CAUSAL NETWORKS

1. INTRODUCTION

A causal network is a directed graph, i.e. a set of nodes connected by directed arcs. The nodes represent events (or propositions), which at their simplest can be either true or false. In addition each node has a certainty associated with it. If we increase the certainty of an event (represented by a node) this increase is transmitted across the net work, according to the nature of the connections and what else has been evidenced, so as to increase/decrease the certainty associated with other nodes.

We can identify three basic types of connection:

1. Serial (Figure 1)
2. Diverging (Figure 2)
3. Converging (Figure 3)

1. SERIAL CONNECTIONS

An example of a serial connection in a causal network is presented in Figure 1. The network states that A causes B with certainty x_B, and B causes C with certainty x_C. Thus:

• If the certainty associated with A is increased (i.e. the evidence for A is increased) then the certainty associated with B should be increased, and
• If the certainty associated withB is increased then the certainty associated with C should also be increased.

We can also reason in the opposite direction. If we have evidence for C then this increases the certainty of B and consequently the certainty of A.

Thus evidence may be transmitted up and down a serial connection. However, if (in Figure 1) the state of B is known (we say that B has been instantiated) then the channel is blocked and A and C become independent, we say that A and C are d-separated.

3. DIVERGING CONNECTIONS

In Figure 2 a diverging connection is presented where A causes both B and C. If the certainty associated with A is increased then this is transmitted to both B and C. Conversely if the certainty of B is increased, this will be transmitted to A, and then from A to C. (And, conversely if the certainty of C is increased, this will be transmitted to A, and then B.)

Thus, if we know nothing about A, then B and C are dependent; any change in the evidence for either B or Cwill affect the other (and also the evidence for A). However, if A is instantiated then B and C will become independent; any change in the evidence for B will not affect C (and vice versa). This is called conditional independence.

4. CONVERGING CONNECTION

A converging connection is presented in Figure 3. Here events A, B and C may D, i.e a change in the certainty for either A, B and C will be transmitted to D. However, A, B and C are independent; for example any change in A will not be transmitted on to B or C through D. However, if we have evidence for D, either directly or through a serial connection to a child node this is transmitted back up to the parents of D and consequently A, B and C become dependent. This is called conditional independence

4.1 Explaining Away

• If the certainty of A is increased this will increase the certainties associated with B and C, but not D because A and D are conditionally dependent.
• If the certainty of D is increased this will increase only the certainty of C (again not D).
• If the certainty C increases this will increase the certainties of A and D; and consequently any increase in the certainty of A will be transmitted on to D.
• If the certainty B increases this will increase the certainty of A, which in turn will increase the certainty of C. A and D are now dependent because C has received evidence transmitted from B. Consequently, because the certainty of Ahas been increased as a result of an increased confidence in B (which is not connected to D), the certainty of D is decreased. This reasoning process is called explaining away.

5. PRIOR CERTAINTY

From the above we have seen that if the certainty associated with an event is changed this has an effect on other events linked through the network (either positive or negative). Thus, prior to any reasoning that my take place, each event must have some "start" certainty associated with it. We can calculate these by assigning prior certainties to the root nodes and then calculating the affect on connected nodes.

6. d-SEPARATION AND d-CONNECTION (SUMMARY)

From the above evidence is transmitted from one node to another if they are dependent, i.e. they are d-connected. A node is d-connected if it is not d-separated. A node A is d-separated from a node B if for all paths between A and B there is a intermediate node C such that the connection is either:

1. Serial and C is instantiated, or
2. diverging and C is instantiated, or
3. converging and C has not received any evidence.

7. REFERENCES

1. Jensen, F.V.(1996). An Introduction to Bayesian Networks, UCL Press, London, (Chapter 2).