We study a notion of positivity of Gaussian directed acyclic graphical models corresponding to a non-negativity constraint on the coefficients of the associated structural equation model. We prove that this constraint is equivalent to the distribution being conditionally increasing in sequence (CIS), a well-known subclass of positively associated random variables. These distributions require knowledge of a permutation, a CIS ordering, of the nodes for which the constraint of non-negativity holds. We provide an algorithm and prove in the noise-less setting that a CIS ordering can be recovered when it exists. We extend this result to the noisy setting and provide assumptions for recovering the CIS orderings. In addition, we provide a characterization of Markov equivalence for CIS DAG models. Further, we show that when a CIS ordering is known, the corresponding class of Gaussians lies in a family of distributions in which maximum likelihood estimation is a convex problem.

翻译：我们研究了高斯有向无环图模型的一种正性概念，该概念对应于相关结构方程模型中系数的非负性约束。我们证明这一约束等价于分布具有序列条件递增性（CIS, conditionally increasing in sequence），这是正相关随机变量中一个著名的子类。此类分布需要已知节点的一个置换（即CIS排序），使得非负性约束成立。我们提出了一种算法，并在无噪声设定下证明：若存在CIS排序，则可恢复该排序。我们将此结果推广至含噪声设定，并给出了恢复CIS排序的假设条件。此外，我们对CIS有向无环图模型的马尔可夫等价性进行了刻画。进一步地，我们表明当CIS排序已知时，相应的高斯分布属于一族分布，其最大似然估计是凸优化问题。