Links 

The homepage of the "A Song for Bayes" arrangment

The homepage of the choir vox11 including music samples

Bayesian net

 

Flu can cause fever. In a bayes net it can be illustrated this way.

 

 

The nodes Flu and Fever represents states, which I can have been in, and the arrow between the nodes represents the causal effect. Flu  may have the values yes or no while fever may have the values no, moderate, high.

This is a qualitative description of the cause-effect relation. The relation is not mechanical because you flu does not always cause fever. Therefore the model must contain probabilities that reflects this fact.

You have to express the condition probability. E.g. the probability for not having Fever given you not have Flu. All in all you have to handle 6 conditioned probabilities for P(Fever | Flu).

 

Flu

Fever

Yes

No

No

 

 

Moderate

 

 

High

 

 

 

By doing so you have described the (subjective) knowledge about the relation between Flu andFever.

To use his knowledge to do reasoning e. g. to diagnostically reasoning, you need also a apriori probability of having Flu. After achieving these probabilities Bayes formula can be used.

Flu

Yes

No

 

 

 

The model of Flu-Fever is a very simple model of the the disease Flu and it is a very simple example of a Bayesian net.

Cause-Effect relations are often chained. If you have Flu, there is a certain probability that you have Fever and if you have Fever you have – again with a certain probability have Headache. The Headache is not caused by the Flu but by the Fever which is caused by the Flu. Graphically you can express this chain in the figure below, where Headache can have values that are either yes or no.

When the conditioned Probability P(Headache | Fever) has been established a Bayesian net has been constructed in which you may reason both ways. Either from Headache to Flu or from Flu to Headache.

A bayesian net can mere more complicated. Below is shown a Bayesian net, which incorporates Cold, Coughing and Arthralgia.

 

 

 

There have been developed mathematical methods and algorithms, which automatically updates the probability in a Bayesian net when evidence has been entered in one node. E.g. when you have Headache and Arthralgia but do not cough the algorithm will calculate the probability for Flu, Fever and both.

A final example:

On a isolated island there is a population of rabbits and foxes. Without the foxes the rabbits would overcrowd the island and without the rabbits the foxes would die. The population of rabbits and foxes in one year have an causal impact of the size of the population of foxes and rabbits the next year. The cause-effect graph is shown below.

Given a granuality of 10 number of intervals for each population you need for Rabbit-1 to suggest a set of probabilities for the 100 combinations(Rabbits, Foxes) in year 0. All in all you have to specify more than 6000 probabilities, but when they are present at hand, you can use them in a Bayesian net to predict the development of the populations and calculate the effect of hunting one or both populations.