Causal inference in hybrid domains, characterized by a mixture of discrete and continuous variables, presents a formidable challenge. We take a step towards this direction and propose Characteristic Interventional Sum-Product Network ($\chi$SPN) that is capable of estimating interventional distributions in presence of random variables drawn from mixed distributions. $\chi$SPN uses characteristic functions in the leaves of an interventional SPN (iSPN) thereby providing a unified view for discrete and continuous random variables through the Fourier-Stieltjes transform of the probability measures. A neural network is used to estimate the parameters of the learned iSPN using the intervened data. Our experiments on 3 synthetic heterogeneous datasets suggest that $\chi$SPN can effectively capture the interventional distributions for both discrete and continuous variables while being expressive and causally adequate. We also show that $\chi$SPN generalize to multiple interventions while being trained only on a single intervention data.

翻译：在兼具离散与连续变量的混合域中进行因果推断是一项艰巨挑战。我们在此方向迈进一步，提出了特征干预求和积网络（χSPN），该模型能够估计来自混合分布的随机变量存在时的干预分布。χSPN通过在干预SPN（iSPN）的叶节点中使用特征函数，借助概率测度的傅里叶-斯蒂尔杰斯变换为离散和连续随机变量提供了统一表征框架。我们采用神经网络基于干预数据来估计学习所得iSPN的参数。在三个合成异构数据集上的实验表明，χSPN能够有效捕捉离散与连续变量的干预分布，同时兼具表达性和因果充分性。我们还证明χSPN仅需在单干预数据上训练即可泛化至多干预场景。