\begin{figure*}[t]
\centering
\hfill
\begin{subfigure}[t]{0.25\linewidth}\centering%(d)
    \begin{tikzpicture}[SCM,scale=1]
        \node (V1) at (0,0) {$X$};
        \node (V2) at (1,0) {$Y$};
        \node (V3) at (2,0) {$Z$};

        \path [arrows = {Circle[fill=white]->}] (V1) edge (V2);
        \path [arrows = {Circle[fill=white]->}] (V3) edge (V2);
    \end{tikzpicture}
\caption{PAG}
\label{fig:pag}
\end{subfigure}\hfill
\begin{subfigure}[t]{0.25\linewidth}\centering%(d)
    \begin{tikzpicture}[SCM,scale=1]
        \node (V1) at (0,0) {$X$};
        \node (V2) at (1,0) {$Y$};
        \node (V3) at (2,0) {$Z$};

        \path [conf-path] (V1) edge [out=45,in=135] (V2);
        \path [<-] (V2) edge (V3);
    \end{tikzpicture}
\caption{MAG}
\label{fig:mag}
\end{subfigure}\hfill
\begin{subfigure}[t]{0.25\linewidth}\centering%(d)
  \begin{tikzpicture}[SCM,scale=1]
        \node (V1) at (0,0) {$X$};
        \node (V2) at (1,0) {$Y$};
        \node (V3) at (2,0) {$Z$};

        \path [->] (V1) edge (V2);
        \path [<-] (V2) edge (V3);
    \end{tikzpicture}
\caption{LEG}
\label{fig:leg}
\end{subfigure}\hfill
\begin{subfigure}[t]{0.25\linewidth}\centering%(d)
  \begin{tikzpicture}[SCM,scale=1]
        \node (V1) at (0,0) {$X$};
        \node (V2) at (1,0) {$Y$};
        \node (V3) at (2,0) {$Z$};

        \path [->] (V1) edge (V2);
        \path [->] (V3) edge (V2);
        \path [conf-path] (V1) edge [out=45,in=135] (V2);
        \path [conf-path] (V2) edge [out=45,in=135] (V3);
    \end{tikzpicture}
\caption{MBD}
\label{fig:mbd}
\end{subfigure}\hfill
\hfill\null
  \caption{Diagrams used in \Cref{sec:enumeration}.}
  \label{fig:mag_leg}
\end{figure*}

\section{The difficulty of enumerating causal diagrams from a PAG}
\label{sec:enumeration}
The techniques explored so far overlook the potential for enumerating (relevant) ME causal diagrams and subsequently applying existing (partial) identification techniques given each diagram separately (that we refer to as ``enumeration strategies''). Enumerating all ME causal diagrams is exponentially costly, and intractable in general even with an assumption of no unobserved confounding, \textit{i.e.} in the space of directed acyclic graphs as shown by \cite{wienobst2023efficient}. However, a number of observations can be made to avoid enumerating all ME causal diagrams which reduces the search space to a (potentially tractable) \emph{subset} of ``relevant'' ME causal diagrams without loss of generality. 

This section explores the definition of sets of ME causal diagrams $\1K \subset \1P$ with the distinctiveness of being equally expressive in the sense that,
\begin{align}
    \label{eq:expressiveness}
    \Big\{P_{\x}(\y; \M)&: \M \in \3M(\1P)\Big\} =\nonumber\\
    &\Big\{P_{\x}(\y; \M): \M \in \3M(\1K)\Big\}.
\end{align}
Let $\3M(\1P)$ denote the set of SCMs compatible with $\1P$, that is the set of SCMs that induce causal diagrams contained in the PAG abstraction $\1P$. Under \Cref{eq:expressiveness}, minimum and maximum values of causal effects remain unchanged, and one may exploit $\1K$ instead of $\1P$ for inference in practice. The hope is that if the set of causal diagrams $\1K$ is small enough then we might be able to apply existing partial identification algorithms on every causal diagram $\G\in\1K$ efficiently.

We start by introducing the notion of a Loyal Equivalent Graph (LEG) (\Cref{def:LEG}), from \cite[Prop. 2]{zhang2012transformational}, that are sets of ME MAGs that retain ``expressiveness'' in the sense of \Cref{eq:expressiveness}. This result is given in \Cref{prop:expressiveness_leg}.

\begin{definition}
    \label{def:LEG}
    Given a MAG $\G$, there exists a ME MAG $\1H$, called a Loyal Equivalent Graph, such that all bi-directed edges in $\1H$ are invariant, and every directed edge in $\G$ is also in $\1H$.
\end{definition}

\begin{proposition}[Expressiveness of LEGs]
\label{prop:expressiveness_leg}
Given a PAG $\1P$, let $\1L$ be the set of ME LEGs. Then, $\3M(\1P) = \3M(\1L)$.
\end{proposition}

For example, the LEG in \Cref{fig:leg} is derived from the MAG in \Cref{fig:mag} by replacing the bi-directed edge with a directed one. \Cref{prop:expressiveness_leg} shows that the set of ME LEGs is as expressive as the set of ME MAGs that are encoded by $\1P$\footnote{Recall, as noted in \Cref{sec:pag_notation}, that MAGs (and LEGs) encode the sets of causal diagrams that would be represented by the same MAG (or LEG).}. The significance of this proposition lies in the fact that ME LEGs are a subset of ME MAGs and that, contrary to ME MAGs, ME LEGs are in principle listable by exhaustively applying a simple criterion for the reversal of directed edges while remaining Markov equivalent, \textit{i.e.} \cite[Lemma 2]{zhang2012transformational}. We review this criterion in more detail in \Cref{sec:app_background} and give an algorithm for enumerating ME LEGs that exploits it in \Cref{sec:app_enumeration_strategies}. 


