Definition of dynamic Bayesian system, and its relation to HMM?

Question

From Wikipedia

A Dynamic Bayesian Network (DBN) is a Bayesian Network which relates variables to each other over adjacent time steps. This is often called a Two-Timeslice BN because it says that at any point in time T, the value of a variable can be calculated from the internal regressors and the immediate prior value (time T-1). DBNs are common in robotics, and have shown potential for a wide range of data mining applications. For example, they have been used in speech recognition, protein sequencing, and bioinformatics. DBN have shown to produce equivalent solutions to Hidden Markov Models and Kalman Filters.

1. I was wondering if "the immediate prior value (time T-1)" means the time index in a DBN is always discrete?
2. Does "at any point in time T, the value of a variable can be calculated from the internal regressors and the immediate prior value (time T-1)" mean a DBN is a discrete-time Markov process?

3. If I understand correctly, a HMM is a discrete-time Markov process too, if ignoring the output from state at the same time. So I wonder if HMM and DBN are the same concept? But another Wikipedia article says

   hidden Markov model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved (hidden) states. An HMM can be considered as the simplest dynamic Bayesian network.

   and there is another quote from the first article:

   DBN have shown to produce equivalent solutions to Hidden Markov Models and Kalman Filters.

Thanks!

Answer 1

I'd recommend looking through these two excellent review papers:

• An Introduction to Hidden Markov Models and Bayesian Networks by Zoubin Gharamani

• Dynamic Bayesian Networks by Kevin Murphy

HMMs are not equivalent to DBNs, rather they are a special case of DBNs in which the entire state of the world is represented by a single hidden state variable. Other models within the DBN framework generalize the basic HMM, allowing for more hidden state variables (see the second paper above for the many varieties).

Finally, no, DBNs are not always discrete. For example, linear Gaussian state models (Kalman Filters) can be conceived of as continuous valued HMMs, often used to track objects in space.