learn AMP CGs , which include BNs as a special case, under the faithfulness assumption and prove its correctness using the minimal separator results. The advantages of this decomposition approach hold in the more general setting: reduced complexity and increased power of computational independence tests. In addition, we show that the PC-like algorithm  is order-dependent, in the sense that the output can depend on the order in which the variables are given. We propose two modifications of the PC-like algorithm that remove part or all of this order-dependence. Simulations under a variety of settings demonstrate the competitive performance of our decomposition-based method, called LCD-AMP, in comparison with the (modified version of) PC-like algorithm. In fact, the decomposition-based algorithm usually outperforms the PC-like algorithm . We empirically show that the results of both algorithms are more accurate and stable when the sample size is reasonably large and the underlying graph is sparse.
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.
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separate a given non-adjacent
separates a given nonadjacent
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We provide an extension of the decomposition approach for learning Bayesian networks (BNs) proposed by (Xie et. al.) to learn AMP CGs , which include BNs as a special case, under the faithfulness assumption and prove its correctness using the minimal separator results. The advantages of this decomposition approach hold in the more general setting: reduced complexity and increased power of computational independence tests. In addition,
To address the problem of learning the structure of AMP CGs from data,
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(Pena, 2012)
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two
several
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Simulations under a variety of settings
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
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version
versions
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In fact, the decomposition-based
The LCD-AMP
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. We
, 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
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stable
stabler