We suppose that each individual in a given population π belongs to one of two mutually exclusive groupsπ
1 and π
2 Our purpose is to classify an individual or individuals randomly drawn from π to either π
1 or π
2 as correctly as possible. However, we can not directly identify each individual as a member of π
1 or π
2 but use the individual's responses to a battery of m dichotomous items to aid in classification.
Let x=(e
1, …, e
m) denote the total response to the given battery of items, where ej=1 if the response on the jth item is “positive” and ej=0 if otherwise, j=1, …, m. Let
fi(x)be the probability function of x in πi, i=1, 2. In this case, if these probability distributions are completely known, as is well known, the best way of this classification should be based on the likelihood ratio L(x)=
f2(x)/
f1(x)or, equivalently, on l(x)=logL(x)=log
f2(x)-log
f1(x).
The optimum solution based on L(x)clearly requires knowledge of the probability distribution of response patterns in each group. But this is a strong requirement if m is large, for both
f1(x)and
f2(x)are distributions with 2m-1 parameters. Bahadur(1961)has shown that if the number of items in the battyery is fairly large and if the items are not highly interdependent, l(x)=log L(x)is approximately normally distributed in πi, with the mean μi=εfi(l(X))and the variance σi2=εfi(l(X)-μi)
2, i=1, 2, respectively, and if f1(x)and f2(x)are not very different, both σ1 and σ2 are approximated by D=√μ2-μ1. But it seems to be an obstacle to classifications that
f;1(x)and
f;2(x)are restricted by the last condition to ensure approximately equal σ1 and σ2. We shall deal with the case that f1(x)and f2(x)are unknown but past observations obtained from π
1 and π
2 are available, respectively, and we shall use Ln1, n2(x)=
fn2(x)/
fn1(x)instead of L(x), where
fn1(x)denotes the relative frequency based on ni observations from π
i.
Let ω
1 and ω
2 be the proporitions of individuals of π belonging to π
1 and π
2, respectively. In the usual situation of classifications, ω=(ω
1, ω
2)will be unknown. We shall regard ω as an unknown prior distribution having frequency interpretation as the chance that an individual randonly drawn from π belongs to π
1 or π
2. Some Bayesians may not approve of frequemcy interpretation for prior distribution in any case. But ω has the clear meaning as frequency in this case.
Suppose now that a new random sample of the size n is obtained from π in order to classify each individual contained in this sample to either π
1 or π
2. The empirical Bayes procedure is considered based on
fn1(x),
fn2(x)and the new sample(x
1, …, x
n)for the classification in the case that x=(e1, …, em)denotes the response pattern of an individual to the m dichotomous items. The prior distribution ω is estimated from (x
1, … x
n) and an empirical Bayes rule which is asymptotically optimal relative to ω in the Robbins' sense is made for our classification problem.
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