“Cause” is an everyday concept that tends to dissolve when looked at too closely. The research on causal analysis of statistical data quite sensibly does not try to define it.
Ah, ok. Applying his definition (“variable X is a probabilistic-cause of variable Y if P(y|do(x)) != P(y) for some values x and y”) to the signal generator, it says that the voltage causes the current; in the current-source version, that the current causes the voltage. That’s exactly what I would say as well.
Of course, his limitation to acyclic relations excludes from his analysis systems that are only slightly more complicated, such as B=AC,C=−k∫B.
That’s what dynamic Bayesian networks are for. The current values of state variables of a system near stable equilibrium are not caused by each other; they are caused by past values. Dynamic Bayesian networks express this distinction with edges that pass forward in time.
excludes from his analysis systems that are only slightly more complicated, such as B=AC,C=−k∫B
The continuous-time limit of a dynamic Bayesian network can be a differential equation such as this.
(ETA) A dynamic Bayesian network is syntactic sugar for an ordinary Bayesian network that has the same structure in each of a series of time slices, with edges from nodes in each time slice to nodes in the next time slice. The Bayesian network that is made by unrolling a dynamic Bayesian network is still completely acyclic. Therefore, Bayesian networks have at least the representation power of finitely iterated systems of explicit recurrence relations and are acyclic, and continuum limits of Bayesian networks have at least the representation power of systems of differential equations and are acyclic. (Some representation powers that these Bayesian networks do not have are the representation powers of systems of implicit recurrence relations, systems of differential algebraic equations without index reduction, and differential games. Something like hybrid Bayesian-Markovian networks would have some of these representation powers, but they would have unphysical semantics (if physics is causal) and would be hard to use safely.)
Except for everyone following Pearl.
Ah, ok. Applying his definition (“variable X is a probabilistic-cause of variable Y if P(y|do(x)) != P(y) for some values x and y”) to the signal generator, it says that the voltage causes the current; in the current-source version, that the current causes the voltage. That’s exactly what I would say as well.
Of course, his limitation to acyclic relations excludes from his analysis systems that are only slightly more complicated, such as B=AC,C=−k∫B.
That’s what dynamic Bayesian networks are for. The current values of state variables of a system near stable equilibrium are not caused by each other; they are caused by past values. Dynamic Bayesian networks express this distinction with edges that pass forward in time.
The continuous-time limit of a dynamic Bayesian network can be a differential equation such as this.
(ETA) A dynamic Bayesian network is syntactic sugar for an ordinary Bayesian network that has the same structure in each of a series of time slices, with edges from nodes in each time slice to nodes in the next time slice. The Bayesian network that is made by unrolling a dynamic Bayesian network is still completely acyclic. Therefore, Bayesian networks have at least the representation power of finitely iterated systems of explicit recurrence relations and are acyclic, and continuum limits of Bayesian networks have at least the representation power of systems of differential equations and are acyclic. (Some representation powers that these Bayesian networks do not have are the representation powers of systems of implicit recurrence relations, systems of differential algebraic equations without index reduction, and differential games. Something like hybrid Bayesian-Markovian networks would have some of these representation powers, but they would have unphysical semantics (if physics is causal) and would be hard to use safely.)
(Dynamic Bayesian networks at the University of Michigan Chemical Engineering Process Dynamics and Controls Open Textbook (“ControlWiki”))