Consider the Neyman-Pearson criterion for two univariate normal distributions: p(x|ωi) ∼ N(µi, σ2i )...

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Consider the Neyman-Pearson criterion for two univariate normal distributions: p(x|ω_i) ∼ N(µ_i, σ2_i ) and P(ωi)=1/2 for i = 1, 2. Assume a zero-one error loss, and for convenience µ₂ > µ₁.

(a) Suppose the maximum acceptable error rate for classifying a pattern that is actually in ω1 as if it were in ω₂ is E₁. Determine the decision boundary in terms of the variables given.

(b) For this boundary, what is the error rate for classifying ω2 as ω1?

(c) What is the overall error rate under zero-one loss?

(d) Apply your results to the specific case p(x|ω₁) ∼ N(−1, 1) and p(x|ω₂) ∼ N(1, 1) and E1 = 0.05.

(e) Compare your result to the Bayes error rate (i.e., without the Neyman-Pearson conditions).

Bhaskar · Accepted Answer

Consider the Neyman-Pearson criterion for two univariate normal distributions: p(x|ωi) ∼ N(µi, 
σ2i ) and P(ωi)=1/2 for i = 1, 2. Assume a zero-one error loss, and for convenience µ2 > µ1. 
(a) Suppose the maximum acceptable error rate for classifying a pattern that is actually in ω1 as 
if it were in ω2 is E1.

Consider the Neyman-Pearson criterion for two univariate normal distributions: p(x|ω i ) ∼ N(µ i , σ2 i ) and P(ωi)=1/2 for i = 1, 2. Assume a zero-one error loss, and for convenience µ 2 > µ 1 . (a)...

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