In this paper we show how to exploit interventional data to acquire the joint conditional distribution of all the variables using the Maximum Entropy principle. To this end, we extend the Causal Maximum Entropy method to make use of interventional data in addition to observational data. Using Lagrange duality, we prove that the solution to the Causal Maximum Entropy problem with interventional constraints lies in the exponential family, as in the Maximum Entropy solution. Our method allows us to perform two tasks of interest when marginal interventional distributions are provided for any subset of the variables. First, we show how to perform causal feature selection from a mixture of observational and single-variable interventional data, and, second, how to infer joint interventional distributions. For the former task, we show on synthetically generated data, that our proposed method outperforms the state-of-the-art method on merging datasets, and yields comparable results to the KCI-test which requires access to joint observations of all variables.

翻译：本文展示了如何利用干预数据，基于最大熵原理获取所有变量的联合条件分布。为此，我们将因果最大熵方法扩展到除观测数据外还能利用干预数据。通过拉格朗日对偶性，我们证明了在介入约束下因果最大熵问题的解属于指数族，这与最大熵解的性质一致。当给定任意变量子集的边际干预分布时，我们的方法能够执行两项重要任务：第一，展示如何从观测数据与单变量干预数据的混合中进行因果特征选择；第二，推断联合干预分布。对于前者，我们在合成生成数据上的实验表明，所提方法在数据集合并上优于当前最先进方法，且其结果与需访问所有变量联合观测值的KCI检验相当。