Determine the threshold value that achieves minimum probability of error, and on the ROC curve, superimpose clearly (using a different color/shape marker) the true positive and false positive values attained by this naive-Bayesian model based minimum-P(error) classifier. Calculate and report an estimate of the minimum probability of error that is achievable by the naive-Bayesian classification rule for this (true) data distribution.
In the third part of this exercise, repeat the same steps as in the previous two cases for the Fisher Linear Discriminant Analysis based classifier. Using the 10000 available samples, with sample average based estimates for mean and covariance matrix for each class, determine the Fisher LDA projection vector w_LDA. For the classification rule w_LDA^T x compared to a threshold τ, which takes values from -∞ to ∞, trace the ROC curve, identify the threshold at which the probability of error (based on sample count estimates) is minimized, and clearly mark that operating point on the ROC curve.
Note: In order for us to have a uniform solution across all submissions, when finding the Fisher LDA projection matrix, do not be concerned about the difference in the class priors. When determining the between-class and within-class scatter matrices, use equal weights for the class means and covariances, like we did in class.