The optimal error rate of Bayes rule for two-class Gaussian homoscedastic model
errorrate(beta0, beta, pi, mu, sigma)A vector of error rate.
An intercept parameter of the discriminant function coefficients.
A \(p \times 1\) vector for the slope parameter of the discriminant function.
A g-dimensional vector for the initial values of the mixing proportions.
A \(p \times g\) matrix for the initial values of the location parameters.
A \(p\times p\) covariance matrix.
The optimal error rate of Bayes rule for two-class Gaussian homoscedastic model can be expressed as $$ err(\beta)=\pi_1\phi\{-\frac{\beta_0+\beta_1^T\mu_1}{(\beta_1^T\Sigma\beta_1)^{\frac{1}{2}}}\}+\pi_2\phi\{\frac{\beta_0+\beta_1^T\mu_2}{(\beta_1^T\Sigma\beta_1)^{\frac{1}{2}}}\} $$ where \(\phi\) is a normal probability function with mean \(\mu_i\) and covariance matrix \(\Sigma_i\).