ROC Curves for Continuous Data

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“k10031final” — 2009/4/6 — 12:30 — page 138 — #146

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138 CHAPTER 7. BAYESIAN METHODS

cases: x

1P

,x

1,N

denote the number of individuals labeled P, N respec-

tively that are classified N by S,andx

KP

,x

KN

denote the number of

individuals labeled P, N respectively that are classified P by S.By

deriving the probability of each occurrence in terms of the model pa-

rameters, the likelihood of the sample can be shown to have the ordered

multinomial form

L∝

+

K+1

i=1

[θ(β

(i)

−β

(i−1)

)(1 − β)+(1− θ)(α

(i−1)

− α

(i)

)α]

x

iP

×

[θ(β

(i)

−β

(i−1)

)β +(1− θ)(α

(i−1)

− α

(i)

)(1 − α)]

x

iN

.

However, a basic problem with this model is that there are 2K +

3 parameters (the α

(·)

,β

(·)

,α,β,andθ), but only 2K + 1 degrees of

freedom for estimation. Hence the model is unidentifiable and not

all parameter ...