By: Lauritzen & Spiegelhalter, 1988
Asia is a small Bayesian network that calculates the probability of a patient having tuberculosis, lung cancer or bronchitis respectively based on different factors - for example whether or not the patient has been to Asia recently.
Shortness-of-breath (dyspnoea) may be due to tuberculosis, lung cancer, bronchitis, more than one of these diseases or none of them. A recent visit to Asia increases the risk of tuberculosis, while smoking is known to be a risk factor for both lung cancer and bronchitis. The results of a single chest X-ray do not discriminate between lung cancer and tuberculosis, as neither does the presence or absence of dyspnoea.
If we learn the fact that a patient is a smoker, we will adjust our beliefs (increased risks) regarding lung cancer and bronchitis. However, our beliefs regarding tuberculosis are unchanged (i.e., tuberculosis is conditionally independent of "smoking" given the empty set of variables). Now, suppose we get a positive X-ray result for the patient. This will affect our beliefs regarding tuberculosis and lung cancer, but not our beliefs regarding bronchitis (i.e., bronchitis is conditionally independent of X-ray given smoking). However, had we also known that the patient suffers from shortness-of-breath, the X-ray result would also have affected our beliefs regarding bronchitis (i.e., "bronchitis" is not conditionally independent of "X-ray" given "smoking" and "dysponea").
A Bayesian belief network for the knowledge described above can look like this:
In a chest clinic, the patients can have tuberculosis, lung cancer, and/or bronchitis (or none of these). To determine the state of the patient, the doctor can make two observations: X-rays, and whether the patient suffers from dyspnoea. Furthermore, the doctor can ask the patient if he has been to asia and whether or not he is a smoker.
Try for yourself to make diagnosis for the patient using the following applet: