On the horizon, algebraic geometry and category theory have opened up new ways to think of posets, and posets are in correspondence with DAGs. We can talk about mappings between Bayesian networks, as well as sheaves which encode the logical restrictions and can be further linearized or categorified.
You're right that it seems like not much attention is paid to this stuff. At a lecture a few years ago, it was pointed out that sheaves on Bayesian networks seem interesting, but that nobody seems to be working directly on applying the maths to existing problems.
I am aware of the categorical treatment and that's how I think about Bayesian nets, but would like to know something about the sheaf-theory aspect of it. Also: it puzzles me that Algebraic Geometry is mentioned in this context, how's that?
Rest assured, people are using this stuff. It's not as sexy as deep learning, but for small/medium data problems where the domain is reasonably well-understood and experts have useful priors (read: the overwhelming majority of business problems), this is pretty much the only unified framework for learning from data while incorporating uncertainty.
Here's a product that claims to be solving lots of problems for lots of people using Bayes nets:
Almost every home-grown search function in existence could benefit from it, in my opinion. Everywhere where you have a useful feedback that imply good relevance (e.g. clickthroughs, customer actions), but do just straight textual keyword searches, a Bayesian approach can do better. Often there are more complex methods that can do even better, but the beauty of using a Bayesian approach is that it's so simple. I've seen so many cases where people spend tons of effort manually tuning product searches etc. when they can trivially have the engine automatically tune itself to what actually produces the best business outcomes.
Do you know of an interactive, easy and free software to build and simulate our own versions of the asia net and example or other custom built ones? I used to like GeNIe but it isn't free anymore.
