Body: <p>You can find the afore-mentioned text <a href="https://causalai.net/r60.pdf" rel="nofollow noreferrer">here</a>. I'm asking a clarification of Definition 17, which is as follows:</p>
  <blockquote>
  <p><strong>Definition 17 (Effect Identifiability)</strong> The causal effect of an action <span class="math-container">$do(X=x)$</span> on a set of variables <span class="math-container">$Y$</span> given a set of observations on variables <span class="math-container">$Z = z$</span>, probability <span class="math-container">$P(Y | do(x), z)$</span> is said to be identifiable from <span class="math-container">$P$</span> and <span class="math-container">$G$</span> if for every two models <span class="math-container">$M^{(1)}$</span> and <span class="math-container">$M^{(2)}$</span> with causal diagram <span class="math-container">$G$</span>, equality <span class="math-container">$P^{(1)}(v)=P^{(2)}(v)&gt;0$</span> implies <span class="math-container">$P^{(1)}(Y | do(x), z) = P^{(2)}(Y | do(x), z)$</span>.</p>
  </blockquote>
  <p>I'm a bit caught off guard by the <span class="math-container">$P^{(1)}(v)=P^{(2)}(v)&gt;0$</span> condition, does it really mean that if equal measure in any single event implies equal interventional probability, then the effect is identifiable? I'm confused, as I would think that <span class="math-container">$P^{(1)}(z)=P^{(2)}(z)&gt;0$</span> would be a more reasonable requirement. I checked several different versions of the text and they all say the former.</p>
  <p><strong>Question:</strong> Can someone please verify if the definition is indeed correct in using any event <span class="math-container">$v$</span> instead of <span class="math-container">$z$</span>? If possible I would also appreciate an intuitive reasoning as to why equality in any event is enough, instead of equality on <span class="math-container">$P^{(1)}(z)=P^{(2)}(z)&gt;0$</span> as a condition.</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-10-08T10:06:23.297Z
Comments.Text: Is this the correct SE to ask this question? I'm more accustomed to stats.SE but I've never quite figured the difference between ai.SE, stats.SE and datascience.SE
Comments.UserDisplayName: AyamGorengPedes
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-10-08T17:58:43.060Z
Comments.Text: Maybe you could remind us what $v$ means in this context? (since it isn't in the definition)
Comments.UserDisplayName: bogovicj
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-10-09T11:43:22.350Z
Comments.Text: @bogovicj, $V$ is the set of observed variables (variables included in the model. In this case, say given $\bar{V}\subseteq V$, then I meant $P^{(1)}(\bar{V}=v)=P^{(2)}(\bar{V}=v)$, so agreement of the two probabilities on any single event.
Comments.UserDisplayName: AyamGorengPedes
Tags: <papers>
Title: Definition of Effect Identifiability in 'On Pearl's Hierarchy and Foundations of Causal Inference'
UserDisplayName: AyamGorengPedes
ThreadText: # Definition of Effect Identifiability in 'On Pearl's Hierarchy and Foundations of Causal Inference'
  
  (Asked by: username_0 on 2024-10-08)
  
  You can find the afore-mentioned text [here](https://causalai.net/r60.pdf). I'm asking a clarification of Definition 17, which is as follows:
  
  > **Definition 17 (Effect Identifiability)** The causal effect of an action $do(X=x)$ on a set of variables $Y$ given a set of observations on variables $Z = z$, probability $P(Y | do(x), z)$ is said to be identifiable from $P$ and $G$ if for every two models $M^{(1)}$ and $M^{(2)}$ with causal diagram $G$, equality $P^{(1)}(v)=P^{(2)}(v)>0$ implies $P^{(1)}(Y | do(x), z) = P^{(2)}(Y | do(x), z)$.
  
  I'm a bit caught off guard by the $P^{(1)}(v)=P^{(2)}(v)>0$ condition, does it really mean that if equal measure in any single event implies equal interventional probability, then the effect is identifiable? I'm confused, as I would think that $P^{(1)}(z)=P^{(2)}(z)>0$ would be a more reasonable requirement. I checked several different versions of the text and they all say the former.
  
  **Question:** Can someone please verify if the definition is indeed correct in using any event $v$ instead of $z$? If possible I would also appreciate an intuitive reasoning as to why equality in any event is enough, instead of equality on $P^{(1)}(z)=P^{(2)}(z)>0$ as a condition.
  
  **Tags:** papers
  
  > **Comments:**
  > **username_0**: Is this the correct SE to ask this question? I'm more accustomed to stats.SE but I've never quite figured the difference between ai.SE, stats.SE and datascience.SE  
  > **username_1**: Maybe you could remind us what $v$ means in this context? (since it isn't in the definition)  
  > **username_0**: @username_1, $V$ is the set of observed variables (variables included in the model. In this case, say given $\bar{V}\subseteq V$, then I meant $P^{(1)}(\bar{V}=v)=P^{(2)}(\bar{V}=v)$, so agreement of the two probabilities on any single event.  
  
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