8.6 () For the model shown in Figure 8.13, we have seen that the number of parameters required to specify the conditional distribution p(y|x1, . . . , xM), where xi ∈ {0, 1}, could be reduced from 2M toM +1 by making use of the logistic sigmoid representation

(8.10). An alternative representation (Pearl, 1988) is given by p(y = 1|x1, . . . , xM) = 1 − (1 − μ0)

M

i=1

(1 − μi)xi (8.104)

where the parameters μi represent the probabilities p(xi = 1), and μ0 is an additional parameters satisfying 0
μ0
1. The conditional distribution (8.104) is known as the noisy-OR. Show that this can be interpreted as a ‘soft’ (probabilistic) form of the logical OR function (i.e., the function that gives y = 1 whenever at least one of the xi = 1). Discuss the interpretation of μ0.