We study the differences arising from merging predictors in the causal and anticausal directions using the same data. In particular we study the asymmetries that arise in a simple model where we merge the predictors using one binary variable as target and two continuous variables as predictors. We use Causal Maximum Entropy (CMAXENT) as inductive bias to merge the predictors, however, we expect similar differences to hold also when we use other merging methods that take into account asymmetries between cause and effect. We show that if we observe all bivariate distributions, the CMAXENT solution reduces to a logistic regression in the causal direction and Linear Discriminant Analysis (LDA) in the anticausal direction. Furthermore, we study how the decision boundaries of these two solutions differ whenever we observe only some of the bivariate distributions implications for Out-Of-Variable (OOV) generalisation.

翻译：本研究探讨使用相同数据在因果与反因果方向上合并预测器所产生的差异。我们通过一个简单模型分析这种不对称性：以单个二元变量为目标变量，两个连续变量为预测变量进行预测器合并。本研究采用因果最大熵（CMAXENT）作为归纳偏置来实现预测器合并，但我们预期当采用其他考虑因果不对称性的合并方法时，也会出现类似的差异。我们证明，若观测所有二元分布，CMAXENT解在因果方向上可简化为逻辑回归，在反因果方向上则简化为线性判别分析（LDA）。此外，我们进一步研究了当仅观测部分二元分布时，这两种解的决策边界如何产生差异，并探讨了这种差异对变量外（OOV）泛化的影响。