We derive an implicit description of the image of a semialgebraic set under a birational map, provided that the denominators of the map are positive on the set. For statistical models which are globally rationally identifiable, this yields model-defining constraints which facilitate model membership testing, representation learning, and model equivalence tests. Many examples illustrate the applicability of our results. The implicit equations recover well-known Markov properties of classical graphical models, as well as other well-studied equations such as the Verma constraint. They also provide Markov properties for generalizations of these frameworks, such as colored or interventional graphical models, staged trees, and the recently introduced Lyapunov models. Under a further mild assumption, we show that our implicit equations generate the vanishing ideal of the model up to a saturation, generalizing previous results of Geiger, Meek and Sturmfels, Duarte and G\"orgen, Sullivant, and others.

翻译：我们推导了半代数集在双有理映射下像的隐式描述，前提是该映射的分母在该集合上为正。对于全局有理可识别的统计模型，这产生了模型定义的约束条件，便于进行模型成员资格测试、表示学习以及模型等价性检验。许多示例说明了我们结果的适用性。这些隐式方程恢复了经典图模型著名的马尔可夫性质，以及其他已深入研究的方程，如Verma约束。它们还为这些框架的推广提供了马尔可夫性质，例如着色或干预图模型、阶段树以及最近引入的Lyapunov模型。在进一步温和的假设下，我们证明了我们的隐式方程通过饱和化生成了模型的消失理想，推广了Geiger、Meek和Sturmfels、Duarte和G\"orgen、Sullivant等人的先前结果。