# Quantum Bayesian Networks

## March 15, 2015

### Bayesian Networks do groups too

Filed under: Uncategorized — rrtucci @ 5:33 pm

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 $\underline{a}_1$, $\underline{a}_2$, and an outgoing arrow $\underline{b}$. Then one assigns to that node a transition matrix/conditional probability $P(b| a_1, a_2)$, where $b\in val(\underline{b}), a_1\in val(\underline{a_1}), a_2\in val(\underline{a_2})$. But what happens if $val(\underline{b}), val(\underline{a_1}), val(\underline{a_2})$ are all equal to a group ${\cal G}$? 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:

http://www.ar-tiste.com/gt-bnets.pdf

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.

Footnotes:
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.

1. >> language based on B nets may seem to some as a step backwards instead of forwards.
The famous rope-a-dope …

Comment by wolfgang — March 15, 2015 @ 8:22 pm

2. Hello,

And congrats for the blog and the interesting post. I just wanted to share a thing, I was reading about artificial intelligence via neural nets, where in some cases belief networks are studied, and came to hear a video lecture from Jason Morton on, among other things, tensor networks and Bayesian networks. Part of his argument was to represent a bayesian net via a factor graph in terms of a tensor network. Not all details were included.

Thinking of bayesian nets, I think one can view the theorem of total probability as a way to calculate the marginal distro of the child (rank-1 tensor), by applying a linear transformation (rank-2 tensor) corresponding to the child’s conditional distro, to the parent’s distro, i. e., doing a tensor product. If the child has two parents, applying also the total probability thm to a joint probability, the marginal of the child results as the tensor product of the conditional distro attached to the child, seen as a rank-3 tensor with the two prior distros of the parents (each rank 1 tensors). I checked an example of this in GeNIe and it works. It looks that one can plug tensors to the nodes of the net and get the tensor net indicating the contractions to do to calculate the marginals. I would love to see a thorough exposition of the equivalence of tensor and bayes networks! In the categorical setting, tensor products are understood inside monoidal categories. I think one needs to augment the categories of probabilistic mappings (markov kernels) with this sort of tensor product, not sure if that’s documented. I would like to see more on stochastic string diagrams.

Causal Theories: A Categorical Perspective on Bayesian Networks – Brendan Fong

Network Theory III: Bayesian Networks, Information and Entropy – John Baez, Brendan Fong, Tobias Fritz, Tom Leinster

Bayesian Machine Learning via Category Theory – Culbertson and Sturtz.

An Algebraic Perspective on Deep Learning, Jason Morton

Comment by jesuslop — March 15, 2015 @ 9:05 pm

3. Jesús, if i understand what you are saying correctly, how to express a tensor node in terms of several B net nodes, I use what I call marginalizer nodes. I’ve been using them since my first paper on quantum B nets, http://arxiv.org/abs/quant-ph/9706039
I also use them in my software Quantum Fog that I wrote about 15 years ago.

Marginalizer nodes are very simple. If a node that has a state composed of 3 substates (a,b,c) is the parent of a marginalizer node, then that marginalizer node with input arrow carrying state (a,b,c) has output arrow carrying state (a), for example

Comment by rrtucci — March 15, 2015 @ 9:49 pm

4. Good stuff. IMHO much more accessible than tensor network notation.

Comment by Henning Dekant — March 16, 2015 @ 2:25 am

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