Explain why the overall bayes risk must be concave down

Reference no: EM132096969

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.

Reference no: EM132096969

Questions Cloud

The actual demand observed : The following table shows the actual demand observed over the last 11 years:

What is the effective annual rate on loan : You want to buy a new sports coupe for $79,500, and the finance office at the dealership has quoted you an APR of 5.8 percent for a 60-month loan to buy the car

For team to be successful over the long run : “For a team to be successful over the long run, it must be structured in a way that both maintains its members’ social well-being and accomplishes its task.”

Does company use crm software for retaining customers : Does the company use CRM software for retaining customers? Discuss how the dilemmas of teamwork vs. individual work might be intensified in a virtual team.

Explain why the overall bayes risk must be concave down : Fill in the steps of the derivation of Eq. 22. Explain why the overall Bayes risk must be concave down as a function of the prior P(?1).

Mistakes that holmes made in interviewing smith and johnson : Name at least three mistakes that Holmes made in interviewing Smith and Johnson

Find the distance between the sun and venus : Venus has a period of revolution of 225 Earth days. Find the distance between the Sun and Venus as a multiple of Earth's orbital radius (an astronomical unit).

Three challenges manager faces in motivating employees : Identify three challenges a manager faces in motivating employees today as opposed to 50 years ago.

Prove that the true error cannot decrease : Prove that the true error cannot decrease if we first project the distributions to a lower dimensional space and then classify them.

User Account

All Pages