# Detection Basics

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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).