Pattern Recognition, 4th Edition

2.8. Problems

1. Show that in a multiclass classification task, the Bayes decision rule minimizes the error probability.Hint: It is easier to work with the probability of correct decision.
2. In a two-class one-dimensional problem, the pdfs are the Gaussians N (0, σ²) and N(1, σ²) for the two classes, respectively. Show that the threshold x⁰ minimizing the average risk is equal towhere λ¹¹ = λ²² = 0 has been assumed.
3. Consider a two equiprobable class problem with a loss matrix L. Show that if ∈¹ is the probability of error corresponding to feature vectors from class ω¹and ∈² for those from class ω², then the average risk r is given by
4. Show that ...