# Different order of insertion - different Bayesian network ? how to prove formally?

I have some Bayesian network which i constructed from some data, say it consists of nodes A, B, C and D and that was the initial order of insertion.

If i reconstruct the network by inserting the nodes in a different order i will get a different structure which will represent the same data and with dependencies (arrows) which are different from the ones in the original network (also almost probably).

How to formally prove that different order of insertion leads to a different structure of a Beyesian network ? Is there some property such proof can be based on ? How can it be proved ?

Well, have a look at the basic connections $$A\rightarrow B\rightarrow C$$ (serial), $$A\leftarrow B\leftarrow C$$ (serial) and $$A\leftarrow B\rightarrow C$$ (diverging). They all give rise to the same joint probability distribution. But the converging connection $$A\rightarrow B\leftarrow C$$ is different.