In this post, I will compare multiple distinct versions of causal effect identification and discuss their equivalence.
We use $\mathtt{Q}$ to denote the causal query of interest, $\mathtt{M}$ to represent a causal model, $\mathcal{G}$ as a causal graph, $X,Y$ to indicate input and output variables, and $x,y$ for their respective realizations.
Pearl (2009) defines the causal effect identifiability as follows:
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Definition (Causal Effect Identifiability) [Pearl, 2009]
$\mathtt{Q}$ is said to be identifiable from $(\mathcal{G},P)$ induced by $\mathtt{M}$ if, for any pairs of models $\mathtt{M}_1$ and $\mathtt{M}2$ such that $P{\mathtt{M}1}$ and $P{\mathtt{M}2}$ are positive and $\mathcal{G}{\mathtt{M}1} = \mathcal{G}{\mathtt{M}_2}$ ,
$$ P_{\mathtt{M}1} = P{\mathtt{M}2} \quad \implies \quad P{\mathtt{M}1}(y \mid \operatorname{do}(x)) = P{\mathtt{M}_2}(y \mid \operatorname{do}(x)). $$
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On the other hands, Shpitser and Pearl (2008) or Huang and Valtota (2006) defines the identifiability in a following manner:
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Definition (Causal Effect Identifiability) [Shpitser and Pearl, Huang and Valtota]
$\mathtt{Q}$ is said to be identifiable from $(\mathcal{G},P)$ induced by $\mathtt{M}$ if $\mathtt{Q}$ can be computed uniquely from any positive probability $P_{\mathtt{M}}$.
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In this document, I will relate these two definitions.
We first recap the one-to-one function.
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Definition (Injective Function or one-to-one function)
A function $f: \mathbb{Q} \mapsto \mathbb{P}$ is said to be injective if and only if $f(q)=f(q')$ implies $q=q'$ for any $q,q' \in \mathbb{Q}$ and $f(q),f(q') \in \mathbb{P}$. Equivalently, a function $f$ is injective if and only if there is a function $g$ such that $g(f(q)) = q$ for any $q \in \mathbb{Q}$.
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To establish a one-to-one relation between these two definitions, let us first fix $(\mathcal{G}, P)$ as our given causal graph and sampling distribution. Then, define the followings:
Consider a mapping $f$ between $\mathcal{Q}{\mathcal{M}{\mathcal{G},P}}$ and $\mathcal{P}{\mathcal{M}{\mathcal{G},P}}$; i.e., a mapping from a query space to the observational distribution space. Since $\mathcal{P}{\mathcal{M}{\mathcal{G},P}} = \{P\}$, $f(\mathtt{Q}) = P$ for all $\mathtt{Q} \in \mathcal{Q}{\mathcal{M}{\mathcal{G},P}}$. Then,
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Claim
The followings are equivalent:
(3) $\Leftrightarrow$ (1)
Suppose (3) holds. Given that $f(\mathtt{Q}1) = f(\mathtt{Q}2)$ because $\mathcal{P}{\mathcal{M}{\mathcal{G},P}} = \{P\}$, the function $f$ is injective if and only if $\mathtt{Q}1 = \mathtt{Q}2$ (by definition). This is precisely stated by Pearl’s definition of identifiability in (1). In other words, $f$ being injective means that $\mathtt{Q}{\mathcal{M}{\mathcal{G},P}}$ contains the same query; i.e., $\mathtt{Q}{\mathcal{M}{\mathcal{G},P}} = \{\mathtt{Q}\}$. That is, all causal models that induces $(\mathcal{G},P)$ induces the same query $\mathtt{Q}$.
(3) $\Leftrightarrow$ (2)
Using the equivalent definition of the injectivity of $f$, there exists a unique function $g$ such that $g(f(\mathtt{Q})) = \mathtt{Q}$ for all $\mathtt{Q} \in \mathcal{Q}{\mathcal{M}{\mathcal{G},P}}$. Since $f(\mathtt{Q}) = P$, for all $\mathtt{Q} \in \mathcal{Q}{\mathcal{M}{\mathcal{G},P}}$, this reduces to state that there exists a unique function $g$ such that $g(P) = \mathtt{Q}$ for all $\mathtt{Q} \in \mathcal{Q}{\mathcal{M}{\mathcal{G},P}}$. This implies that $\mathtt{Q}{\mathcal{M}{\mathcal{G},P}} = \{\mathtt{Q}\}$. That is, there exists a unique mapping from $P$ to $\mathtt{Q}$.