A Bayesian network is a DAG (directed acyclic graph) for which each of its nodes is assigned a transition matrix. Say a node has 2 incoming arrows , , and an outgoing arrow . Then one assigns to that node a transition matrix/conditional probability , where . But what happens if are all equal to a group ? Then one can define group multiplication as a B net node. Then one can translate any group theory statement to a statement about B nets. This means that there is some justification in saying that the vast collection of facts that we call Group Theory can all be viewed as a subset of the discipline of Bayesian Networks. Wow!
I’ve prepared a 3 page appendix to this blog post which you can find here:
In the appendix, I give a more detailed and formal explanation of how one can define some elementary group theory concepts (for instance, invariant groups, cosets, coset multiplication) in terms of B nets.
I like to think of B nets as being a very BROAD CANVAS, and this observation that Group theory is a mere subset of B nets, sets my heart aflutter 🙂
Of course, there is always someone bigger. B nets are themselves a subset (but a powerful and physicist-friendly one) of category theory (I know nothing about category theory, with my apologies to my friends who do).
P.S. After my religious conversion/epiphany, I continue to learn more and more about group theory. I feel my GT superpowers increasing day by day. I am becoming a lean, mean, group theory thinking machine. My GT thinking is beginning to float like a butterfly and sting like a bee. My inspiration is the greatest, wisest boxer of all time, Muhammad Ali.
Of course, the conventional notation for group theory is more than adequate, and very efficient, so expressing the same concepts in a less succinct language based on B nets may seem to some as a step backwards instead of forwards. But that is not the point of this blog post. The point is to show that group theory can be expressed in terms of B nets.
I should mention that using DAGS to describe group theory operations is a well known idea. The purpose of this post is not to communicate an original idea of mine, but to disseminate an idea that I find really cool. The idea in question concerns B nets and therefore aligns well with the main theme of this blog.