This looks really interesting. It is still rather sketchy, though, and my physics is not good enough to be confident about how to fill in the details. Would you mind sending along your pdf?
One minor quibble and then a more general remark:
The somewhat harder bullet to swallow is the assumption that the random variables are Markovian; that is, they are “memoryless” in the sense that only the present state determines the future.
This is not quite correct as a description of the SGS-Pearl system. For SGS-Pearl, a causal system is Markovian relative to a graph, not necessarily relative to time. That is, the graphical parents of a variable screen that variable off from all of its non-descendants. Assume a discrete-time model. There is no requirement that the graphical parents of a given variable live in the immediately previous time-slice. We could have delayed, direct causation. For example, if we had X(t=T) --> X(t=T+2) and also Y(t=T+1) --> X(t=T+2), where no other variable is a direct cause of X(t=T+2). The variable X(t=T+2) is independent of its non-descendants given X(t=T) and Y(t=T+1).
If you assume that all genuine, fully unrolled causal connections have the same time-interval, then your version of the Markov condition lines up with SGS-Pearl.
On its face, stating the usual global causal Markov axiom—but assuming densely-ordered times, rather than non-densely-ordered times, is non-obvious. Since in such a system no variable (graph-vertex) will have any specifiable direct causes (parents), you can’t simply say that a variable is independent of its non-descendants conditional on its parents.
You also can’t just condition on a time-slice of ancestors (even assuming that such a slice screens off the past from the future), since there might be branching from a variable that lives between whatever slice you pick and the target variable. That is, suppose the times are densely ordered, and your target variable Z lives at time T2. Now, suppose that you condition on all the variables X at time T1 < T2. Since the ordering is dense, there are times between T1 and T2. For all we know, there might be a variable X(unlucky) that lives at one of the times between T1 and T2 such that X(unlucky) is a common cause of Z and some non-ancestral, non-descendant of Z, call it Y. In that case, we expect Y and Z to be associated in virtue of the common cause X(unlucky).
I’ll find somewhere to stick it and link it to the post.
The somewhat harder bullet to swallow is the assumption that the random variables are Markovian; that is, they are “memoryless” in the sense that only the present state determines the future.
This is not quite correct as a description of the SGS-Pearl system.
Yeah, sloppy writing on my part; the “time” that appears here is only an observer’s sequence of observations of the system’s state. I agree with what you say about discrete-time models.
On its face, stating the usual global causal Markov axiom—but assuming densely-ordered times, rather than non-densely-ordered times, is non-obvious.
What was assumed there is not that, because in this development I have not gotten to the point of finding a directed graph of variables anywhere. Presumably I’ll need a local corollary of Pearl’s theorem 1.4.1, i.e., every causal flow model (probably subject to some technical restrictions) locally induces a graph model with a compatible joint probability distribution. This has some hope of succeeding; if a distribution is consistent with a graph model, then small perturbations of it are also consistent with it.
Sorry, my real life job intervened and killed most of this week. There’s a slight kink in the current draft that I need to rewrite (it’s not a game-breaker), but that’ll have to wait until I get some free time. I also need to find some free webspace somewhere; it seems that the Megaupload debacle killed all the free filesending platforms.
My attempts at finding semi-stable webspace failed. In the meanwhile, for the sake of Nisan’s razor, here is a temporary link to the .pdf. It hasn’t been fixed yet; the ending is not very rigorous. I probably got a bit too excited near the end.
This looks really interesting. It is still rather sketchy, though, and my physics is not good enough to be confident about how to fill in the details. Would you mind sending along your pdf?
One minor quibble and then a more general remark:
This is not quite correct as a description of the SGS-Pearl system. For SGS-Pearl, a causal system is Markovian relative to a graph, not necessarily relative to time. That is, the graphical parents of a variable screen that variable off from all of its non-descendants. Assume a discrete-time model. There is no requirement that the graphical parents of a given variable live in the immediately previous time-slice. We could have delayed, direct causation. For example, if we had X(t=T) --> X(t=T+2) and also Y(t=T+1) --> X(t=T+2), where no other variable is a direct cause of X(t=T+2). The variable X(t=T+2) is independent of its non-descendants given X(t=T) and Y(t=T+1).
If you assume that all genuine, fully unrolled causal connections have the same time-interval, then your version of the Markov condition lines up with SGS-Pearl.
On its face, stating the usual global causal Markov axiom—but assuming densely-ordered times, rather than non-densely-ordered times, is non-obvious. Since in such a system no variable (graph-vertex) will have any specifiable direct causes (parents), you can’t simply say that a variable is independent of its non-descendants conditional on its parents.
You also can’t just condition on a time-slice of ancestors (even assuming that such a slice screens off the past from the future), since there might be branching from a variable that lives between whatever slice you pick and the target variable. That is, suppose the times are densely ordered, and your target variable Z lives at time T2. Now, suppose that you condition on all the variables X at time T1 < T2. Since the ordering is dense, there are times between T1 and T2. For all we know, there might be a variable X(unlucky) that lives at one of the times between T1 and T2 such that X(unlucky) is a common cause of Z and some non-ancestral, non-descendant of Z, call it Y. In that case, we expect Y and Z to be associated in virtue of the common cause X(unlucky).
Thoughts?
I’ll find somewhere to stick it and link it to the post.
Yeah, sloppy writing on my part; the “time” that appears here is only an observer’s sequence of observations of the system’s state. I agree with what you say about discrete-time models.
What was assumed there is not that, because in this development I have not gotten to the point of finding a directed graph of variables anywhere. Presumably I’ll need a local corollary of Pearl’s theorem 1.4.1, i.e., every causal flow model (probably subject to some technical restrictions) locally induces a graph model with a compatible joint probability distribution. This has some hope of succeeding; if a distribution is consistent with a graph model, then small perturbations of it are also consistent with it.
This is what I meant to assume: that X is a continuous-time markov process.
Ah! Yes, that makes sense. I’m looking forward to reading the paper.
Sorry, my real life job intervened and killed most of this week. There’s a slight kink in the current draft that I need to rewrite (it’s not a game-breaker), but that’ll have to wait until I get some free time. I also need to find some free webspace somewhere; it seems that the Megaupload debacle killed all the free filesending platforms.
My attempts at finding semi-stable webspace failed. In the meanwhile, for the sake of Nisan’s razor, here is a temporary link to the .pdf. It hasn’t been fixed yet; the ending is not very rigorous. I probably got a bit too excited near the end.