Maybe the individual trajectories of such a system could be described as solutions to some stochastic differential equation?

Also by analogy with continuous-time Markov processes, it might be easier to forget about individual trajectories and instead think about the "flow" of probability density in phase space, which can probably be described by a partial differential equation without needing to define your own Riemann sums, path integrals and such. Or maybe I'm missing something here and you really need customized machinery?

Also Cosma Shalizi is an expert on both causal models and continuous-time stochastic processes, so maybe you could ask him or look at his work if you haven't seen it already.

May I suggest working out the graphical model version of continuous time Markov chains as an intermediate step

(e.g. something like this: http://boa.unimib.it/bitstream/10281/19575/1/phd_unimib_040750.pdf)

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).

Thoughts?

LaTeX in Less Wrong

Also, the wiki page on using LaTeX in Less Wrong.

It would be ideal if there were a script somewhere that eats an austere LaTeX file and spits out an html file.

Time is real, so I'm not a fan of timelessness. However, while you have a conservative flow in your model, you can still work your way back to histories and thus to the reality of time within a history.

Consider how Bohmian mechanics does it. You have the Schrodinger evolution of a wavefunction, with a conserved flow of probability density. If you chart a trajectory through configuration space according to the gradient of the phase of the wavefunction, you end up with a timelike foliation of (configuration space x time) with nonintersecting trajectories. Add the usual probability measure, and voila, you have a multiverse theory of self-contained worlds which neither split nor merge, and in which the Born rule applies.

In principle you can do the same thing with one of those timeless-looking "wavefunctions of the universe" which show up in quantum cosmology. Here, instead of H psi = i.hbar dpsi/dt, you just have H psi = 0 (where the Hamiltonian is general relativity coupled to other fields). So instead of an evolving wavefunction on a "configuration space of the universe", you just have a static wavefunction. But you can still take the gradient of psi's phase, everywhere in that configuration space, and so you can figure out Bohmian trajectories that divide up the universal configuration space into disjoint self-contained histories.

In practice, things are more complicated. In general relativity, you distinguish between coordinate time and physical time (proper time). The proper time which elapses along a specific timelike curve is an invariant, an objective quantity. But it is calculated from a metric, the exact form of which depends on the coordinate system. You can rescale coordinate time, according to some diffeomorphism, but then you adjust the metric accordingly, so that distances, angles, and durations remain the same. If you actually try to follow the program of Bohmian quantum gravity that I outlined, it's hard to define the wavefunction of the universe without reifying a particular coordinate time, a step which is just like having a preferred frame in special relativity. I suspect that the answer lies in string theory's holographic principle, which says that a quantum theory containing gravity is equivalent to another quantum theory that doesn't contain gravity, and which is defined on the boundary of the space inhabited by the first theory. In terms of this second theory, the space away from the boundary is emergent, it's made of composite degrees of freedom from the boundary theory. In the real world, it's going to be time which is "emergent", from the "renormalization group flow" of a Euclidean field theory defined at "past infinity". In fact, excuse me while I run away and study the Bohmian trajectories for such a theory...

Anyway, bringing this back to Judea Pearl: As soon as mathematics represents a history as a "trajectory" in a state space, it is already becoming a little "timeless" in a formal sense. Consider something as simple as a time series. You can plot it on a graph and now it's a shape rather than a process. You can specify its properties in a timeless geometrical fashion, even though one of the directions on the graph represents time. In talking about flows on state spaces, I don't think you're doing more than this. So what you're doing is harmless, from the perspective of a time-realist like myself, but it also doesn't really embody the full revolution of Julian Barbour's ontological timelessness, which necessarily involves both general relativity and quantum mechanics. General relativity makes proper time a physical variable, and quantum mechanics matters by way of many worlds: Barbour's multiverse is one of "many moments" (he calls them time capsules). In order to interpret an unevolving wavefunction of the universe, rather than divide it up into trajectories, he completely pulverizes it into moments, one moment for each point in configuration space.

If you want to imitate Barbour's timelessness, then the crucial step is the ontological one of denying that the moments have a unique past or future. But if you have a conservative causal flow, you can always string the moments together into specific histories, like the Bohmian trajectories. Technical difficulties for the definition of trajectories only enter for relativistic systems, because you want to avoid reifying a particular coordinate time. But for nonrelativistic systems, it looks like formal timelessness in a causal model (in the sense you describe) is just a change of perspective that's always available and can always be reversed.