Solved – Definition of dynamic Bayesian system, and its relation to HMM

graphical-modelhidden markov model

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!

Best Answer

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

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.

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