Reasoning with Probabilities

They are uncertain because there is uncertainty in the rules  used and uncertainty in the data.

The uncertainty in the data may be expressed with the probability of a certain input variable to be in a certain state. For example, we may say "Since the forest is in this area, the rock is probably permeable. The geological map I have, which has scale 1:200,000, marks all that region as permeable."

Then we may think of our network as a probabilistic network: An attribute may be in a certain state with certain probability, given the probabilities with which its causes are in the relevant states, and the conditional probabilities that relate causes and effects.

Then we shall have to answer questions like:

"If the slope of a field is gentle with probability 20% and the rock permeable with probability 50% and the soil depth shallow with probability 30%, what is the probability of erosion to be high?"

This implies that the arrows that lead from causes to effects must be endowed with the values of the conditional probabilities that express the probability of a certain state of an effect arising given the necessary conditions of that state are fulfilled.

P(Erosion=high)=

P(Erosion=high | slope=gentle, rock=permeable, soil depth=shallow) X P(slope=gentle) X P(rock=permeable) X P(soil depth=shallow)

+P(Erosion=high | slope=gentle, rock=permeable, soil depth=deep) X P(slope=gentle) X P(rock=permeable) X P(soil depth=deep)

+...

(All the combinations of states of the conditioning variables must be considered.)