DBN and HMM and KFM and MRF and IsingM

BN is probability DAG, the node is variable, the edge is relationship of probability.

DBN is the dynamic version, which means it is used for dynamic variables, does not mean it is changing dynamic. so

1) the relationship is common law never changed, but you may only use part of the common law in one time slice. for example, in HMM, the state variable is changing. In one time , you are in stateA, then you use the observing law about stateA. In another time, you are in stateB, then you use the observing law about stateB.

2) In the DAG, there are nodes coming from the past time. given you all the time slices, the DBN can be viewed as BN, which has edges from nodes of time k-m to nodes of time k. For example, in KFM and HMM, m is always 1 or 0.

HMM is a special case of DBN, in which,

1)a can-not-see variable(state variable) is discrete variable.

2)the nodes from the past of time k-1 is used only.

KFM is a special case of DBN, in which,

1) a can-not-see variable(state variable) is continuous variable.

2) the nodes from the past of time k-1 is used only.

In a way, the BN or DBN is trying to utilize the independence of variables to make the graph look simple, have few edges，also called “Sparsity”. So you can avoid too much contional rules, like P(A|B|C|D). Given you a BN, you can know who and who are independent, who and who are not, and the probability path from one variable to another.

As for undirected graphic, there are MRF, which is a special case of Ising Model. They can represent cycling, but can not represent induced dependencies.