Are Bayesian networks typically faithful?
贝叶斯网络通常满足忠实性吗？AI 解读 ↓
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贝叶斯网络通常满足忠实性吗？
- 研究问题
- 因果发现中常用的忠实性假设，在一般贝叶斯网络及受限模型类中是否具有拓扑或测度意义上的典型性？
- 方法与思路
- 作者分别研究总变差与弱拓扑中的忠实分布集合，分析条件指数族、受限非参数密度及含潜变量网络，并联系条件独立检验。
- 材料陈述的贡献
- 摘要称，在所述模型类和正则条件下获得开放稠密性或零测度结果，并据此论证 PC、FCI 等算法在典型集合上的一致性。
- 阅读关注点
- 应核对不同模型类使用的拓扑、正则条件和潜投影定义，并区分开放稠密、几乎处处与算法一致性各自的含义。
- 阅读边界
- 这是 arXiv 预印本，同行评审及接收状态未核实；典型性不等于有限样本可靠性，摘要也未说明接近不忠实分布时的检验表现。
原文摘要
Faithfulness is a common assumption in causal inference, often motivated by the fact that the faithful parameters of linear Gaussian and discrete Bayesian networks are typical, and the folklore belief that this should also hold for other classes of Bayesian networks. We address this open question by showing that among all Bayesian networks over a given DAG, the faithful Bayesian networks are indeed `typical': they constitute a dense, open set with respect to the total variation metric. This does not directly imply that faithfulness is typical in restricted classes of Bayesian networks that are often considered in statistical applications. To this end we consider the class of Bayesian networks parametrised by conditional exponential families, for which we show that under regularity conditions, the faithful parameters constitute a dense and open set, the unfaithful parameters have Lebesgue measure zero, and the induced faithful distributions are open and dense in the weak topology. This extends the existing results for linear Gaussian and discrete Bayesian networks. We also show for nonparametric classes of Bayesian networks with uniformly equicontinuous and uniformly bounded conditional densities that the faithful Bayesian networks are open and dense in the weak topology. All these results also hold for Bayesian networks with latent variables, if faithfulness is only required to hold with respect to the latent projection. Finally, for the considered conditional exponential family parametrisations and nonparametric conditional density models, the topological properties of conditional independence imply the existence of a consistent conditional independence test. Together with the topological properties of faithfulness, this implies that sound constraint-based causal discovery algorithms like PC and FCI are consistent on an open and dense -- and hence `typical' -- set of Bayesian networks.