[论文解读] Uniform generation of random acyclic digraphs

Jack Kuipers, Giusi Moffa|arXiv (Cornell University)|Feb 29, 2012

Bayesian Methods and Mixture Models参考文献 22被引用 2

一句话总结

本文提出了一种基于递归枚举的方法，用于对有向无环图（DAG）进行精确的均匀随机生成，避免了马尔可夫链蒙特卡洛（MCMC）方法固有的收敛问题。通过利用DAG的组合结构及其极限分布，该方法能够高效地对任意大小的DAG进行精确采样，同时支持各种结构约束。

ABSTRACT

AbstractWe show how to sample acyclic digraphs uniformly at random through recursive enumeration. This provides an exactmethod which avoids the convergence issues of the alternative Markov chain methods. The limiting behaviour of thedistribution of acyclic digraphs also allows us to sample arbitrarily large acyclic digraphs. Finally we discuss howto include various restrictions in the combinatorial enumeration for efficient uniform sampling of the correspondinggraphs.Keywords: Random graph generation, acyclic digraphs, recursive enumeration, Bayesian networks1. IntroductionDirected acyclic graphs (DAGs) are the basic representation of the structure underlying Bayesian networks, whichin turn represent multivariate probability distributions (Lauritzen, 1996; Neapolitan, 2004). They are largely used inmany fields of applied statistics with especially important applications in biostatistics, such as the learning of epistaticrelationships (Jiang et al., 2011). The estimation of DAGs or their equivalence class is a hard problem and methodsfor their efficient reconstruction from data is a very active field of resea rch: a recent review is given by Daly et al.,2011 while some new methodological developments for estimating high dimensional sparse DAGs are discussed byKalisch and Bu¨hlmann, 2007; Colombo et al., 2012. For simulation studies aimed at assessing the performance oflearning algorithms which reconstruct a graph from data, it is crucial to be able to generate uniform samples fromthe space of DAGs so that any structure related bias is removed. The only currently available method relies on theconstruction of a Markov chain whose properties ensure that the limiting distribution is uniform over all DAGs witha given number of vertices n. The strategy is based on a well known idea first suggested by M adigan and York (1995)as a Markov Chain Monte Carlo (MCMC) scheme in the context of Bayesian graphical models to sample from theposterior distribution of graphs conditional on the data. A specific algorithm for uniform sampling of DAGs wasfirst provided by Melanc¸on et al. (2001), with the advantage over the standard MCMC scheme of not requiring theevaluation of the sampled graphs’ neighbourhood size, at the expense of a slower convergence. The method waslater extended by Ide and Cozman (2002); Ide et al. (2004); Melanc¸on and Philippe (2004) to limit the sampling torestricted sets of DAGs. An R implementation was also recently provided by Scutari (2010). Since Markov chainbased algorithms pose non-negligible convergence and computational issues, in practice random upper or lower tri-angular adjacency matrices are often sampled to generate random ensembles for simulation studies [as for exampleimplemented in the pcalg R package of Kalisch et al. (2012)]. This method however does not provide uniformly dis-tributed graphs on the space of DAGs and could for example perform poorly to obtain starting points for hill-climbingalgorithms or slowly converging Markov chains by increasing the risk of remaining within a small neighbourhood ofcertain graphs and more inefficiently exploring the space. Likewise uniform sampling allows the correct evaluationof reconstructing algorithms. Finally, when evaluating the prevalence of certain features in a population, a uniformsample is essential. Here we therefore present a sampling strategy based on the recursive enumeration of DAGs butwhere no explicit listing is required.

研究动机与目标

• 解决在模拟研究和算法评估中，有向无环图（DAG）缺乏精确、均匀采样方法的问题。
• 克服先前DAG采样方法中使用马尔可夫链蒙特卡洛（MCMC）方法所导致的收敛与混合问题。
• 开发一种方法，通过利用DAG分布的极限行为，实现对任意大小DAG的均匀采样。
• 在不牺牲均匀性的前提下，将结构约束（如度数上限、边数限制）整合到采样过程中。
• 提供一种计算效率更高的替代方案，以替代随机上三角/下三角矩阵采样，后者无法生成均匀的DAG集合。

提出的方法

• 该方法基于DAG的组合结构，特别是拓扑序和源点选择，进行递归枚举。
• 通过按拓扑序递归添加顶点，逐步构建DAG，通过受控的边插入确保无环性。
• 通过根据完成部分DAG为指定大小完整DAG的方式数对部分DAG进行加权，保持采样均匀性。
• 通过类似动态规划的计数方法引导采样概率，避免显式枚举所有DAG。
• 利用已知的DAG分布渐近性质，实现对大规模图的采样而无需完整枚举。
• 通过在构建过程中限制递归选择，将最大入度或边数等约束整合进来。

实验结果

研究问题

• RQ1 是否可以不依赖马尔可夫链收敛性，实现对有向无环图的均匀采样？
• RQ2 如何将递归组合枚举方法适配以高效生成均匀随机的DAG？
• RQ3 在保持均匀性的同时，结构约束在多大程度上可以嵌入采样过程？
• RQ4 能否通过利用分布的极限性质，使该方法扩展到生成任意大小的DAG？
• RQ5 与现有的MCMC和矩阵采样方法相比，该方法在效率和均匀性方面表现如何？

主要发现

• 所提出的方法在给定顶点数的所有DAG空间中实现了精确的均匀采样，消除了MCMC收敛问题带来的偏差。
• 递归枚举框架通过利用DAG计数的渐近行为，使得对任意大小DAG的采样成为可能。
• 通过限制递归选择，该方法能高效支持有界入度或边数限制等结构约束下的采样。
• 通过避免显式枚举所有DAG，该方法在中等规模图上仍保持计算可行性。
• 与随机上三角/下三角矩阵采样相比，该方法在均匀性方面表现更优，更适合生成多样化且无偏的图集合。
• 该方法通过提供均匀分布的训练数据，使DAG学习算法的正确评估成为可能。