Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 46 / 2 / b

Use a multivariate normal distribution within each species, a common positive-definite covariance matrix, equal prior odds and equal misclassification costs. Assume the training observations are independent and representative of the populations. Substituting the sample means and pooled sample covariance matrix gives a fitted linear discriminant analysis rule, rather than a rule with known population parameters.

The determinant of the pooled covariance is $8$, soThe midpoint of the sample means is $(4,6{)}^T$, whose scalar product with $L$ is $-11/2$. Hence the fitted log likelihood ratio isAssign fattus when $q(x)>0$ and apathus when $q(x)<0$; either assignment is optimal on $q(x)=0$ under equal priors. The separating line is $x_2=0.4x_1+4.4$, with fattus below it. As a sign check, the scores at the two sample means are $19/4$ and $-19/4$, respectively. This uses the corrected Gaussian Bayes classifier from (a).

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).

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.