# Identifying Causal Direction via Dense Functional Classes

## TL;DR

This paper introduces LCUBE, a fast and interpretable method for causal inference between two variables using the MDL principle and cubic splines, outperforming existing methods on real-world datasets.

## Contribution

The paper proposes a novel MDL-based causal score using cubic splines, with proven identifiability and practical advantages over prior approaches.

## Key findings

- LCUBE achieves superior precision on Tuebingen cause-effect dataset.
- It outperforms state-of-the-art methods on multiple benchmarks.
- The method is simple, interpretable, and fast with only one hyperparameter.

## Abstract

We address the problem of determining the causal direction between two univariate, continuous-valued variables, X and Y, under the assumption of no hidden confounders. In general, it is not possible to make definitive statements about causality without some assumptions on the underlying model. To distinguish between cause and effect, we propose a bivariate causal score based on the Minimum Description Length (MDL) principle, using functions that possess the density property on a compact real interval. We prove the identifiability of these causal scores under specific conditions. These conditions can be easily tested. Gaussianity of the noise in the causal model equations is not assumed, only that the noise is low. The well-studied class of cubic splines possesses the density property on a compact real interval. We propose LCUBE as an instantiation of the MDL-based causal score utilizing cubic regression splines. LCUBE is an identifiable method that is also interpretable, simple, and very fast. It has only one hyperparameter. Empirical evaluations compared to state-of-the-art methods demonstrate that LCUBE achieves superior precision in terms of AUDRC on the real-world Tuebingen cause-effect pairs dataset. It also shows superior average precision across common 10 benchmark datasets and achieves above average precision on 13 datasets.