My whole argument rests on a weaker reed than I first appreciated, because the definition of mutual information I linked is for univariate random variables. When I searched for a definition of mutual information for stochastic processes, all I could really find was various people writing that it was a generalization of mutual information for random variables in “the natural way”. But the point you bring up is actually a step in the direction of a stronger argument, not a weaker one. Sampling the function to get a time series makes a vector-valued random variable out of a stochastic process, and numerical differentiation on that random vector is still deterministic. My argument then follows from the definition of multivariate mutual information.
Sampling the function to get a time series makes a vector-valued random variable out of a stochastic process, and numerical differentiation on that random vector is still deterministic.
This is not correct. Given the vector of all values of A sampled at intervals dt, the derivative of that vector—that is, the time series for B—is not determined by the vector itself, only by the complete trajectory of A. The longer dt is, the less the vector tells you about B.
My whole argument rests on a weaker reed than I first appreciated, because the definition of mutual information I linked is for univariate random variables. When I searched for a definition of mutual information for stochastic processes, all I could really find was various people writing that it was a generalization of mutual information for random variables in “the natural way”. But the point you bring up is actually a step in the direction of a stronger argument, not a weaker one. Sampling the function to get a time series makes a vector-valued random variable out of a stochastic process, and numerical differentiation on that random vector is still deterministic. My argument then follows from the definition of multivariate mutual information.
This is not correct. Given the vector of all values of A sampled at intervals dt, the derivative of that vector—that is, the time series for B—is not determined by the vector itself, only by the complete trajectory of A. The longer dt is, the less the vector tells you about B.
True. I was also assuming that