Question: Consider the minimax criterion for a two-category classification problem.
(a) Fill in the steps of the derivation of Eq. 22.
(b) Explain why the overall Bayes risk must be concave down as a function of the prior P(ω1), as shown in Fig. 2.4.
(c) Assume we have one-dimensional Gaussian distributions p(x|ωi) ∼ N(µi, σ2 i ), i = 1, 2 but completely unknown prior probabilities. Use the minimax criterion to find the optimal decision point x∗ in terms of µi and σi under a zero-one risk.
(d) For the decision point x∗ you found in (??), what is the overall minimax risk? Express this risk in terms of an error function erf(·).
(e) Assume p(x|ω1) ∼ N(0, 1) and p(x|ω2) ∼ N(1/2, 1/4), under a zero-one loss. Find x∗ and the overall minimax loss.
(f) Assume p(x|ω1) ∼ N(5, 1) and p(x|ω2) ∼ N(6, 1). Without performing any explicit calculations, determine x∗ for the minimax criterion. Explain your reasoning.