Those methods compute a weighted bi-criteria aggregation between fit and complexity $\mathcal{A}_{\hat{\mathcal{B}}}(\theta) =S_{\hat{\mathcal{B}}}(\theta) + \lambda C_{\hat{\mathcal{B}}}(\theta)$. We use the same notation $\hat{\theta}$ for the set of parameters that minimizes the score $\mathcal{A}_{\hat{\mathcal{B}}}(\theta)$.

If $\mathcal{A}_{X\rightarrow Y}(\theta) < \mathcal{A}_{Y\rightarrow X}(\theta)$, it is decided that X → Y , otherwise it is decided that Y → X .

This bi-objective aggregation corresponds for example to cause-effect inference based on the MML principle. In this case, the total score is directly $S_{\hat{\mathcal{B}}}(\theta) + C_{\hat{\mathcal{B}}}(\theta)$, without any aggregation parameter λ . However this aggregation parameter is “implicitly” set in the chosen prior π ( θ ).