LDA with unequal priors

Topic: Statistical classifiers and discriminant analysis · more questions on this topic
‹ Unequal costs A quadratic boundary ›

Redo the two-class LDA example with priors \widehat\pi_1=\tfrac23 and \widehat\pi_2=\tfrac13. Give the new boundary and classify (3.4,3.2).

Worked answer (click to show)

A change in priors shifts the intercept and leaves the shared-covariance slope unchanged. Now

\alpha=\log 2+9.75=0.693+9.75=10.443,

and \boldsymbol\beta is unchanged, so the boundary is 10.443-1.5(x_1+x_2)=0, that is

x_1+x_2=6.962.

The boundary has moved away from class 1, the more probable class, by 6.962-6.5=0.462. For (3.4,3.2) the sum is 6.6<6.962, so the point is class 1. Under equal priors, 6.6>6.5 gave class 2. The prior changes the decision at this point.