Approaches to bivariate causal discovery based on the minimum description length (MDL) principle approximate the (uncomputable) Kolmogorov complexity of the models in each causal direction, selecting the one with the lower total complexity. The premise is that nature's mechanisms are simpler in their true causal order. Inherently, the description length (complexity) in each direction includes the description of the cause variable and that of the causal mechanism. In this work, we argue that current state-of-the-art MDL-based methods do not correctly address the problem of estimating the description length of the cause variable, effectively leaving the decision to the description length of the causal mechanism. Based on rate-distortion theory, we propose a new way to measure the description length of the cause, corresponding to the minimum rate required to achieve a distortion level representative of the underlying distribution. This distortion level is deduced using rules from histogram-based density estimation, while the rate is computed using the related concept of information dimension, based on an asymptotic approximation. Combining it with a traditional approach for the causal mechanism, we introduce a new bivariate causal discovery method, termed rate-distortion MDL (RDMDL). We show experimentally that RDMDL achieves competitive performance on the Tübingen dataset. All the code and experiments are publicly available at github.com/tiagobrogueira/Causal-Discovery-In-Exchangeable-Data.

翻译：基于最小描述长度（MDL）原则的双变量因果发现方法近似计算每个因果方向中模型的（不可计算的）柯尔莫哥洛夫复杂度，并选择总复杂度较低的方向。其前提是自然机制在其真实因果顺序中更为简单。本质上，每个方向的描述长度（复杂度）包括对原因变量和因果机制的描述。在本工作中，我们论证当前最先进的基于MDL的方法未能正确解决原因变量描述长度的估计问题，实际上将决策权交由因果机制的描述长度。基于率失真理论，我们提出了一种衡量原因变量描述长度的新方法，该方法对应于为达到代表底层分布的失真级别所需的最小速率。该失真级别通过直方图密度估计规则推导得出，而速率则利用信息维的相关概念基于渐近近似进行计算。将此法与传统因果机制方法相结合，我们提出了一种新的双变量因果发现方法，称为率失真MDL（RDMDL）。实验证明，RDMDL在Tübingen数据集上取得了具有竞争力的性能。所有代码和实验均公开于github.com/tiagobrogueira/Causal-Discovery-In-Exchangeable-Data。