While I have actually done some programming with Boltzmann machines in a physics class, I am not familiar with their theoretical characterization. In contrast, I know a modest amount about the theory of graphical models (about the first few chapters of Lauritzen's book Graphical Models).
Question: Is there any meaningful relationship between graphical models and the Boltzmann machine? Is the Boltzmann machine a type of graphical model?
Obviously the Boltzmann machine is a type of neural network. I have heard that some neural networks are mathematically related to graphical models and that some are not.
Related Questions on CrossValidated which don't answer my question:
This is similar to a previous question which has been asked before: What's the relation between hierarchical models, neural networks, graphical models, bayesian networks? but is more specific.
Moreover, the accepted answer to that question does not clarify my confusion -- even if the nodes in the standard graphical representation of a neural network do not represent random variables, that does not necessarily mean that no such representation exists. Specifically, I am thinking about how the nodes in the typical graphical representation of Markov chains represent the set of possible states rather than the random variables $X_i$, but one could also create a graph showing the conditional dependence relationships between the $X_i$, which shows that every Markov chain is in fact a Markov random field. The answer also says that neural networks (presumably including Boltzmann machines) are "discriminative", but doesn't go into more detail to explain what that claim means, nor is the obvious follow-up question "are graphical models not discriminative?" addressed. Likewise, the accepted answer links to Kevin Murphy's website (I actually read some of his PhD thesis when learning about Bayesian networks), but this website discusses only Bayesian networks and does not mention neural networks at all -- thus it fails to illuminate how they are different.
This other question is probably most similar to mine: Mathematically modeling neural networks as graphical models However, none of the answers were accepted, and likewise only give references but do not explain the references (e.g. this answer). While one day I will hopefully be able to understand the references, right now I am at a basic level of knowledge and would most appreciate an answer which is as simplified as possible. Also, the Toronto course linked to in the top answer (http://www.cs.toronto.edu/~tijmen/csc321/lecture_notes.shtml) addresses this, but not in very much detail. Furthermore, the notes for the one lecture which might answer my question are not available to the public.
March 25 Lecture 13b: Belief Nets 7:43. For this slide, keep in mind Boltzmann Machines. There, too, we have hidden units and visible units, and it's all probabilistic. BMs and SBNs have more in common than they have differences. 9:16. Nowadays, "Graphical Models" are sometimes considered as a special category of neural networks, but in the history that's described here, they were considered to be very different types of systems.