A second redundancy result is given in \Cref{prop:expressiveness_causal_graphs_in_leg} by introducing so called \emph{maximally bi-directed} (MBD) diagrams.

\begin{definition}
    \label{def:mbd}
    \textit{A causal diagram $\G$ is said to be maximally bi-directed if no further bi-directed edges can be added without breaking a $d$-separation.}
\end{definition}

\begin{proposition}[Expressiveness of MBD diagrams]
\label{prop:expressiveness_causal_graphs_in_leg}
Given a PAG $\1P$, let $\1D$ be the set of ME MBD diagrams. Then, $\3M(\1P) = \3M(\1D)$.
\end{proposition}

MDB causal diagrams can be constructed from an LEG by adding bi-directed edges on top of invisible directed edges wherever possible. For example, the causal diagram $\G$ in \Cref{fig:mbd}, compatible with the LEG $L$ in \Cref{fig:leg}, is said to be maximally bi-directed as no further bi-directed edges can be added while remaining Markov equivalent. $\G$ has the distinctiveness of inducing a family of SCMs which includes all SCMs that are compatible with any causal diagram compatible with the corresponding LEG $L$ in \Cref{fig:leg}, i.e. $\3M(\G)=\3M(L)$. More specifically, in this example, $\G$ induces a class of SCMs given by: $x:=f_X(\u_{X,Y}), z:=f_Z(\u_{X,Z}), y:=f_Y(x, z, \u_{X,Z},\u_{X,Y})$, with deterministic functions and exogeneous distributions arbitrarily defined. We can see that this parameterization is flexible enough to represent any SCM induced by causal diagrams compatible with $L$. As a consequence, in general, \Cref{prop:expressiveness_causal_graphs_in_leg} implies that the set of ME MBD causal diagrams is as expressive as the set of \emph{all} ME causal diagrams. \Cref{sec:app_enumeration_strategies} gives an algorithm for generating all ME MBD diagrams $\1D$ from the set of all ME LEGs $\1L$. 

\begin{figure}[t]
\vspace{-0.1cm}
\captionsetup{skip=5pt}
\centering
    \begin{tikzpicture}[SCM,scale=1]
        \node (X) at (0,0) {$X$};
        \node (V1) at (1,-0.4) {$W$};
        \node (V2) at (1,0.4) {$V$};
        \node (Y) at (2,0) {$Y$};

        \path [->] (X) edge (V1);
        \path [->] (X) edge (V2);
        \path [->] (V1) edge (Y);
        \path [->] (V2) edge (Y);
        \path [conf-path] (X) edge[out = -70, in=200] (V1);
        
        \node (X) at (3,0) {$X$};
        \node (V1) at (4,-0.4) {$W$};
        \node (V2) at (4,0.4) {$V$};
        \node (Y) at (5,0) {$Y$};

        \path [->] (X) edge (V1);
        \path [->] (X) edge (V2);
        \path [->] (V1) edge (Y);
        \path [->] (V2) edge (Y);
        \path [conf-path] (X) edge[out = 70, in=160] (V2);
    \end{tikzpicture}
\caption{Diagrams used in \Cref{prop:nonredundancy_causal_graphs_in_leg}.}
\label{fig:non_uniqueness_MBD}
\end{figure}

Additionally, note that in general multiple MBD causal diagrams can be derived from with a single LEG. For example, \Cref{fig:non_uniqueness_MBD} gives two MBD causal diagrams induced by the same LEG (both bi-directed edges could not appear simultaneously as that would violate the independence $(V\indep W \mid X)_P$). Therefore, unfortunately, a reduction of the set of ME MBD diagrams $\1D$ while preserving the space of SCMs $\3M(\1D)$ is, in general, not possible. Multiple MBD diagrams for each LEG may have to be considered to properly characterize causal effects given a PAG $\1P$.

\begin{proposition}[Non-redundancy]
    \label{prop:nonredundancy_causal_graphs_in_leg}
    Given a PAG $\1P$, let $\G$ and $\1H$ be two MBD diagrams constructed from a ME LEG. In general, $\3M(\G)\not\subseteq \3M(\1H)$ and $\3M(\1H)\not\subseteq \3M(\1G)$. 
\end{proposition}

In other words, bounds for a given causal effect computed given two MBD diagrams will in general be different and one has to consider both diagrams to correctly characterize bounds on causal effects given an equivalence class. The proof proceeds by exploiting the MBD causal diagrams in \Cref{fig:non_uniqueness_MBD} as a counter-example. As a consequence of this result, we conjecture that enumeration techniques, in the worst-case, would have to consider all MBD diagrams separately. 

To better understand the computation cost of enumerating LEGs and MBD causal diagrams, we propose the first enumeration algorithms for this purpose in \Cref{alg:depth_first_search} and \Cref{alg:causal_graphs_from_LEG} in \Cref{sec:app_enumeration_strategies} (proceeding similarly to how one might enumerate Markov equivalent DAGs as done by \cite{wienobst2023efficient}). These (potentially sub-optimal) procedures suggest that doing enumerating ``relevant'' causal diagrams requires a polynomial cost in the number of edges of LEGs, in addition to the computational cost of the bounding algorithms themselves\footnote{For example, in \cite{zhang2021partial}, inference of bounds requires optimization over a number of parameters that is exponential in the number of variables.}. Overall, this analysis suggests that existing bounding techniques, even with the consideration of redundancies presented in this section, are not practical beyond a handful of variables \citep{zeitler2022causal}. Still, given that the proposed bounds (\Cref{sec:partial-identification}) have not been shown to be tight in general, one might still be interested in enumerating the MBD causal diagrams to get more informative bounds, despite computational costs.