Additive Noise Model is an extension of LiNGAM model to deal with nonlinear mechanism derived by Hoyer et al. 
A bivariate additive noise model (ANM) X → Y is defined as: 
$Y:=f_Y(X) + N_Y$ with $X \perp \! \! \! \perp N_Y$
where $f_Y : \mathbb{R}\rightarrow \mathbb{R}$ is a Borel measurable function.

Additive Noise Model (ANM)

In causal modeling, the interaction between the noise variable and the cause is primarily categorized in two ways:

- **Additive noise interaction**: $$Y := \hat{f}_Y(X) + N_Y$$
- **Complex noise interaction**: $$Y := \hat{f}_Y(X, N_Y)$$, where the cause variable and the noise are mixed through a non-linear mechanism.

University of Michigan - Ann Arbor

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When modeling causal mechanisms within a candidate bivariate generative model, two primary characterizations must be specified:

1. The type of interaction between the noise variable and the cause.
2. The class of functions used to define the estimated mechanism $$\hat{f}$$.

Causal Mechanisms in a Candidate Generative Model

 Isabelle Guyon, Alexander Statnikov, Berna Bakir Batu (2019)
Cause Effect Pairs in Machine Learning (The Springer Series on Challenges in Machine Learning)
https://www.amazon.com/Effect-Machine-Learning-Springer-Challenges/dp/3030218090

Cause Effect Pairs in Machine Learning

Now we have all the ingredients required to sample data points of the estimated distribution $Q_{X , Y}$ with the generative model by proceeding as follow: 

1. Draw $\{(n_{X,j}, n_{Y,j})\}_{j=1}^n$ , n samples independent and identically distributed from the joint distribution $Q_{N_X}(\theta_{N_X})\times Q_{N_Y}(\theta_{N_Y})$ of independent noise variables $( N_X , N_Y )$. 
2. Generate n samples $\hat{\mathcal{D}} = \{(\hat{x}_j, \hat{y}_j) \}_{j=1}^n$, where each estimated sample $\hat{x}_j$ of variable $X$ is computed from $\hat{f}_X(\theta_X)$ with the j-th estimated samples $n_{X , j}$; then each estimated sample $\hat{y}_j$ of variable $Y$ is computed from $\hat{f}_Y(\theta_Y)$ with the j-th estimated samples $n_{Y , j}$ and $\hat{x}_j$. 

Generative candidate models supports both interventions and counterfactual reasoning.

Sampling Data Points with a Candidate Model

The class of functions used to define the estimated mechanism $$\hat{f}$$ may range from linear mechanisms, as in the LiNGAM algorithm, to complex non-linear mechanisms modeled with Gaussian processes or neural networks. In general, more complex mechanisms allow the candidate generative model to fit the data better and apply to a wider variety of cases. However, increased complexity can make it more difficult to assess causal orientation, as the model has a higher chance of fitting the data equally well in both directions.

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