Uncertainty relations and fluctuation theorems for Bayes nets

The pioneering paper [Ito and Sagawa, 2013] analyzed the non-equilibrium statistical physics of a set of multiple interacting systems, S, whose joint discrete-time evolution is specified by a Bayesian network. The major result of [Ito and Sagawa, 2013] was an integral fluctuation theorem (IFT) governing the sum of two quantities: the entropy production (EP) of an arbitrary single v in S, and the transfer entropy from v to the other systems. Here I extend the analysis in [Ito and Sagawa, 2013]. I derive several detailed fluctuation theorems (DFTs), concerning arbitrary subsets of all the systems (including the full set). I also derive several associated IFTs, concerning an arbitrary subset of the systems, thereby extending the IFT in [Ito and Sagawa, 2013]. In addition I derive "conditional" DFTs and IFTs, involving conditional probability distributions rather than (as in conventional fluctuation theorems) unconditioned distributions. I then derive thermodynamic uncertainty relations relating the total EP of the Bayes net to the set of all the precisions of probability currents within the individual systems. I end with an example of that uncertainty relation.