Tied covariance matrices -> Linear decision boundaries

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

$= \log π_c − µ^T_c Σ^{−1} µ_c +x^T Σ^−1 µ_c +const − x^T Σ^{−1} x$

$=\gamma_c + x^T\beta_c + \kappa$

So the decision boundaries occur when

$\gamma_c + x^T\beta_c + \kappa = \gamma_{c'} + x^T\beta_{c'} + \kappa$

$x^T(\beta_c - \beta_{c'}) = \gamma_{c'} - \gamma_{c}$

GOTO workbook