Fig. 1.

Causal modeling of MAPK signaling pathway. Circles are variables, double circles are variables intervened upon, squares are deterministic functional assignments, gray nodes are observed variables, and white nodes are hidden variables. (a) Structural causal model. $N_{Raf}$, $N_{Mek}$ and $N_{Erk}$ are statistically independent noise variables. Root node $Raf$ is only dependent on noise variable $N_{Raf}$. Non-root nodes $Mek$ and $Erk$ are dependent on their parent and on the associated noise variable. (b) Counterfactual model. The intervention fixes the count of phosphorylated $Mek$ to $m^{\text{'}}$, such that $Mek$ is no longer dependent on $Raf$ and $N_{Mek}$. Given an observed data point, counterfactual inference infers the noise variables ${\widehat{N}}_{Raf}$, and ${\widehat{N}}_{Erk}$.