Thermodynamic uncertainty relations and fluctuation theorems for Bayes nets

The pioneering paper [1] analyzed the non-equilibrium statistical physics of a set S of multiple interacting systems whose joint discrete-time evolution is specified by a Bayesian network. Their major result was an integral fluctuation theorem (IFT) governing the sum of two quantities: the entropy production (EP) of an arbitrary single one of the systems, v ∈ S, and the transfer entropy from v to the other systems in S.
Here I derive a detailed fluctuation theorem (DFT) for an arbitrary subset of the systems in S (including the full set). I then derive an IFT for an arbitrary subset of the systems in S, extending the IFT in [1]. I also derive thermodynamic uncertainty relations for the precision of probability currents among joint states of the systems, and for the precision of the work performed by the full set of systems. Rather than the EP generated by a single system in S and the transfer entropy from it to the other systems, these results involve the sum of the EPs generated by all of the systems in S and the change in the multi-information of the joint system as it evolves.

[1] - S. Ito and T. Sagawa, "Information thermodynamics on causal networks", PRL, 111 (2013), no. 18, 180603