I no longer think the setup above is viable, for reasons that connect to why I think Critch’s operationalization is incomplete and why boundaries should ultimately be grounded in Pearlian Causality and interventions.
(Note: I am thinking as I’m writing, so this might be a bit rambly.)
The world-trajectory distribution is ambiguous.
Intuition: Why does a robust glider in Lenia intuitively feel like a system possessing boundary? Well, I imagine various situations that happen in the world (like bullets) and this pattern mostly stays stable in face of them.
Now, notice that the measure of infiltration/exfiltration depends on ϕ∈Δ(Wω), a distribution over world history. Infil(ϕ):=Aggt≥0MutWω∼ϕ((Vt+1,At+1);Et∣(Vt,At,Pt))
So, for the above measure to capture my intuition, the approximate Markov condition (operationalized by low infil & exfil) must consider the world state Wω that contains the Lenia pattern with it avoiding bullets.
Remember, W is the raw world state, no coarse graining. So ϕ is the distribution over the raw world trajectory. It already captures all the “potentially occurring trajectories under which the system may take boundary-preserving-action.” Since everything is observed, our distribution already encodes all of “Nature’s Intervention.” So in some sense Critch’s definition is already causal (in a very trivial sense), by the virtue of requiring a distribution over the raw world trajectory, despite mentioning no Pearlian Causality.
Issue: Choice of ϕ
Maybe there is some canonical true ϕ for our physical world that minds can intersubjectively arrive at, so there’s no ambiguity.
But when I imagine trying to implement this scheme on Lenia, there’s immediately an ambiguity as to which distribution (representing my epistemic state on which raw world trajectories that will “actually happen”) we should choose:
Perhaps a very simple distribution: assigning uniform probability over world trajectories where the world contains nothing but the glider moving in a random direction with some initial point offset.
I suspect many stances other the one factorizing the world into gliders would have low infil/exfil, because the world is so simple. This is the case of “accidental boundary-ness.”
Perhaps something more complicated: various trajectories where e.g., the Lenia patterns encounters bullets, evolves with various other patterns, etc.
This I think rules out “accidental boundary-ness.”
I think the latter works. But now there’s a subjective choice of the distribution, and what are the set of possible/realistic “Nature’s Intervention”—all the situations that can ever be encountered by the system under which it has boundary-like behaviors—that we want to implicitly encode into our observational distribution. I don’t think it’s natural for ϕ assign much probability to a trajectory whose initial conditions are set in a very precise way such that everything decays into noise. But this feels quite subjective.
Hints toward a solution: Causality
I think the discussion above hints at a very crucial insight:
ϕ must arise as a consequence of the stable mechanisms in the world.
Suppose the world of Lenia contains various stable mechanisms like a gun that shoots bullets at random directions, scarce food sources, etc.
We want ϕ to describe distributions that the boundary system will “actually” experience in some sense. I want the “Lenia pattern dodges bullet” world trajectory to be considered, because there is a plausible mechanism in the world that can cause such trajectories to exist. For similar reasons, I think the empty world distributions are impoverished, and a distribution containing trajectories where the entire world decays into noise is bad because no mechanism can implement it.
Thus, unless you have a canonical choice of ϕ, a better starting point would be to consider the abstract causal model that encodes the stable mechanisms in the world, and using Discovering Agents-style interventional algorithms that operationalize the notion “boundaries causally separate environment and viscera.”
Well, because of everything mentioned above on how the causal model informs us on which trajectories are realistic, especially in the absence of a canonical ϕ. It’s also far more efficient, because the knowledge of the mechanism informs the algorithm of the precise interventions to query the world for, instead of having to implicitly bake them in ϕ.
There are still a lot more questions, but I think this is a pretty clarifying answer as to how Critch’s boundaries are limiting and why DA-style causal methods will be important.
I think the update makes sense in general, isn’t there however some way mutual information and causality is linked? Maybe it isn’t strong enough for there to be an easy extrapolation from one to the other.
Also I just wanted to drop this to see if you find it interesting, kind of on this topic? Im not sure its fully defined in a causality based way but it is about structure preservation.
Yeah I’d like to know if there’s a unified way of thinking about information theoretic quantities and causal quantities, though a quick literature search doesn’t show up anything interesting. My guess is that we’d want separate boundary metrics for informational separation and causal separation.
