Multinomial Discriminant Analysis

The DA model of Section 9.2.8 can be generalized to yield a multinomial DA model defined by

X? l(y< =j) ~ Mjij, 2;)



Р(Уі =j) = Qj


for / = 1, 2,. . ., n and j = 0, 1,. .

., m. By Bayes’s rule we obtain


Р(Уі=М?)= m8j(xT)qj, X g^f)Qk

where gj is the density function of 2,). Just as we obtained (9.2.48) from

(9.2.46) , we can obtain from (9.3.48)

^р^-^Дц+^ + хГАх,*), (9.3.49)

where РЛ1), fiA2), and A are similar to (9.2.49), (9.2.50), and (9.2.51) except that the subscripts 1 and 0 should be changed to j and 0, respectively.

As before, the term xf’Axf drops out if all the 2’s are identical. If we write Дко fi’x2)xT = the DA model with identical variances can be written exactly in the form of (9.3.34), except for a modification of the subscripts of /? and x.

Examples of multinomial DA models are found in articles by Powers et al.

(1978) and Uhler (1968), both of which are summarized by Amemiya (1981).

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