Gaussian discriminants -> Quadratic decision boundaries

Log posterior is “The discriminant function”

\(\log p(y = c|x, θ) = \log π_c − (1/2)\log|2πΣ_c | − (x − µ_c )^T Σ^{−1}_c (x − µ_c ) + const\)

Let \(p(y=c|x,\theta)=p(y=c'|x,\theta)\)

Then

\((x − µ_c )^TΣ^{−1}_c(x − µ_c) - (x − µ_{c'} )^T Σ^{−1}_{c'} (x − µ_{c'})=f(\pi_c,\pi_c',\Sigma_c,\Sigma_c')\)

So the decision boundaries between classes are quadratic in \(x\).

GOTO workbook