Algorithm 2 2-variable causal inference

Input: Finite sample data $\mathit{d}\equiv (\mathit{x},\mathit{y})\in {\mathbb{R}}^{N\times 2}$, Hyperparameters $P_{\beta },P_f,{\varsigma }^2,r$ Output: Predicted causal direction ${\mathcal{D}}_{X\to Y}\in \{X\to Y,Y\to X\}$ Rescale the data to the $[0,1]$ interval. That is, $min\{x_1,\dots ,x_N\}=min\{y_1,\dots ,y_N\}=0$ and $max\{x_1,\dots ,x_N\}=max\{y_1,\dots ,y_N\}=1$ Define an equally spaced grid of $(z_1,\dots ,z_{n_{\mathrm{bins}}})$ in the interval $[0,1]$ Calculate matrices $\mathit{B},\mathit{F}$ representing the covariance operators B and F evaluated at the positions of the grid, i.e., ${\mathit{B}}_{ij}=B(z_i,z_j)$ Find the ${\beta }_0\in {\mathbb{R}}^{[0,1]}$ for which $\gamma$, as defined in Appendix A.1 (Equation (A2)), becomes minimal Calculate the $\mathit{d}$-dependent terms of the information Hamiltonian in Equation (7) (i.e., all terms except ${\mathcal{H}}_0$) Repeat steps 4 and 5 with $\mathit{y}$ and $\mathit{x}$ switched Calculate the Bayes factor ${\mathcal{O}}_{X\to Y}$ If ${\mathcal{O}}_{X\to Y}>1$, return $X\to Y$, else return $Y\to X$