Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs
稀疏有向无环图马尔可夫等价类上的可逆MCMCAI 解读 ↓
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稀疏有向无环图马尔可夫等价类上的可逆MCMC
- 研究问题
- 如何在稀疏有向无环图的马尔可夫等价类空间中有效抽样，并探索大规模等价类的结构，突破既有算法处理顶点数量的限制？
- 方法与思路
- 作者通过一组满足相应条件的转移算子构造可逆、不可约马尔可夫链，给出可计算的平稳分布及用于稀疏等价类的加速算法。
- 材料陈述的贡献
- 摘要称算法可探索具有数千顶点的稀疏图等价类；实验还发现多数边有方向、无向子图通常较小，其数量随顶点数近似线性增长。
- 阅读关注点
- 应核对算子的有效性条件、不可约性证明和平稳分布的抽样权重，并检查稀疏性的定义及实验结构规律所对应的图生成机制。
- 阅读边界
- 仅依据OpenAlex索引摘要，稿件版本未注明。实验规律不等于一般定理，摘要也未说明抽样结果如何支持具体因果识别。
原文摘要
Graphical models are popular statistical tools which are used to represent dependent or causal complex systems. Statistically equivalent causal or directed graphical models are said to belong to a Markov equivalent class. It is of great interest to describe and understand the space of such classes. However, with currently known algorithms, sampling over such classes is only feasible for graphs with fewer than approximately 20 vertices. In this paper, we design reversible irreducible Markov chains on the space of Markov equivalent classes by proposing a perfect set of operators that determine the transitions of the Markov chain. The stationary distribution of a proposed Markov chain has a closed form and can be computed easily. Specifically, we construct a concrete perfect set of operators on sparse Markov equivalence classes by introducing appropriate conditions on each possible operator. Algorithms and their accelerated versions are provided to efficiently generate Markov chains and to explore properties of Markov equivalence classes of sparse directed acyclic graphs (DAGs) with thousands of vertices. We find experimentally that in most Markov equivalence classes of sparse DAGs, (1) most edges are directed, (2) most undirected subgraphs are small and (3) the number of these undirected subgraphs grows approximately linearly with the number of vertices.