One thing I am wondering is how you think about time vs. space in your analysis. If we think of all of physics as a very nonlinear dynamical system, then how do you move from that to these large causal networks you are drawing?
The equations of physics are generally local in both time and space. That actually makes causal networks a much more natural representation, in some ways, than nonlinear dynamical systems; dynamical systems don’t really have a built-in notion of spatial locality, whereas causal networks do. Indeed, one way to view causal networks is as the bare-minimum model in which we have both space-like and time-like interactions. So I don’t generally think about moving from dynamical systems to causal networks; I think about starting from causal networks.
This also fits well with how science works in a high-dimensional world like ours. Scientists don’t look at the state of the whole universe and try to figure out how that evolves into the next state. Rather, they look at spatially-localized chunks of the universe, and try to find sets of mediators which make the behavior of that chunk of the universe “reproducible”—i.e. independent of what’s going on elsewhere. These are Markov blankets.
Just to make sure I’m understanding the concept of causal networks with symmetry correctly, since I’m more used to thinking of dynamical systems: I could in principle think of a dynamical system that I simulate on my computer as a DAG with symmetry, ie using Euler’s method to simulate dx/dt = f(x) I get a difference equation x(t+1) = /delta T * f(x(t)) that I then use to simulate my dynamical system on a computer, and I can think of that as a DAG where x(t) /arrow x(t+1), for all t, and of course theres a symmetry over time since f(x(t)) is constant over time. If I have a spatially distributed dynamical system, like a network, then there might also be symmetries in space. In this way your causal networks with symmetry can capture any dynamical system (and I guess more since causal dependencies need not be deterministic)? Does that sound right?
The most fundamental ones are not. The useful , but not fundamental ones, are because there is not much you can do practically with nonlocal equations.
The equations of physics are generally local in both time and space. That actually makes causal networks a much more natural representation, in some ways, than nonlinear dynamical systems; dynamical systems don’t really have a built-in notion of spatial locality, whereas causal networks do. Indeed, one way to view causal networks is as the bare-minimum model in which we have both space-like and time-like interactions. So I don’t generally think about moving from dynamical systems to causal networks; I think about starting from causal networks.
This also fits well with how science works in a high-dimensional world like ours. Scientists don’t look at the state of the whole universe and try to figure out how that evolves into the next state. Rather, they look at spatially-localized chunks of the universe, and try to find sets of mediators which make the behavior of that chunk of the universe “reproducible”—i.e. independent of what’s going on elsewhere. These are Markov blankets.
The main piece which raw causal networks don’t capture is symmetry, e.g. the laws of physics staying the same over time. I usually picture the world in terms of causal networks with symmetry or, equivalently, causal submodels organized like programs.
Just to make sure I’m understanding the concept of causal networks with symmetry correctly, since I’m more used to thinking of dynamical systems: I could in principle think of a dynamical system that I simulate on my computer as a DAG with symmetry, ie using Euler’s method to simulate dx/dt = f(x) I get a difference equation x(t+1) = /delta T * f(x(t)) that I then use to simulate my dynamical system on a computer, and I can think of that as a DAG where x(t) /arrow x(t+1), for all t, and of course theres a symmetry over time since f(x(t)) is constant over time. If I have a spatially distributed dynamical system, like a network, then there might also be symmetries in space. In this way your causal networks with symmetry can capture any dynamical system (and I guess more since causal dependencies need not be deterministic)? Does that sound right?
That is exactly correct.
The most fundamental ones are not. The useful , but not fundamental ones, are because there is not much you can do practically with nonlocal equations.