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Under equal costs for either wrong assignment, the Bayes classifier chooses the class of greatest posterior probability. With positive prior probabilities, it assigns class one when ${\pi }_1{f}_1(x)>{\pi }_2{f}_2(x)$; ties can be resolved arbitrarily. For the two multivariate normal densities with common positive-definite covariance matrix $V$, their normalizing factors cancel. The log posterior odds areTherefore the Gaussian Bayes classifier has the required linear form, withA larger prior probability for class one decreases $b$, expanding its decision region. The prior-dependent Gaussian discriminant boundary is the hyperplane $a^{T}x=b$.

Now the prior odds are ${\pi }_f/{\pi }_a=2$. The fitted log posterior odds are $q(x)+\log 2$, so the Gaussian Bayes classifier assigns fattus when $q(x)>-\log 2$. The prior-dependent Gaussian discriminant boundary becomesIt moves upwards parallel to the original line by $0.8\log 2\simeq 0.5545$, expanding the fattus region. This is the dashed line in the sketch. The fifth individual's fitted log likelihood ratio remains zero, but its posterior odds become $2:1$, giving posterior fattus probability $2/3$.

The fifth individual is now classified as fattus. For completeness, the other four classifications remain apathus, fattus, apathus and fattus: their scores lie on the same side of the new threshold as before.

Evaluating the fitted linear discriminant analysis score givesThe fifth individual lies exactly on the discriminant boundary. Its two fitted posterior probabilities are equal, so the equal-prior Bayes classifier cannot prefer either species. A specified tie convention could assign it to one species, but such an assignment would not be evidence favoring that species.

In the sketch the solid line is the equal-prior boundary. The dashed line is the prior-dependent Gaussian discriminant boundary for the altered priors in (iii).