Just finished the local causal states paper, it’s pretty cool! A couple of thoughts though:
I don’t think the causal states factorize over the dynamical bayes net, unlike the original random variables (by assumption). Shalizi doesn’t claim this either.
This would require proving that each causal state is conditionally independent of its nondescendant causal states given its parents, which is a stronger theorem than what is proved in Theorem 5 (only conditionally independent of its ancestor causal states, not necessarily all the nondescendants)
Also I don’t follow the Markov Field part—how would proving:
if we condition on present neighbors of the patch, as well as the parents of the patch, then we get independence of the states of all points at time t or earlier. (pg 16)
… show that the causal states is a markov field (aka satisfies markov independencies (local or pairwise or global) induced by an undirected graph)? I’m not even sure what undirected graph the causal states would be markov with respect to. Is it the …
… skeleton of the dynamical Bayes Net? that would require proving a different theorem: “if we condition on parents and children of the patch, then we get independence of all the other states” which would prove local markov independency
… skeleton of the dynamical Bayes Net + edges for the original graph for each t? that would also require proving a different theorem: “if we condition on present neighbors, parents, and children of the patch, then we get independence of all the other states” which would prove local markov independency
Also for concreteness I think I need to understand its application in detecting coherent structures in cellular automata to better appreciate this construction, though the automata theory part does go a bit over my head :p
Just finished the local causal states paper, it’s pretty cool! A couple of thoughts though:
I don’t think the causal states factorize over the dynamical bayes net, unlike the original random variables (by assumption). Shalizi doesn’t claim this either.
This would require proving that each causal state is conditionally independent of its nondescendant causal states given its parents, which is a stronger theorem than what is proved in Theorem 5 (only conditionally independent of its ancestor causal states, not necessarily all the nondescendants)
Also I don’t follow the Markov Field part—how would proving:
… show that the causal states is a markov field (aka satisfies markov independencies (local or pairwise or global) induced by an undirected graph)? I’m not even sure what undirected graph the causal states would be markov with respect to. Is it the …
… skeleton of the dynamical Bayes Net? that would require proving a different theorem: “if we condition on parents and children of the patch, then we get independence of all the other states” which would prove local markov independency
… skeleton of the dynamical Bayes Net + edges for the original graph for each t? that would also require proving a different theorem: “if we condition on present neighbors, parents, and children of the patch, then we get independence of all the other states” which would prove local markov independency
Also for concreteness I think I need to understand its application in detecting coherent structures in cellular automata to better appreciate this construction, though the automata theory part does go a bit over my head :p