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.
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.