@katastrofa Excellent question.

You are probably familiar with the double slit experiment, which Richard Feynman considered to be at the heart of QM.

I figure you also know about the famous Monty Hall problem, which perfectly illustrates the counter-intuitive nature of conditional probabilities. The latter can be described by the simplest Bayesian Network, i.e. a V-structure.

If you complexify B-Nets the same structure encodes the collapse of the wave-function in the double slit experiment.

It drives home that algebraically retrieving the which-way information of the particles is the very same thing as getting the conditionally information that becomes available when opening the extra door in the Monty Hall set-up.

B-Nets are a very rich and well researched topic in data science and statistics, so in making this connection there is a lot to learn for quantum information science from that domain.

Full disclaimer: I am a co-founder of rrtucci's start-up.

What's the motivation for making the conditional probabilities in the nodes of Bayesian networks complex? What's the idea behind it and what advantages does it give you? (One could put there quaternions or any vectors, but what for?)

The motivation is Quantum Mechanics itself.

In classical probability, you can represent any probability distribution P(a_1, a_2, a_3, ...a_n) (where sum over all a_i of P equals 1) in terms of a network (DAG) of nodes, where each node stands for the conditional probability P(a_i| parents of a_i). These networks were named Bayesian Networks by Judea Pearl, my idol, who has made significant contributions to their theory

In  Quantum Mechanics, you can represent any pure state with state vector A(a_1, a_2, ...a_n)  (this is a probability amplitude, composed of complex numbers A, not a probability P between 0 and 1, and sum over all a_i of |A|^2 equals 1)  in terms of a network (DAG) of nodes, where each node stands for the conditional probability amplitude A(a_i| parents of a_i). This result can be generalized easily from pure states to density matrices. These networks are called Quantum Bayesian Networks.

People like Steve Adler of Princeton have tried to replace the complex numbers of quantum mechanics by quaternions, but have not found any new physics. Of course, quaternions are used very frequently in quantum mechanics, where they are replaced by an equivalent construct called Pauli matrices

@katastrofa, Oh, one more thing (  Steve Jobs's spirit must have reminded me)


If you replace the probabilities of classical Neural Nets by complex numbers, you get quantum neural nets QNN, which can be run on quantum computers. That is the goal of many people, including myself with my software https://github.com/artiste-qb-net/Quantum_Edward

People hope that QNN's will be better than classical NN in some respects, but this is still uncertain

What's the motivation for making the conditional probabilities in the nodes of Bayesian networks complex? What's the idea behind it and what advantages does it give you? (One could put there quaternions or any vectors, but what for?)

The motivation is Quantum Mechanics itself.

In classical probability, you can represent any probability distribution P(a_1, a_2, a_3, ...a_n) (where sum over all a_i of P equals 1) in terms of a network (DAG) of nodes, where each node stands for the conditional probability P(a_i| parents of a_i). These networks were named Bayesian Networks by Judea Pearl, my idol, who has made significant contributions to their theory

In  Quantum Mechanics, you can represent any pure state with state vector A(a_1, a_2, ...a_n)  (this is a probability amplitude, composed of complex numbers A, not a probability P between 0 and 1, and sum over all a_i of |A|^2 equals 1)  in terms of a network (DAG) of nodes, where each node stands for the conditional probability amplitude A(a_i| parents of a_i). This result can be generalized easily from pure states to density matrices. These networks are called Quantum Bayesian Networks.

People like Steve Adler of Princeton have tried to replace the complex numbers of quantum mechanics by quaternions, but have not found any new physics. Of course, quaternions are used very frequently in quantum mechanics, where they are replaced by an equivalent construct called Pauli matrices

OK, but what do you want to achieve by this and what for? Or, in other words, what advantage over old methods does it give you?

The major contribution of Judea Pearl to statistics is causal inference (the arrows in DAGs) - how do you model causality? As a fan of Judea Pearl you probably read his recent paper discussing the inability of ML to describe it. I'm curious how you've overcome this problem.

You wrote that in another project you calculate quantum entanglement. Do I understand correctly that you calculate the measure of entanglement? Which one? I'm asking because physicists (especially in quantum optics) try to find a good and useful (experimentally measureable) one.

Relativistic quantum mechanics certainly relies on quaternions, but good to know some new studies are going on - thanks for the reference.

