Before revision: 220-34, 2007 (omitting constants): The variational means that minimize the (path integral) of free energy can now be recovered by solving the generalized filter: where , is a block matrix derivative operator of identify matrices such that Variational basis Generalized filtering is based on the following lemma: The self-consistent solution to satisfies the variational principle of stationary action, where action is the path integral of variational free energy Proof: self-consistency requires the motion of the mean to be the mean of the motion and (by the fundamental lemma of variational calculus) Put simply, small perturbations to the path of the mean do not change variational free energy and it has the least action of all possible (local) paths.
After revision: 220-34 2007 (omitting constants): The variational means that minimize the (path integral) of free energy can now be recovered by solving the generalized filter: where , is a block matrix derivative operator of identify matrices such that Variational basis Generalized filtering is based on the following lemma: The self-consistent solution to satisfies the variational principle of stationary action, where action is the path integral of variational free energy Proof: self-consistency requires the motion of the mean to be the mean of the motion and (by the fundamental lemma of variational calculus) Put simply, small perturbations to the path of the mean do not change variational free energy and it has the least action of all possible (local) paths.