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I'm just starting to learn about causal inference methods, focused on Pearl's do-calculus. So the point between Pearl's causal graphs and rules for manipulating causal graphs appears to be to turn a causal graph into a statistical model (e.g. a linear regression).

So you might have a causal graph such as $Z \rightarrow X, X \rightarrow M, M \rightarrow Y, Z \rightarrow Y$ (Z is a confounder of Z and Y, but X also partially causes Y through M). If my aim is to figure out the causal effect of X on Y and I just did the naive thing and setup a linear regression with $Y = aX + e$ and tried to estimate a regression coefficient, then of course I would get a biased estimate due to the presence of the confounder Z. On the other hand, if I do $Y = aX + bM + cZ + e$, then I will block the effect of X by conditioning on the mediator M. So again, knowing the causal graph will tell me to condition on Z but not on M, i.e. $Y = aX + cZ + e$ is the correct statistical model that allows me to estimate a causal effect.

But is $Y = aX + cZ + e$ (as a regression model, not a math equation) also a causal model (albeit a "wrong" causal model)? If I manipulate $X$ it tells me what happens to $Y$. Doesn't it correspond to the causal graph $X \rightarrow Y, Z \rightarrow Y$ ? If so, then is the Pearl method just finding a transform of a causal graph into another causal graph that is easier to work with or represent as a regression?

edit: I think my causal graph analysis in this simple example is wrong, but hopefully the broader point is still clear

edit #2:

If I write a model $Y + aX + bZ + e$ in a programming language, I could do so as a function, e.g. in Python

 def model(a,X,c,Z):
     return a*X + b*Z + np.random.randn()

So if I change the input $X$, it will cause the output of $model(...)$ to change, but I cannot do the opposite. Isn't that a causal model?

I'm just starting to learn about causal inference methods, focused on Pearl's do-calculus. So the point between Pearl's causal graphs and rules for manipulating causal graphs appears to be to turn a causal graph into a statistical model (e.g. a linear regression).

So you might have a causal graph such as $Z \rightarrow X, X \rightarrow M, M \rightarrow Y, Z \rightarrow Y$ (Z is a confounder of X and Y, but X also partially causes Y through M).

If my aim is to figure out the causal effect of X on Y and I just did the naive thing and setup a linear regression with $Y = aX + e$ and tried to estimate a regression coefficient, then of course I would get a biased estimate due to the presence of the confounder Z. On the other hand, if I do $Y = aX + bM + cZ + e$, then I will block the effect of X by conditioning on the mediator M. So again, knowing the causal graph will tell me to condition on Z but not on M, i.e. $Y = aX + cZ + e$ is the correct statistical model that allows me to estimate a causal effect.

But is $Y = aX + cZ + e$ (as a regression model, not a math equation) also a causal model (albeit a "wrong" causal model)? If I manipulate $X$ it tells me what happens to $Y$. Doesn't it correspond to the causal graph $X \rightarrow Y, Z \rightarrow Y$ ? If so, then is the Pearl method just finding a transform of a causal graph into another causal graph that is easier to work with or represent as a regression?

edit: I think my causal graph analysis in this simple example is wrong, but hopefully the broader point is still clear

edit #2:

If I write a model $Y + aX + bZ + e$ in a programming language, I could do so as a function, e.g. in Python

 def model(a,X,c,Z):
     return a*X + b*Z + np.random.randn()

So if I change the input $X$, it will cause the output of $model(...)$ to change, but I cannot do the opposite. Isn't that a causal model?

I'm just starting to learn about causal inference methods, focused on Pearl's do-calculus. So the point between Pearl's causal graphs and rules for manipulating causal graphs appears to be to turn a causal graph into a statistical model (e.g. a linear regression).

So you might have a causal graph such as $Z \rightarrow X, X \rightarrow M, M \rightarrow Y, Z \rightarrow Y$ (Z is a confounder of Z and Y, but X also partially causes Y through M). If my aim is to figure out the causal effect of X on Y and I just did the naive thing and setup a linear regression with $Y = aX + e$ and tried to estimate a regression coefficient, then of course I would get a biased estimate due to the presence of the confounder Z. On the other hand, if I do $Y = aX + bM + cZ + e$, then I will block the effect of X by conditioning on the mediator M. So again, knowing the causal graph will tell me to condition on Z but not on M, i.e. $Y = aX + cZ + e$ is the correct statistical model that allows me to estimate a causal effect.

