a Markov blanket represents a probabilistic fact about the model without any knowledge you possess about values of specific variables, so it doesn’t matter if you actually do know which way the agent chooses to go.
The usual definition of Markov blankets is in terms of the model without any knowledge of the specific values as you say, but I think in Critch’s formalism this isn’t the case. Specifically, he defines the ‘Markov Boundary’ of Wt (being the non-abstracted physics-ish model) as a function of the random variableWt (where he writes e.g. Bt:=fWB(Wt) ), so it can depend on the values instantiated at Wt.
it would just not make sense to try to represent agent boundaries in a physics-ish model if we were to use the usual definition of Markov blankets—the model would just consist of local rules that are spacetime homogeneous, so there is no reason to expect one can apriori carve out an agent from the model without looking at its specific instantiated values.
fWB can really be anything, so Bt doesn’t necessarily have to correspond to physical regions (subsets) of Wt, but they can be if we choose to restricting our search of infiltration/exfiltration-criteria-satisfying fWB to functions that only return boundaries-in-the-sense-of-carving-the-physical-space.
e.g.Bt can represent which subset of Wt the physical boundary is, like 0, 0, 1, 0, 0, … 1, 1, 0
So I think under this definition of Markov blankets, they can be used to denote agent boundaries, even in physics-ish models (i.e. ones that relate nicely to causal relationships). I’d like to know what you think about this.
Critch’s formalism isn’t a markov blanket anyway, as far as I understand it, since he cares about approximate information boundaries rather than perfect Markov properties. Possibly he should not have called his thing “directed markov blankets” although I could be missing something.
If I take your point in isolation, and try to imagine a Markov blanket where the variables of the boundary Btcan depend on the value of Wt, then I have questions about how you define conditional independence, to generalize the usual definition of Markov blankets. My initial thought is that your point will end up equivalent to John’s comment. IE we can construct random variables which allow us to define Markov blankets in the usual fixed way, while still respecting the intuition of “changing our selection of random variables depending on the world state”.
I think something in the style of abstracting causal models would make this work—defining a high-level causal model such that there is a map from the states of the low-level causal model to it, in a way that’s consistent with mapping low-level interventions to high-level interventions. Then you can retain the notion of causality to non-low-level-physical variables with that variable being a (potentially complicated) function of potentially all of the low-level variables.
The usual definition of Markov blankets is in terms of the model without any knowledge of the specific values as you say, but I think in Critch’s formalism this isn’t the case. Specifically, he defines the ‘Markov Boundary’ of Wt (being the non-abstracted physics-ish model) as a function of the random variable Wt (where he writes e.g. Bt:=fWB(Wt) ), so it can depend on the values instantiated at Wt.
it would just not make sense to try to represent agent boundaries in a physics-ish model if we were to use the usual definition of Markov blankets—the model would just consist of local rules that are spacetime homogeneous, so there is no reason to expect one can apriori carve out an agent from the model without looking at its specific instantiated values.
fWB can really be anything, so Bt doesn’t necessarily have to correspond to physical regions (subsets) of Wt, but they can be if we choose to restricting our search of infiltration/exfiltration-criteria-satisfying fWB to functions that only return boundaries-in-the-sense-of-carving-the-physical-space.
e.g.Bt can represent which subset of Wt the physical boundary is, like 0, 0, 1, 0, 0, … 1, 1, 0
So I think under this definition of Markov blankets, they can be used to denote agent boundaries, even in physics-ish models (i.e. ones that relate nicely to causal relationships). I’d like to know what you think about this.
Critch’s formalism isn’t a markov blanket anyway, as far as I understand it, since he cares about approximate information boundaries rather than perfect Markov properties. Possibly he should not have called his thing “directed markov blankets” although I could be missing something.
If I take your point in isolation, and try to imagine a Markov blanket where the variables of the boundary Btcan depend on the value of Wt, then I have questions about how you define conditional independence, to generalize the usual definition of Markov blankets. My initial thought is that your point will end up equivalent to John’s comment. IE we can construct random variables which allow us to define Markov blankets in the usual fixed way, while still respecting the intuition of “changing our selection of random variables depending on the world state”.
I think something in the style of abstracting causal models would make this work—defining a high-level causal model such that there is a map from the states of the low-level causal model to it, in a way that’s consistent with mapping low-level interventions to high-level interventions. Then you can retain the notion of causality to non-low-level-physical variables with that variable being a (potentially complicated) function of potentially all of the low-level variables.