a) The probabilistic decision process of the multi-class predictor, for an example classification problem of four classes: 1, 2, 3 and 4. Firstly, one of the binary classifiers in a set *B *is selected with uniform probability 1#*B*, where #*B *is the number of binary targets in *B*. Secondly, class subset 1 or 2,3 is selected with probability *q*_{[1|23]}(*x*) or 1 - *q*_{[1|23]}(*x*), respectively. Thirdly, class 2 or 3 is selected with a probability of 1/2. Accordingly, one of the classes is selected with a certain probability. b) Calculation of class probability by *SHS*. For a binary classifier [*l*|*m*] in *B *and an input x which is a member of classes in *l *or *m*, we define *q*_{[l|m]}(*x*) as an estimated probability where x is a member of class(es) in *l*, and the complement probability 1 - *q*_{[l|m]}(*x*) where x is a member of class(es) in *m*. For example, *q*_{[1|23]}(*x*) and 1 - *q*_{[1|23]}(*x*) indicate the probability that *x *belongs to the class 1, and that *x *belongs to the class 2 or 3 provided that x belongs to the class 1, 2 or 3. In the *SHS *procedure, the probabilistic outputs by the multiple classifiers are shared and integrated by multiple classes, leading to the estimated class membership probabilities: *p*_{1},*p*_{2},*p*_{3 }and *p*_{4}. When *l *and/or *m *are set of multiple classes, the corresponding probabilistic outputs are shared equally to each of the members. For example, *q*_{[1|23]}(*x*) is added to *p*_{1}, 1 - *q*_{[1|23]}(*x*) is shared equally and added to *p*_{2 }and *p*_{3}, *q*_{[1|234]}(*x*) is added to *p*_{1}, 1 - *q*_{[1|234]}(*x*) is shared equally and added to *p*_{2}, *p*_{3 }and *p*_{4}, and so on for all members of *B*. Consequently, we obtain an estimation of multiple class probabilities *p*_{1},*p*_{2}, *p*_{3 }and *p*_{4 }by normalizing them so that the summation *p*_{1}, *p*_{2}, *p*_{3 }and *p*_{4 }would be one.

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