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The Epistemic Paradigm h97

The cogency requirement of Postulate 4.1 may seek a map Xk tnat max-
imizes probability ProbjJxJ, i-e., the mode of the probability law. Other
representations of the cogency requirement may lead to a map Xk tnat 's tne
median of Prob^lx*], the conditional mean, etc. Equation 4.4 relates this
new probability function at the integration stage with the probability function
at the prior stage. The ProbjJxJ is a monadic function of Xk (because its
value depends on just one issue—whether Xk occurs or not). But the condi-
tional probability Prob?[xk|Xdata(S)] is a dvadic function of xk and X dato(S)
because its value depends on the two issues that it relates.
There are various notational ways of representing the effects of knowledge
bases § and ^ on the probability functions considered in each one of the
mapping stages. In order to familiarize the reader with the conditional proba-
bilities used in BME mapping, in the following example we choose to reexamine
Proposition 4.1 above using a slightly different notation.

EXAMPLE 4.5: Let us denote the prior probability function relative to the
general knowledge § as the conditional probability

Consider the probability function Prob'[-], such that Prob'[^] = 1. This
implies that

Furthermore, given that Prob

At the posterior stage we define the probability function Prob" [•]
such that Prob" [ Xdata(5)} = 1. This means that Prob" \\ k\X^(S}\ =
a
Prob"[x* and Xdata(S)} = Prob"[xJ; nd since Prob" [ Xk\X data(S)} =
Prob' [Xk\Xdata(S), we Set tne following expression

Furthermore, using Equation 4.12, we find

or