@katastrofa
These are very physiky questions and I'm just a very poor businessman now. The viability of Bayesforge does not depend on my ramblings about Bayesian networks. But let me give some brief answers

"how do you model causality? "
I like to define Judea Pearl's "do" operations using entropies. Such a definition can be generalized to quantum mechanics by replacing the joint probability distributions by density matrices. I have written only one paper on this, so i am just an amateur about this topic. https://arxiv.org/abs/1307.5837 This is obviously not the only way to go about this.  There is much to learn yet about this topic. On the software front, there already exist programs (for example in the CRAN repository, and Bayesialab) that do Judea Pearl do calculus calculations. So I can see a day when someone will generalize that software to quantum bayesian networks.

"Do I understand correctly that you calculate the measure of entanglement? Which one?"
 I want to keep that secret for now. My "Q Entanglement Lab" is already finished in its first version. And it works nicely. But I still haven't written docs or jupyter notebooks for it. It will be open source "eventually" (I haven't decided yet when), but I want others to write their own Entanglement labs. I'm afraid that if they see what I have done before writing their own, they might be less original when writing their own. 

If you replace the probabilities of classical Neural Nets by complex numbers

Possible, but what is the rationale? It is a physicist's intuition and is it mathematically bone fide? Is it responsible behaviour, mathematically?

At the end of the day, NNS and HMMs etc. are just directed graphs (matrix), so making the edge' transition probabilities complex is a no-brainer in that sense.

Is it the same as in Markov Chains?
http://bidabad.com/doc/complex-prob.pdf

I would tend to trust complex probabilities more than the negative ones, the latter being - IMO- a fix when models break down.

"Is it the same as in Markov Chains?

http://bidabad.com/doc/complex-prob.pdf"

Both a classical and a quantum Markov chain are modeled by the bayesian network 
x(t)->x(t+1)
for t=0, 1, 2, ..., except in the classical case node x(t+1) for t>=0 represents a probability P[x(t+1) | x(t)]  satisfying 
sum_k P[x(t+1)=k | x(t)]  = 1 
whereas in the quantum case, it represents a probability AMPLITUDE A[x(t+1) | x(t)] which is a complex number satisfying 
sum_k |A[x(t+1)=k | x(t)]|^2  = 1

The reason for this definition is quantum mechanics itself. QM is a theory that is 90 years old, has been immensely successful, and has infected the vast majority of physics since it was invented. In my opinion, the reason that complex numbers are prevalent in quantum mechanics is because complex numbers are ideally suited for describing wave phenomena ( a complex number encodes perfectly the wave's amplitude and phase), and it so happens that quantum mechanics describes a phenomenon which has both wave and particle characteristics. 

So, no, this is not the same as what this nice paper you cite is saying. i just glanced at it, but I think they are interested in doing an analytic continuation in the discrete time variable t=0, 1, 2, 3,... Then once they decide to use continuous time, they find it convenient to use complex probabilities too. That is just my preliminary interpretation, i might be wrong.
Analytic continuations in time, either from discrete real or continuous real to complex is a venerable technique very fruitful in physics. For example, 
https://en.wikipedia.org/wiki/Imaginary_time

@Cuchulainn Complex numbers are algebraically closed, which seems a non-brainer when you want to describe nature, and the way Bob (rrtucci) complexified B-Nets, the resulting QB ones are isomorphic to quantum mechanical density matrices, which is another proper formulation of QM. So in this sense they a are at least as mathematically proper as QM itself  

@Cuchulainn Complex numbers are algebraically closed, which seems a non-brainer when you want to describe nature, and the way Bob (rrtucci) complexified B-Nets, the resulting QB ones are isomorphic to quantum mechanical density matrices, which is another proper formulation of QM. So in this sense they a are at least as mathematically proper as QM itself  

For the record, I am a mathematician. So I can handle the maths part. Complex numbers are very simple beasts, I fear.

Complex numbers are algebraically closed,
Do you have an example or the relevance to anything? 

What's the motivation for making the conditional probabilities in the nodes of Bayesian networks complex? What's the idea behind it and what advantages bedoes it give you? (One could put there quaternions or any vectors, but what for?)

Now you're talking.ons themselves.”

" No one uses quaternions  AFAIK, except maybe for graphics."

No!

See https://en.wikipedia.org/wiki/Pauli_matrices
The quaternions (1, i, j, k) can be represented by the matrices (1, i*sigx, i*sigy, i*sigz), where sigx, sigy, sigz are the 3  Pauli matrices. Pauli matrices arise extremely frequently in quantum mechanics. The Pauli matrices are representations of the lie algebra of the group SU(2); exp(i*sum_{k=1,2,3}a_k*J_k)  for reals a_1, a_2, a_3,  rotates a quantum state in 3-D space, where J_1, J_2, J_3 are representations of the lie algebra of  SU(2)

" No one uses quaternions  AFAIK, except maybe for graphics."

