Graphical models for correlation of random variables and prediction of hidden observations

I am studying about Graphical Models and I came up with a simple example but I am not sure which kind of technique (HMM, DGM, MRF) would be able to help me with that.

Imagine we have three balls that can move along this one dimensional space. I can only measure their coordinates:

I collect some observations through time and I notice that A always move together with B, in other words, A is correlated with B:

What I would like Graphical Models to tell me is that $A ⫫ C$ (independent) but A and B are not, there is a dependence with them. Moreover, would it be possible to estimate B given that I only measured A? In short:

• Is it possible to find which Random Variables are dependent (and which ones are not) given only their observations over time t? (e.g. coordinates)
• Given that I solved the previous problem and I now know which ones have dependence. Assuming also that at a certain time $t_k$ some variables are not observed, how can I estimate the location of a non-observed variable (such as B) given that I observed a dependent a variable A?

I created some possible observations for this example. If somebody could show me concretely how I compute that, it would be great:

A,B,C
0,0,0
0,0,5
1,1,5
3,3,3
2,2,5
4,4,0
2,2,0

So, when I query what is the position of B given that A is 6? How can I use the previous observations to come up with a solution?

Thank you

• Do you want to test for dependence or just correlation? If dependence, then this question is related: stats.stackexchange.com/questions/25132/… Commented Feb 2, 2014 at 23:06
• Dependence. What I wanted to do is to learn the graph from data. I thought about calculating Pearson correlate coefficient between all pairs of variables and use a threshold on those coefficients to decide if they are dependent or not, but it does not sound like a correct solution. Thanks for the link. Commented Feb 4, 2014 at 10:49
• I think there is a number of issues here. First HMM's as the name suggests are used to model hidden states of a system, your system seems to contain 3 observable states. Second, if you are interested in discovering structure of models, it's feasible but non-trivial, see openmarkov.org/docs/tutorial/tutorial.html#toc-Chapter-2 for a nice explanation. Third, a graphical model is merely an encoding of a dependency structure. At one point you will have to assume a probability distribution somewhere. If prediction is your goal, you better get it right, or use some other algorithm - nnet. Commented Feb 6, 2014 at 21:46