Algorithm 1 Sampling of causal data via forward model

Input: Power spectra $P_{\beta },P_f$, noise variance ${\varsigma }^2$, number of bins $n_{\mathrm{bins}}$, desired number (As we draw the number of samples from Poisson distribution in each bin, we do not deterministically control the total number of samples) of samples $\tilde{N}$ Output:N samples $(d_i)=(x_i,y_i)$ generated from a causal relation of either $X\to Y$ or $Y\to X$ Draw a sample field $\beta \in {\mathbb{R}}^{[0,1]}$ from the distribution $\mathcal{N}(\beta |0,B)$ Set an equally spaced grid with $n_{\mathrm{bins}}$ points in the interval $[0,1]$: $\mathit{z}=(z_1,\dots ,z_{n_{\mathrm{bins}}}),z_i=\frac{i-0.5}{n_{\mathrm{bins}}}$ Calculate the vector of Poisson means $\mathit{\lambda }=({\lambda }_1,\dots {\lambda }_{n_{\mathrm{bins}}})$ with ${\lambda }_i\propto e^{\beta (z_i)}$ At each grid point $i\in \{1,\dots ,n_{\mathrm{bins}}\}$, draw a sample $k_i$ from a Poisson distribution with mean ${\lambda }_i$: $k_i\sim {\mathcal{P}}_{{\lambda }_i}(k_i)$ Set $N={\sum }_{i=1}^{n_{\mathrm{bins}}}{k}_i$ For each $i\in \{1,\dots ,n_{\mathrm{bins}}\}$ add $k_i$ times the element $z_i$ to the set of measured $x_j$. Construct the vector $\mathit{x}=(\dots ,\underset{k_i\mathrm{times}}{\underbrace{z_i,z_i,z_i}},\dots )$ Draw a sample field $f\in {\mathbb{R}}^{[0,1]}$ from the distribution $\mathcal{N}(f|0,F)$. Rescale f s.th. $f\in {[0,1]}^{[0,1]}$ Draw a multivariate noise sample $\mathbf{ϵ}\in {\mathbb{R}}^N$ from a normal distribution with zero mean and variance ${\varsigma }^2$, $\mathbf{ϵ}\sim \mathcal{N}(\mathbf{ϵ}|0,{\varsigma }^2)$ Generate the effect data $\mathit{y}$ by applying f to $\mathit{x}$ and adding $\mathbf{ϵ}$: $\mathit{y}=f(\mathit{x})+\mathbf{ϵ}$ With probability $\frac{1}{2}$ return $\mathit{d}=({\mathit{x}}^T,{\mathit{y}}^T)$, otherwise return $\mathit{d}=({\mathit{y}}^T,{\mathit{x}}^T)$