But is $Y = aX + cZ + e$ (as a regression model, not a math equation) also a causal model (albeit a "wrong" causal model)? If I manipulate $X$ it tells me what happens to $Y$. Doesn't it correspond to the causal graph $X \rightarrow Y, Z \rightarrow Y$ ? If so, then is the Pearl method just finding a transform of a causal graph into another causal graph that is easier to work with or represent as a regression?

edit: I think my causal graph analysis in this simple example is wrong, but hopefully the broader point is still clear

I'm just starting to learn about causal inference methods, focused on Pearl's do-calculus. So the point between Pearl's causal graphs and rules for manipulating causal graphs appears to be to turn a causal graph into a statistical model (e.g. a linear regression).

So you might have a causal graph such as $Z \rightarrow X, X \rightarrow M, M \rightarrow Y, Z \rightarrow Y$ (Z is a confounder of Z and Y, but X also partially causes Y through M). If my aim is to figure out the causal effect of X on Y and I just did the naive thing and setup a linear regression with $Y = aX + e$ and tried to estimate a regression coefficient, then of course I would get a biased estimate due to the presence of the confounder Z. On the other hand, if I do $Y = aX + bM + cZ + e$, then I will block the effect of X by conditioning on the mediator M. So again, knowing the causal graph will tell me to condition on Z but not on M, i.e. $Y = aX + cZ + e$ is the correct statistical model that allows me to estimate a causal effect.

But is $Y = aX + cZ + e$ (as a regression model, not a math equation) also a causal model (albeit a "wrong" causal model)? If I manipulate $X$ it tells me what happens to $Y$. Doesn't it correspond to the causal graph $X \rightarrow Y, Z \rightarrow Y$ ? If so, then is the Pearl method just finding a transform of a causal graph into another causal graph that is easier to work with or represent as a regression?

edit: I think my causal graph analysis in this simple example is wrong, but hopefully the broader point is still clear

edit #2:

If I write a model $Y + aX + bZ + e$ in a programming language, I could do so as a function, e.g. in Python

 def model(a,X,c,Z):
     return a*X + b*Z + np.random.randn()

So if I change the input $X$, it will cause the output of $model(...)$ to change, but I cannot do the opposite. Isn't that a causal model?

I'm just starting to learn about causal inference methods, focused on Pearl's do-calculus. So the point between Pearl's causal graphs and rules for manipulating causal graphs appears to be to turn a causal graph into a statistical model (e.g. a linear regression).

So you might have a causal graph such as $Z \rightarrow X, X \rightarrow M, M \rightarrow Y, Z \rightarrow Y$ (Z is a confounder of Z and Y, but X also partially causes Y through M). If my aim is to figure out the causal effect of X on Y and I just did the naive thing and setup a linear regression with $Y = aX + e$ and tried to estimate a regression coefficient, then of course I would get a biased estimate due to the presence of the confounder Z. On the other hand, if I do $Y = aX + bM + cZ + e$, then I will block the effect of X by conditioning on the mediator M. So again, knowing the causal graph will tell me to condition on Z but not on M, i.e. $Y = aX + cZ + e$ is the correct statistical model that allows me to estimate a causal effect.

But is $Y = aX + cZ + e$ (as a regression model, not a math equation) also a causal model (albeit a "wrong" causal model)? If I manipulate $X$ it tells me what happens to $Y$. Doesn't it correspond to the causal graph $X \rightarrow Y, Z \rightarrow Y$ ? If so, then is the Pearl method just finding a transform of a causal graph into another causal graph that is easier to work with or represent as a regression?

I'm just starting to learn about causal inference methods, focused on Pearl's do-calculus. So the point between Pearl's causal graphs and rules for manipulating causal graphs appears to be to turn a causal graph into a statistical model (e.g. a linear regression).

So you might have a causal graph such as $Z \rightarrow X, X \rightarrow M, M \rightarrow Y, Z \rightarrow Y$ (Z is a confounder of Z and Y, but X also partially causes Y through M). If my aim is to figure out the causal effect of X on Y and I just did the naive thing and setup a linear regression with $Y = aX + e$ and tried to estimate a regression coefficient, then of course I would get a biased estimate due to the presence of the confounder Z. On the other hand, if I do $Y = aX + bM + cZ + e$, then I will block the effect of X by conditioning on the mediator M. So again, knowing the causal graph will tell me to condition on Z but not on M, i.e. $Y = aX + cZ + e$ is the correct statistical model that allows me to estimate a causal effect.

But is $Y = aX + cZ + e$ (as a regression model, not a math equation) also a causal model (albeit a "wrong" causal model)? If I manipulate $X$ it tells me what happens to $Y$. Doesn't it correspond to the causal graph $X \rightarrow Y, Z \rightarrow Y$ ? If so, then is the Pearl method just finding a transform of a causal graph into another causal graph that is easier to work with or represent as a regression?

edit: I think my causal graph analysis in this simple example is wrong, but hopefully the broader point is still clear