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Recall Example Questions Detection Basics H.Birkan YILMAZ1 1 Bogazici University, Department of Computer Engineering, Turkey CMPE591 Recall Example Questions Outline 1 Recall Optimality Frameworks 2 Example Questions Bayesian Optimality Minimax Optimality NP Optimality Recall Example Questions Optimality Frameworks Binary Hypothesis Testing (again) Goal Main goal is to decide between two hypothesis H0 Y ∼ p0 H1 Y ∼ p1 Recall Example Questions Optimality Frameworks Bayesian Optimality (again) Known Parameters π0 , π1 cij pi Goal Minimizing the average cost. Recall Example Questions Optimality Frameworks MiniMax Optimality (again) Known Parameters cij pi Goal Minimizing maximum cost or equivalently finding the equalizer rule. Min Max{R0 (δ), R1 (δ)} δ Recall Example Questions Optimality Frameworks Neyman-Pearson Optimality (again) Known Parameters pi Goal Z Maximizing PD = Z p1 (y )dy Γ1 p0 (y )dy ≤ α s.t.PF = Γ1 Remark Maximum PD is achived when PF = α for continuous conditional pdfs. Recall Example Questions Bayesian Optimality V. H. Poor II.F.2.a Example Suppose Y is a random variable that, under H0 , has pdf 2 3 (y + 1) 0 ≤ y ≤ 1 p0 (y ) = 0 otherwise and, under hypothesis H1 , has pdf 1 0≤y ≤1 p1 (y ) = 0 otherwise Find the Bayes rule and minimum Bayes risk for testing H0 versus H1 with uniform costs and equal priors. Recall Example Questions Minimax Optimality V. H. Poor II.F.2.b Example Suppose Y is a random variable that, under H0 , has pdf 2 3 (y + 1) 0 ≤ y ≤ 1 p0 (y ) = 0 otherwise and, under hypothesis H1 , has pdf 1 0≤y ≤1 p1 (y ) = 0 otherwise Find the Minimax rule and Minimax risk for uniform costs. Recall Example Questions NP Optimality V. H. Poor II.F.2.c Example Suppose Y is a random variable that, under H0 , has pdf 2 3 (y + 1) 0 ≤ y ≤ 1 p0 (y ) = 0 otherwise and, under hypothesis H1 , has pdf 1 0≤y ≤1 p1 (y ) = 0 otherwise Find the Neyman-Pearson rule and the corresponding detection probability for false-alarm probability α ∈ (0, 1).