大型无圈有向图的均匀随机生成
Uniform random generation of large acyclic digraphs
摘要：
有向无环图是贝叶斯网络结构的基本表示，它表示多元概率分布。在许多实际应用中，如基因调控网络的逆向工程，不仅模型参数的估计，但结构本身的重建是非常感兴趣的。以及评估不同的结构学习算法在模拟研究中，一个统一的样本从空间的有向无环图是必需的，以评估某些结构特征的流行。在这里，我们分析了如何通过递归枚举，以前认为太计算参与的方法，在随机一致的无环有向图的样本。基于复杂性的考虑，我们特别讨论了如何枚举直接提供一个精确的方法，它避免了其他马尔可夫链方法的收敛性问题，实际上是计算更快。非循环有向图的分布的极限行为允许我们对任意大的图进行采样。基于递归枚举采样的思想的基础上，我们还介绍了一种新的混合马尔可夫链比目前的替代品更快的收敛，同时仍然很容易适应各种限制。最后，我们讨论了如何将这些限制包括在组合计数和新的混合马尔可夫链方法中，以实现相应图的有效均匀采样。
英文摘要：
Directed acyclic graphs are the basic representation of the structure underlying Bayesian networks, which represent multivariate probability distributions. In many practical applications, such as the reverse engineering of gene regulatory networks, not only the estimation of model parameters but the reconstruction of the structure itself is of great interest. As well as for the assessment of different structure learning algorithms in simulation studies, a uniform sample from the space of directed acyclic graphs is required to evaluate the prevalence of certain structural features. Here we analyse how to sample acyclic digraphs uniformly at random through recursive enumeration, an approach previously thought too computationally involved. Based on complexity considerations, we discuss in particular how the enumeration directly provides an exact method, which avoids the convergence issues of the alternative Markov chain methods and is actually computationally much faster. The limiting behaviour of the distribution of acyclic digraphs then allows us to sample arbitrarily large graphs. Building on the ideas of recursive enumeration based sampling we also introduce a novel hybrid Markov chain with much faster convergence than current alternatives while still being easy to adapt to various restrictions. Finally we discuss how to include such restrictions in the combinatorial enumeration and the new hybrid Markov chain method for efficient uniform sampling of the corresponding graphs.
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