Discriminant informant

A term in discriminant analysis denoting a variable used to establish a rule for assigning an object with measurements $x=(x_1,\dotsc,x_p)$, drawn from a mixture of $k$ sets with distribution densities $p_1(x),\dotsc,p_k(x)$ and a priori probabilities $q_1,\dotsc,q_k$, to one of these sets. The $i$-th discriminant informant of the object with measurement $x$ is defined as

$$S_i=-[q_1p_1(x)r_1i+\dotsb+q_kp_k(x)r_{ki}],\quad i=1,\dotsc,k,$$

where $r_{ij}$ is the loss due to assigning an element from the distribution $i$ to the distribution $j$. The rule for assigning an object to the distribution with the largest discriminant informant has minimum mathematical expectation of the loss. In particular, if all $k$ distributions are normal and have identical covariance matrices, all discriminant informants are linear. Then, if $k=2$, the difference $S_1-S_2$ is Fisher's linear discriminant function.

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