It is very convenient to consider true values and observed values as causes and effects (see Fig. , imagining also a continuous set of causes and many possible effects). The process of going from causes to effects it is called `deduction'.2.11 The possible values which may be observed are classified in belief by
Once the likelihood is determined (we have the performance of the detector under control) we can build , under the hypothesis that will be observed.2.12 In order to arrive at the general formula in an heuristic way, let us consider only two values of . If they seem to us equally possible, it will seem natural to be in favour of the value which gives the highest likelihood that will be observed. For example, assuming , , considering a normal likelihood with , and having observed , one tends to believe that the observation is most likely caused by . If, on the other hand, the quantity of interest is positively defined, then switches from most probable to impossible cause; becomes certain. There are, in general, intermediate cases in which, because of previous knowledge (see, e.g., Fig. and related text), one tends to believe a priori more in one or other of the causes. It follows that, in the light of a new observation, the degree of belief of a given value of will be proportional to