What is a dynamic Bayesian network

In one Dynamic Bayesian Network (DBN), not only one observation made at a time is processed over a number of features, but a whole one Time series of observations. Therefore, the same features occur multiple times for the required number of times. The nodes of the DBN are so-called corresponding to these points in time Time slices assigned. In a DBN, edges exist not only under the nodes of a time slice, but also across time slices.
The following simplifying basic assumptions are usually made:
  1. Edges across time slices always point in the direction of future time slices, i.e. the future depends (among other things) on the past, but not the other way around (this assumption should probably be indisputable).
  2. The problem is stationary, i.e. the structure and the conditional probabilities are identical in all time slices (except for the first, the so-called Prior time slice)
  3. The Markov condition of the first order is fulfilled, i.e. there are only edges within a time slice t and to next time slice t + 1. This is a time slice t-1 regardless of the time slice t + 1when all nodes are in t are given (see also section "Dependency structure (d-separation)")
This means that only the prior time slice (t = 0) and the following time slice (t = 1) necessary.



DBN: node of the prior time slice (t = 0) end on "0" this DBN has (per time slice) a "dynamic" edge (red)

The figure shows the simplest meaningful structure of a DBN. The knot State remains unobserved (hidden) and is estimated by observations made in the knot Observation be introduced. Such a model is the DBN equivalent of one Hidden Markov model (HMM). The conditional probability table of the state node contains the transition probabilities. Maintaining the current state is usually the most likely option. There are low probabilities for the transition to certain "neighboring states", while the transition to other states is completely excluded (e.g. P (State (t) = s3 | State (t-1) = s1) = 0). The values ​​of the transition probabilities depend in particular on the Recording rate i.e. the time difference between two time slices.





Unrolled DBN with 7 time slices (incl. A priori time slice)

An obvious way to calculate in such a network is this:
The time slice t = 1 is used for the required number of steps (T) is automatically copied and added to the network. This procedure is called Unroll. Then, as for a static BN, a complete junction tree can be created for the overall network obtained in this way.
However, this method has several disadvantages:

  • Before starting the calculations, it must be known how many time slices are required.
  • A huge junction tree is created for a large number of time slices. It should also be noted that at the first point in time t = 0 (only features of the time slice t = 0 observed) the influence is calculated up to the nodes of the last time slice. The reverse is also true for the last point in time (features from the time slices t = 0 ... T observed) nor the influence on the unobserved characteristics of the previous one traced back to the prior time slice. As a rule, however, the influence is only minimal over some time slices.
Often only the characteristics within one are of interest anyway Time window a "current" time slice t around.
There are methods to create junction trees that, like the DBN itself, can be shortened in slices towards the past and expanded towards the future. One is the dHugin method [1]. The prerequisite for this is a junction tree that has an identical structure for all time slices (except for the prior time slice) (see definition of stationary DBNs). This can only be guaranteed if the entire information comes from a time slice t to the following time slice t + 1 is passed on via just one separator is passed on. To do this, all nodes in the moral graph Connections across the disk boundary t / t + 1 away have to be united in a clique. This is easy to achieve by inserting these nodes with fill-in- Edges are connected in pairs (see also section "Inference (Junction Tree)").
This process also has a decisive disadvantage:
If several edges are used over time, but they are "far" apart in the network, an oversized clique is created that quickly exceeds the limits of manageability.
[1] "dHugin: A computational system for dynamic time-sliced ​​Bayesian networks", U. Kjærulff, International Journal of Forecasting, 1995