I no longer think the setup above is viable, for reasons that connect to why I think Critch’s operationalization is incomplete and why boundaries should ultimately be grounded in Pearlian Causality and interventions.
(Note: I am thinking as I’m writing, so this might be a bit rambly.)
The world-trajectory distribution is ambiguous.
Intuition: Why does a robust glider in Lenia intuitively feel like a system possessing boundary? Well, I imagine various situations that happen in the world (like bullets) and this pattern mostly stays stable in face of them.
Now, notice that the measure of infiltration/exfiltration depends on ϕ∈Δ(Wω), a distribution over world history. Infil(ϕ):=Aggt≥0MutWω∼ϕ((Vt+1,At+1);Et∣(Vt,At,Pt))
So, for the above measure to capture my intuition, the approximate Markov condition (operationalized by low infil & exfil) must consider the world state Wω that contains the Lenia pattern with it avoiding bullets.
Remember, W is the raw world state, no coarse graining. So ϕ is the distribution over the raw world trajectory. It already captures all the “potentially occurring trajectories under which the system may take boundary-preserving-action.” Since everything is observed, our distribution already encodes all of “Nature’s Intervention.” So in some sense Critch’s definition is already causal (in a very trivial sense), by the virtue of requiring a distribution over the raw world trajectory, despite mentioning no Pearlian Causality.
Issue: Choice of ϕ
Maybe there is some canonical true ϕ for our physical world that minds can intersubjectively arrive at, so there’s no ambiguity.
But when I imagine trying to implement this scheme on Lenia, there’s immediately an ambiguity as to which distribution (representing my epistemic state on which raw world trajectories that will “actually happen”) we should choose:
Perhaps a very simple distribution: assigning uniform probability over world trajectories where the world contains nothing but the glider moving in a random direction with some initial point offset.
I suspect many stances other the one factorizing the world into gliders would have low infil/exfil, because the world is so simple. This is the case of “accidental boundary-ness.”
Perhaps something more complicated: various trajectories where e.g., the Lenia patterns encounters bullets, evolves with various other patterns, etc.
This I think rules out “accidental boundary-ness.”
I think the latter works. But now there’s a subjective choice of the distribution, and what are the set of possible/realistic “Nature’s Intervention”—all the situations that can ever be encountered by the system under which it has boundary-like behaviors—that we want to implicitly encode into our observational distribution. I don’t think it’s natural for ϕ assign much probability to a trajectory whose initial conditions are set in a very precise way such that everything decays into noise. But this feels quite subjective.
Hints toward a solution: Causality
I think the discussion above hints at a very crucial insight:
ϕ must arise as a consequence of the stable mechanisms in the world.
Suppose the world of Lenia contains various stable mechanisms like a gun that shoots bullets at random directions, scarce food sources, etc.
We want ϕ to describe distributions that the boundary system will “actually” experience in some sense. I want the “Lenia pattern dodges bullet” world trajectory to be considered, because there is a plausible mechanism in the world that can cause such trajectories to exist. For similar reasons, I think the empty world distributions are impoverished, and a distribution containing trajectories where the entire world decays into noise is bad because no mechanism can implement it.
Thus, unless you have a canonical choice of ϕ, a better starting point would be to consider the abstract causal model that encodes the stable mechanisms in the world, and using Discovering Agents-style interventional algorithms that operationalize the notion “boundaries causally separate environment and viscera.”
Well, because of everything mentioned above on how the causal model informs us on which trajectories are realistic, especially in the absence of a canonical ϕ. It’s also far more efficient, because the knowledge of the mechanism informs the algorithm of the precise interventions to query the world for, instead of having to implicitly bake them in ϕ.
There are still a lot more questions, but I think this is a pretty clarifying answer as to how Critch’s boundaries are limiting and why DA-style causal methods will be important.
I think the update makes sense in general, isn’t there however some way mutual information and causality is linked? Maybe it isn’t strong enough for there to be an easy extrapolation from one to the other.
Also I just wanted to drop this to see if you find it interesting, kind of on this topic? Im not sure its fully defined in a causality based way but it is about structure preservation.
https://youtu.be/1tT0pFAE36c?si=yv6mbswVpMiywQx9
Active Inference people also have the boundary problem as core in their work so they have some interesting stuff on it.
Yeah I’d like to know if there’s a unified way of thinking about information theoretic quantities and causal quantities, though a quick literature search doesn’t show up anything interesting. My guess is that we’d want separate boundary metrics for informational separation and causal separation.