% !TEX root =  ../main.tex
\section{On the invertibility assumption}\label{app:invertibility}
Here we illustrate that invertibility of \mymixing holds for the main classes of Markovian SCMs.

\spara{Additive Noise Model.}
In this case, considering $n$ causal variables, the structural equations read as
\begin{equation}\label{eq:anm}
    X_i = f_i(\parents_i) + Z_i, \quad \forall\, i \in [n]\,.
\end{equation}
It is immediate to obtain that the components of $\mymixing^{-1}: \myendogenousvals \rightarrow \myexogenousvals$ are $m^{-1}_i = X_i - f_i(\parents_i) $, $\forall\, i \in [n]$.

\spara{Post-nonlinear Model.}
In this case, the structural equations read as
\begin{equation}\label{eq:pnl}
    X_i = g_i(f_i(\parents_i) + Z_i), \quad \forall\, i \in [n]\,;
\end{equation}
where $g_i$ is invertible \cite{zhang2009identifiability,kaltenpoth2023nonlinear}.
It is immediate to obtain that the components of $\mymixing^{-1}: \myendogenousvals \rightarrow \myexogenousvals$ are $m^{-1}_i = g_i^{-1}(X_i) - f_i(\parents_i) $, $\forall\, i \in [n]$.

\spara{Location-scale Noise Models.}
In this case, the structural equations are \cite{immer2023identifiability}
\begin{equation}\label{eq:lsnm}
    X_i = f_i(\parents_i) + h_i(\parents_i)Z_i, \quad \forall\, i \in [n]\,;
\end{equation}
where $f_i: \myendogenousvals_{\parents_i} \rightarrow \reall$ and $h_i: \myendogenousvals_{\parents_i} \rightarrow \reall_+$ is strictly positive.
It is immediate to obtain that the components of $\mymixing^{-1}: \myendogenousvals \rightarrow \myexogenousvals$ are $m^{-1}_i = \left(X_i - f_i(\parents_i) \right)/h_i(\parents_i)$, $\forall\, i \in [n]$.