I often describe QB nets (Quantum Bayesian Networks) as being a graphical method for representing any quantum density matrix (see, for example, http://arxiv.org/abs/1204.1550 for more info on why I say that). Because of this description of mine, some might be led to believe that QB nets do not shed much additional light onto theories that stick to pure states for the most part.
In most of my papers, I use QB nets with nodes that have a finite number of states. Once again, this might lead some to believe that QB nets do not apply to continuum theories, yet most theories in High Energy Physics are continuum theories
I wanted to write a brief blog post to emphasize that QB nets can also be useful for both pure state and continuum theories.
In fact, the same dichotomy already exists for CB nets (Classical Bayesian nets). Judea Pearl is a very famous researcher that likes his CB nets with nodes that have a finite number of states, and Andrew Gelman is a very famous reseacher that likes his CB nets with nodes that have a continuum of states, and partly for that reason, instead of calling them Bayesian networks, Andrew calls them hierarchical models, but they are basically the same thing. Of course, you can go further in the continuum direction and use not just some nodes with an infinite number of states, but also use graphs with an infinite number of nodes (and perhaps periodic boundary conditions).
In my 1995 paper entitled “Quantum Bayesian Nets”, I give a QB net which yields a Feynman path integral (FPI) that in turn yields the Schroedinger equation for a single mass particle in an arbitrary potential. My proof is a simple adaptation of a proof that I learned from the Feynman & Hibbs book “Quantum Mechanics and Path Integrals”.
It should be fairly straightforward to generalize my derivation to find a QB net for any quantum field theory that is defined in terms of a CONTINUOUS or discrete or hybrid sum over exp(i*action) (and such FPIs are really sums over PURE quantum states)