No!

See https://en.wikipedia.org/wiki/Pauli_matrices
The quaternions (1, i, j, k) can be represented by the matrices (1, i*sigx, i*sigy, i*sigz), where sigx, sigy, sigz are the 3  Pauli matrices. Pauli matrices arise extremely frequently in quantum mechanics. The Pauli matrices are representations of the lie algebra of the group SU(2); exp(i*sum_{k=1,2,3}a_k*J_k)  for reals a_1, a_2, a_3,  rotates a quantum state in 3-D space, where J_1, J_2, J_3 are representations of the lie algebra of  SU(2)

Lie groups I've studied as undergrad. In the intervening years I have seen no real engineering nor finance applications for them. Even Lie theory for partial differential equations (PDE) seems to be very elegant but computationally not easy to implement. I'll put it down to a lack of insight.


On a follow-on remark, it is not clear to me what the problem you are trying to solve, promote, sell.. 

"On a follow-on remark, it is not clear to me what the problem you are trying to solve, promote, sell.. "

I guess my original and primary goal was to get you interested in BayesForge. BF is a cloud hosted, free (for limited use, if free tier is available), personal file system and development environment for classical and quantum AI software development

Then I started to explain to you the mermaid call and wonders of quantum mechanics, how it is parallel in many ways to classical probability. And, it is not a pipe dream, QM has been verified thoroughly for almost a hundred years. In the last 10, it has also been verified in  quantum computing. Some people believe that in the future,  we will be able to use quantum computers to do AI more efficiently than with classical computers, although such a goal has not been reached yet and may prove unattainable. Bayesforge allows you to do both classical and quantum AI, and hybrids, so you can use it even if you are not interested in quantum computing.

@
katastrofa
These are very physiky questions and I'm just a very poor businessman now. The viability of Bayesforge does not depend on my ramblings about Bayesian networks. But let me give some brief answers

Your presentation mentions physical concepts which have rigorous definitions - maybe except the picture of apes jumping around Josephson junctions(?) I wanted to know how exactly they are related to your project. Is someone helping you with the physiky part? If there's no hard core quantum physicist on board, who's doing the entanglement project? You didn't answer if my interpretation of what "calculating entanglement" means is correct, but assuming it is, experts in the field struggle with that problem - not to mention noncognoscenti.

"how do you model causality? "
I like to define Judea Pearl's "do" operations using entropies. Such a definition can be generalized to quantum mechanics by replacing the joint probability distributions by density matrices. I have written only one paper on this, so i am just an amateur about this topic. https://arxiv.org/abs/1307.5837 This is obviously not the only way to go about this.  There is much to learn yet about this topic. On the software front, there already exist programs (for example in the CRAN repository, and Bayesialab) that do Judea Pearl do calculus calculations. So I can see a day when someone will generalize that software to quantum bayesian networks.

Joint probabilities or density matrices won't model causality/Pearl's "do". You need something more for directional correlations... Anyway, you will have some "quantum" model with lots of degrees of freedom (possibly not measurable ones) - where would you apply it? would you be able to calibrate it? That's why I've been asking you several times where's the need for your model. Whatever I can think of doesn't hold. There indeed are many ways to build more expressive neural networks, but will it be of any practical or scientific value or just for an umpteenth arxiv paper?
BTW, I believe a natural candidate for such alrgorithms is quantum molecular dynamics (maybe even already running on D-waves).

"Do I understand correctly that you calculate the measure of entanglement? Which one?"
I want to keep that secret for now. My "Q Entanglement Lab" is already finished in its first version. And it works nicely. But I still haven't written docs or jupyter notebooks for it. It will be open source "eventually" (I haven't decided yet when), but I want others to write their own Entanglement labs. I'm afraid that if they see what I have done before writing their own, they might be less original when writing their own.

If you don't mind staying poor, I think the easiest way for you to get funding would be to go back to academia and suck up to some influential professor, who will back your proposal as a grant application. I knew completely clueless people who got degrees from top unis and hefty grants for research on "quantum" nonsense in this way. Just make it sound complex and mysterious - as I was once told by an official advisor when applying a fellowship at a British uni, "you need to learn to bullshit" I'm not saying your project is bad - I don't understand it enough to criticise.


Have fun and good luck!

@rrtucci,
do you provide complimentary Quantum computers and Q compilers?