The Bayesian Belief Network methodology provides a mathematical technique which can be used to assess quantitatively the effect which different pieces of information will have on our belief in a variety of different scenarios. In doing this, it merely automates and makes precise an activity which we all constantly carry out. The following example will illustrate the concept:
You wish to guess the nationality of a tourist sitting at a neighbouring table in a cafe in Glasgow. What nationality would you suspect if realised that he was speaking in Finnish? How would the conclusion differ if instead you heard him speaking in French?
Finnish is a language which has no relation to most of the other languages of Europe. It is correspondingly more difficult for non-Finns to learn the language than to learn other Indo- European languages. This is reflected in the statement that
Prob(He can speak Finnish | He is not Finnish) is close to zero.
This is a conditional probability. It tells us about the chances of something happening conditional on something else being true.
However, it is true that
Prob(He can speak Finnish | He is Finnish) will be close to one (since most citizens of Finland can speak their national language).
It is intuitively obvious that if these two probability statements are true then our observation that the tourist can speak Finnish is strong evidence that he is, indeed, a Finn.
By contrast, French is an international language which is taught as a second language in many other countries. In addition, both Belgium and Switzerland contain sizeable populations whose mother-tongue is French. Hence it follows that
Prob(He can speak French | He is not French) may be relatively high. Hence, although
Prob(He can speak French | He is French) will be close to one, it is intuitively clear that since speaking French is not such a distinctive mark of nationality as is speaking Finnish, an observation that the tourist is speaking French is not particularly strong proof that he is French.
Life rarely involves the isolated consideration of single pieces of information. We might hear the tourist speaking English to the waiter, and recognise that he is speaking with a Scandinavian accent. If we had previously heard him speak in Finnish, this would add little to our knowledge. However, if we had earlier heard him speaking in French, we might adjust our ideas to conclude that he is probably a Scandinavian who happens to speak excellent French (though he might be a Frenchman who was taught English by a Swede!)
Examples such as the above can get very complicated! This merely illustrates the point that the human brain becomes overloaded when asked to evaluate large quantities of conditional information. In particular, we tend to give too much weight to new information which confirms earlier evidence, even where the new information follows as a consequence of the earlier evidence. e.g. If the tourist had earlier spoken in Finnish, we was very likely to also have a Scandinavian accent. our observation of this fact does not add much to our earlier knowledge of his nationality.
The Bayesian Belief Network takes a number of pieces of evidence and consistently evaluates their joint significance to determine the relative plausibility of different hypotheses. It does this using a variant of Bayes's Rule (a mathematical rule to relate conditional probabilities) acting on a network of relationship between different observations and hypotheses. The numbers which the Bayesian network generates are known as beliefs because they describe the belief which we might have in different hypotheses (such as nationality of tourist, or disease of cow) given the observation of particular pieces of evidence (such as language spoken or clinical sign observed).