Thanks for taking the time to think/comment. It may help us to fix a reference which describes Pearl's thinking and his calculus. There are several of his papers available online, but this one is pretty comprehensive: ftp://ftp.cs.ucla.edu/pub/stat_ser/r284-reprint.pdf "Bayesianism and Causality, Or, Why I am only a Half-Bayesian".
Now onto your points:
1) You are correct that nothing in Pearl's calculus varies depending on the number of variables Yi which causally depend on X. For any number of Yi, the intervention do(Z = 1) breaks all the links between X and the Yi and doesn't change the vale of X at all. Also, there is no "paradox" within Pearl's calculus here: it is internally consistent.
The real difficulty is that the calculus just doesn't work as a full conceptual analysis of counterfactuals, and this becomes increasingly clear the more Yi variables we add. It is a bit unfortunate, because while the calculus is elegant in its own terms, it does appears that conceptual analysis is what Pearl was attempting. He really did intend his "do" calculus to reflect how we usually understand counterfactuals, only stated more precisely. Pearl was not consciously proposing a "revisionist" account to the effect: "This is how I'm going to define counterfactuals for the sake of getting some math to work. If your existing definition or intuition about counterfactuals doesn't match that definition, then sorry, but it still won't affect my definition." Accordingly, it doesn't help to say "Regular intuitions say one thing, Pearl's calculus says another, but the calculus is better, therefore the calculus is right and intuitions are wrong". You can get away with that in revisionist accounts/definitions but not in regular conceptual analysis.
2) The structural equations do indeed imply there is a causal link from the X to the Yi. But there is NO causal link in the opposite direction from the Yi to the X, or from any Yi to any Yj. The causal graph is directed, and the structural equations are asymmetric. Note that in Pearl's models, the structural equation Yi = X is different from the reverse structural equation X = Yi, even though in regular logic and probability theory these are equivalent. This point is really quite essential to Pearl's treatment, and is made clear by the referenced document.
3) See point 1. Pearl's calculus is trying to analyse counterfactuals (and causal relations) as we usually understand them, not to propose a revisionist account. So evidence about how we (naturally) interpret counterfactuals (in both the Gore case and the X, Y case) is entirely relevant here.
Incidentally, if you want my one sentence view, I'd say that Pearl is correctly analysing a certain sort of counterfactual but not the general sort he thinks he is analysing. Consider these two counterfactuals:
If A were to happen, then B would happen.
If A were to be made to happen (by outside intervention) then B would happen.
I believe that these are different counterfactuals, with different antecedents, and so they can have different truth values. It looks to me like Pearl's "do" calculus correctly analyses the second sort of counterfactual, but not the first.
(Edited this comment to fix typos and a broken reference.)
Okay. So according to Causality (first edition, cause I'm poor), Theorem 7.1.7, the algorithm for calculating the counterfactual P( (Y= y)_(X = x) | e) -- which represents the statement "If X were x, then Y would be y, given evidence e" -- has three stages:
Abduction; use the probability distribution P(x, y| E = e).
Action; perform do(X = x).
Calculate p(Y = y) relative to the new graph model and its updated joint probability distribution.
In our specific case, we want to calculate P (X = 0_(Z = 1)). There's no evidence to condition on, so a...
Followup to: Probability is Subjectively Objective
The classic explanation of counterfactuals begins with this distinction:
In ordinary usage we would agree with the first statement, but not the second (I hope).
If, somehow, we learn the definite fact that Oswald did not shoot Kennedy, then someone else must have done so, since Kennedy was in fact shot.
But if we went back in time and removed Oswald, while leaving everything else the same, then—unless you believe there was a conspiracy—there's no particular reason to believe Kennedy would be shot:
We start by imagining the same historical situation that existed in 1963—by a further act of imagination, we remove Oswald from our vision—we run forward the laws that we think govern the world—visualize Kennedy parading through in his limousine—and find that, in our imagination, no one shoots Kennedy.
It's an interesting question whether counterfactuals can be true or false. We never get to experience them directly.
If we disagree on what would have happened if Oswald hadn't been there, what experiment could we perform to find out which of us is right?
And if the counterfactual is something unphysical—like, "If gravity had stopped working three days ago, the Sun would have exploded"—then there aren't even any alternate histories out there to provide a truth-value.
It's not as simple as saying that if the bucket contains three pebbles, and the pasture contains three sheep, the bucket is true.
Since the counterfactual event only exists in your imagination, how can it be true or false?
So... is it just as fair to say that "If Oswald hadn't shot Kennedy, the Sun would have exploded"?
After all, the event only exists in our imaginations—surely that means it's subjective, so we can say anything we like?
But so long as we have a lawful specification of how counterfactuals are constructed—a lawful computational procedure—then the counterfactual result of removing Oswald, depends entirely on the empirical state of the world.
If there was no conspiracy, then any reasonable computational procedure that simulates removing Oswald's bullet from the course of history, ought to return an answer of Kennedy not getting shot.
"Reasonable!" you say. "Ought!" you say.
But that's not the point; the point is that if you do pick some fixed computational procedure, whether it is reasonable or not, then either it will say that Kennedy gets shot, or not, and what it says will depend on the empirical state of the world. So that, if you tell me, "I believe that this-and-such counterfactual construal, run over Oswald's removal, preserves Kennedy's life", then I can deduce that you don't believe in the conspiracy.
Indeed, so long as we take this computational procedure as fixed, then the actual state of the world (which either does include a conspiracy, or does not) presents a ready truth-value for the output of the counterfactual.
In general, if you give me a fixed computational procedure, like "multiply by 7 and add 5", and then you point to a 6-sided die underneath a cup, and say, "The result-of-procedure is 26!" then it's not hard at all to assign a truth value to this statement. Even if the actual die under the cup only ever takes on the values between 1 and 6, so that "26" is not found anywhere under the cup. The statement is still true if and only if the die is showing 3; that is its empirical truth-condition.
And what about the statement ((3 * 7) + 5) = 26? Where is the truth-condition for that statement located? This I don't know; but I am nonetheless quite confident that it is true. Even though I am not confident that this 'true' means exactly the same thing as the 'true' in "the bucket is 'true' when it contains the same number of pebbles as sheep in the pasture".
So if someone I trust—presumably someone I really trust—tells me, "If Oswald hadn't shot Kennedy, someone else would have", and I believe this statement, then I believe the empirical reality is such as to make the counterfactual computation come out this way. Which would seem to imply the conspiracy. And I will anticipate accordingly.
Or if I find out that there was a conspiracy, then this will confirm the truth-condition of the counterfactual—which might make a bit more sense than saying, "Confirm that the counterfactual is true."
But how do you actually compute a counterfactual? For this you must consult Judea Pearl. Roughly speaking, you perform surgery on graphical models of causal processes; you sever some variables from their ordinary parents and surgically set them to new values, and then recalculate the probability distribution.
There are other ways of defining counterfactuals, but I confess they all strike me as entirely odd. Even worse, you have philosophers arguing over what the value of a counterfactual really is or really means, as if there were some counterfactual world actually floating out there in the philosophical void. If you think I'm attacking a strawperson here, I invite you to consult the philosophical literature on Newcomb's Problem.
A lot of philosophy seems to me to suffer from "naive philosophical realism"—the belief that philosophical debates are about things that automatically and directly exist as propertied objects floating out there in the void.
You can talk about an ideal computation, or an ideal process, that would ideally be applied to the empirical world. You can talk about your uncertain beliefs about the output of this ideal computation, or the result of the ideal process.
So long as the computation is fixed, and so long as the computational itself is only over actually existent things. Or the results of other computations previously defined—you should not have your computation be over "nearby possible worlds" unless you can tell me how to compute those, as well.
A chief sign of naive philosophical realism is that it does not tell you how to write a computer program that computes the objects of its discussion.
I have yet to see a camera that peers into "nearby possible worlds"—so even after you've analyzed counterfactuals in terms of "nearby possible worlds", I still can't write an AI that computes counterfactuals.
But Judea Pearl tells me just how to compute a counterfactual, given only my beliefs about the actual world.
I strongly privilege the real world that actually exists, and to a slightly lesser degree, logical truths about mathematical objects (preferably finite ones). Anything else you want to talk about, I need to figure out how to describe in terms of the first two—for example, as the output of an ideal computation run over the empirical state of the real universe.
The absence of this requirement as a condition, or at least a goal, of modern philosophy, is one of the primary reasons why modern philosophy is often surprisingly useless in my AI work. I've read whole books about decision theory that take counterfactual distributions as givens, and never tell you how to compute the counterfactuals.
Oh, and to talk about "the probability that John F. Kennedy was shot, given that Lee Harvey Oswald didn't shoot him", we write:
And to talk about "the probability that John F. Kennedy would have been shot, if Lee Harvey Oswald hadn't shot him", we write:
That little symbol there is supposed to be a box with an arrow coming out of it, but I don't think Unicode has it.
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