Consider a classification problem with C classes for which the feature vector $$ has

M components each of which can take L discrete states. Let the values of the components be represented

by a $1-$of$-$L binary coding scheme. Further suppose that, conditioned on the class c$,$ the M components of

$$ are independent, so that the class$-$conditional density factorizes with respect to the feature vector components. Show that the quantities ac given by

ac $=$ ln

p$($x $|$ y $=$ c$)$ p$($y $=$ c$)$

$,$

which appear in the argument to the softmax function describing the posterior class probabilities, are linear functions of the components of $.$ Note that this represents an example of the naive Bayes model.

In an alternative notation, we would write this as

ak $=$ ln

p$($x $|$ Ck$)$ p$($Ck$)$

$.$