What is a Bayesian Belief Network?

Briefing note version 2.2, 26 September 2013

By: Roy Haines-Young, David N Barton, Ron Smith and Anders L Madsen.

Bayesian Belief Networks (BBNs) have been identified as one of the cross-cutting themes within OpenNESS. The purpose of this example is to introduce the concept and the kinds of analysis it can support within the Project, and to stimulate discussion and collaboration across the work packages. More detailed explanations and general can be found in Kjærulff and Madsen (2013) and Cain (2001).

Cain (2001) defines a Bayesian Belief Network as a graphical tool for building decision support systems to help make decisions under uncertain conditions. The key phrase to focus on in this definition is uncertain conditions. As Cain points out, BBNs were originally developed to allow the impact of uncertainty about management systems to be accounted for, so that decision makers could balance the desirability of an outcome against the chance that the management option selected might fail. The representation of a system in terms of a set of relationships that have probabilities associated with them is at the heart of the Bayesian approach.

An example of a simple model that might form the basis of a BBN is shown in Figure 1. If we think about the sorts of things that might influence agricultural yield, for example, then these might include water supply and fertiliser applications. The amount of water applied to the crop might, in turn, be influenced by such factors as soil type and the level of irrigation. Figure 1 shows this diagrammatically, in what is technically called a .Directed Acyclic Graph. (DAG). Constructing such a graph is usually one of the first steps in building a BBN.

The variables in the DAG shown in Figure 1 are also called nodes; the term variable and node mean the same thing. The relationships between the variables are shown as a set of arrows or links. These simply set out the connections between the variables; they show what influences what. The direction of the arrows describes what we think the probabilistic relationships are within the system. In Figure 1 each of the nodes are shown as being able to take various states. In the HUGIN software used to construct this network, the states can be seen in the associated monitor windows. This network is obviously a very basic one. Thus yield is simply represented as good or poor, or soil type can be sandy or clay. The monitor windows show the state of each node as a belief bar, which shows the probability that the variable (node) is in a particular state. What the BBN allows us to do through the links, is to assign conditional probabilities to the states of the different nodes, showing their dependencies on the nodes that feed into them. These probabilities are held in Conditional Probability Tables (CPTs) that underlie each node. In the example the tables that define the response of yield and crop water application are shown in Figure 2.

The effect of assigning these probabilities on these two nodes can be investigated below. If we have evidence, for example, that the correct amount of fertiliser was used and the irrigation application was high then the BBN shows the probability of a good yield to have increased to 0.75. We specify that the correct amount of fertiliser was added by setting the probability of enough to 100% - this added evidence is indicated by the red bar. If on the other hand we know the soil type, and this extra evidence would propagate through the network to change the outcome. On sand the probability would be 0.74, while for clay it is 0.76.

One of the other interesting features of a BBN is that we can look at the kinds of condition that would lead to a particular outcome. Thus, we can select poor yield (by setting it to 100%, meaning we are only interested in this outcome state) and the network can help us identify what kinds of condition might lead to such a result.

Below is a set of HUGIN widgets for interacting with the model (click on the probability bar to instantiate a node or remove evidence):

---

---

Contact information

For further details contact: Roy Haines-Young@Nottingham.ac.uk

References

Cain, J. (2001) Planning improvements in natural resources management: Guidelines for using Bayesian networks to support the planning and management of development programmes in the water sector and beyond. CEH, Wallingford.

Kjærulff, U.B and Madsen, A.L. (2013): Bayesian Networks and Influence Diagrams: A Guide to Construction and Analysis. Second Edition. Springer.

Haines-Young, R., Barton, D.N., Smith, R. and A. Madsen (2013): Bayesian Belief Networks, a cross-cutting methodology in OpenNESS: Briefing note version 2.2, 26 September 2013.

---