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I have a Bayes Net with 20 variables, but I found out that one of the Parent variables is dependent on the previous value of its Child as:

C(t-1)->P(t)->C(t)

C and P are binary (True or False). All of the other variables are time-invariant.

To simplify I have been assuming that this temporal dependency does not exist and treating these nodes as time invariant as: P->C

If I decide to consider the temporal dependency does this mean the whole Bayes net is suddenly a Dynamic Bayes Net even if all the other variables are time invariant? Then my Static Bayes Net is essentially a single time slice of a Dynamic Bayes Net? Is there a nice way to convert between a Static Bayes net and Dynamic Bayes Net?

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If I decide to consider the temporal dependency does this mean the whole Bayes net is suddenly a Dynamic Bayes Net even if all the other variables are time invariant?

In short, yes, because a dynamic Bayesian network (DBN) is an umbrella term for Bayesian networks that "relates variables to each other over adjacent time steps." So technically, your network satisfies this (rather general) definition. There are various types of DBNs based on various things, such as whether one considers deterministic or stochastic processes, etc.

Here's an example (in python) from BayesServer of a network that they call a DBN but it only has one time-dependent vertex/variable.

The term "dynamic" refers to time evolution of the state space of a (abstract graph theoretic) representation of a system, but it does not mean that the graph structure itself is changing in time (this is another field of study, e.g. see here.), i.e., the conditional dependencies (edges) are not changing in time.

See Kevin Murphy's PhD thesis for a technical treatment of some special cases of DBNs for stochastic processes.

Then my Static Bayes Net is essentially a single time slice of a Dynamic Bayes Net?

Pretty much! However, there are some subtitles, e.g. there are differences in how learning is accomplished in static and dynamic networks (see chapter 6 of 3).

Is there a nice way to convert between a Static Bayes net and Dynamic Bayes Net?

I honestly have no idea :) it is probably a context-specific issue, but might result in some cool new insights!

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