# Causal Inference on Discrete Data using Additive Noise Models

## TL;DR

This paper extends additive noise models to discrete variables, providing a theoretical foundation and an efficient algorithm for causal inference from finite samples of discrete data, validated on synthetic and real datasets.

## Contribution

It generalizes additive noise models to discrete data and introduces a new algorithm for causal inference that is theoretically sound and practically effective.

## Key findings

- The model is identifiable for discrete variables under generic conditions.
- The proposed algorithm accurately distinguishes cause from effect in finite samples.
- Experimental results demonstrate effectiveness on both synthetic and real data.

## Abstract

Inferring the causal structure of a set of random variables from a finite sample of the joint distribution is an important problem in science. Recently, methods using additive noise models have been suggested to approach the case of continuous variables. In many situations, however, the variables of interest are discrete or even have only finitely many states. In this work we extend the notion of additive noise models to these cases. We prove that whenever the joint distribution $\prob^{(X,Y)}$ admits such a model in one direction, e.g. $Y=f(X)+N, N \independent X$, it does not admit the reversed model $X=g(Y)+\tilde N, \tilde N \independent Y$ as long as the model is chosen in a generic way. Based on these deliberations we propose an efficient new algorithm that is able to distinguish between cause and effect for a finite sample of discrete variables. In an extensive experimental study we show that this algorithm works both on synthetic and real data sets.