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A lecture note from the university of new mexico, department of electrical and computer engineering, for the course ece642: detection and estimation theory. The notes cover the topic of neyman-pearson hypothesis testing, focusing on false alarms and misses. The concept of false alarms as the probability of declaring the presence of a target when it is actually absent, and misses as the probability of failing to detect a target when it is present. The document also discusses the trade-off between false alarms and misses and the neyman-pearson optimality criterion for making this trade-off.
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ECE642: Detection and Estimation Theory
Dr. Sudharman K. Jayaweera
Assistant Professor
Department of Electrical and Computer Engineering
University of New Mexico
Lecture 04 - August
th , Thursday
Fall 2007
ECE642: Detection and Estimation Theory
Neyman Pearson (classical) Hypothesis Testing
as the average risk).was defined in terms of minimizing the overall expected cost (definedIn the Bayesian formulation for the hypothesis testing the optimality
knowledge of the prior probabilities of the two hypotheses.This required a specific cost structure on the decisions and also the
probabilities are not possible or desirable.of a specific cost structure on the decisions made and the priorIn many practical problems of interest, however, such an imposition
In such cases another optimality criteria named
Neyman Pearson
optimality is often used.
ECE642: Detection and Estimation Theory
False Alarms
In radar or sonar problems, the two hypothesis
0 and
1 usually
correspond to the absence and the presence of a target.
(acceptingThus, Type I errors corresponds to declaring there is a target
1 ) when there is no target
For this reason, Type I errors are called the
false alarms
For a decision rule
δ , the probability of a type I error is known as the
size of
δ
or the
false alarm probability
or the
false alarm rate
of
δ ,
and is denoted by
F (^) ( δ ) .
ECE642: Detection and Estimation Theory Misses
present (i.e. acceptingSince a Type II error corresponds to declaring that there is no target
0 as true) when in fact there is a target (i.e
1
is the true hypothesis), this represents a
miss
Thus, type II errors are called misses
For a decision rule
δ , the probability of type II errors is called the
miss probability
and denoted by
P M ( δ ).
In the above terminology, a correct acceptance of
1 , (i.e declaring
that there is a target when in fact there is one.) is called a
detection
We denote by
D ( δ ) the
detection probability
This is called the
power of
δ
Clearly,
P D ( δ ) = 1
M
( δ )
ECE642: Detection and Estimation Theory
Neyman-Pearson Optimality Criterion
bound on the false alarm probabilityThe Neyman-Pearson criterion for making this trade off is to place a
F (^) ( δ ) and then to minimize the
miss probability
M ( δ ) subjected to this constraint.
Using (2), the Neyman-Pearson design criterion is
max δ P D ( δ )
subject to
F (^) ( δ ) ≤
(^) α
where
α
is the upper bound on the false alarm probability.
α
is known as the
level of the test
or the
significance level of the test
most powerful Thus, according to (3), the Neyman-Pearson design goal is to find the
α -level test
of
0
versus
1 .
ECE642: Detection and Estimation Theory
Few Remarks
criterion)asymmetry in importance of the two hypotheses (unlike the BayesThe Neyman-Pearson criterion allows to recognize the basic
Neyman-Pearson hypothesis testing is also called the
classical
hypothesis testing
radar and sonar applications whileTraditionally it is the most commonly used optimality criteria in
Bayes criterion
is the common
choice in communication systems.
ECE642: Detection and Estimation Theory
Randomize and Non-randomized Tests
Thus, according to the above definition, the
non-randomized
decision rules
that we used earlier are a special case of the
randomized decision rules
In particular, a non-randomize rule
δ corresponds to the randomize
rule
δ˜ ( y ) =
δ ( y )
ECE642: Detection and Estimation Theory
False Alarm Probability of a Test
probability with which it acceptsRecall that, the false alarm probability of a decision rule is the
1 given that
0 is the true
hypothesis.
Since
δ˜ ( y ) of a randomize test is the conditional probability of
accepting
1 given
(^) , we may obtain the false alarm probability of a
randomize test by averaging
δ˜ ( y ) over the distribution of
under
0 .
Since the density of
under
0 is
p 0 ( y ) , the false alarm probability
of the test
δ˜ P is: F (^) ( δ˜ )
0 { δ˜ ( Y (^) ) }
Z Γ δ ˜ ( y ) p 0 ( y ) μ (
dy
where
0 (^) { . }
denotes expectation under hypothesis
0 (we may
sometime write this more informatively as
Y (^) | H 0 (^) {
. } )
ECE642: Detection and Estimation Theory
Proposition 4.1: The Neyman-Pearson Lemma
Consider the hypothesis pair of (1) in which distribution
j has
density
p j^ for
(^) 0 and
(^) 1. Suppose that
α (^) >
following statements are true: i)
Optimality:
Let
δ˜
be any decision rule satisfying
F (^) ( δ˜ ) ≤
(^) α
and
let
δ˜ ′ be any decision rule of the form
δ˜ ′ ( y ) =
if
p 1 ( y ) > η p 0 ( y )
γ (y)
if
p 1 ( y ) =
η p 0 ( y )
if
p 1 ( y ) < η p 0 ( y )
where
η (^) ≥
(^) 0 and 0
(^) γ ( y ) (^) ≤
1 are such that
F (^) ( δ˜ ′ ) =
α
. Then
D ( δ˜ ′ ) ≥
D ( δ˜ )
That is any size-
α
decision rule of the form of (17) is an N-P rule.
ECE642: Detection and Estimation Theory
Proposition 4.1: The Neyman-Pearson Lemma (ctd...)
ii)
Existence:
For every
α (^) ∈
there is a decision rule
δ˜ NP
of the
form of (17) with
γ ( y ) =
γ 0 (a constant), for which
F (^) ( δ˜ NP
α .
iii)
Uniqueness:
Suppose that
δ˜ ” is any
α -level Neyman-Pearson
optimal decision rule for
0 versus
1
. Then
δ˜ ” must be of the form
of (17), except possibly on a subset of
having zero probability
under
0
and
1
ECE642: Detection and Estimation Theory
Proof of the Neyman-Pearson Lemma: (i) Optimality (ctd...)
Using (4) and (5) in (9)
D ( δ˜ ′ ) (^) −
(^) P D ( δ˜ ) ≥ η ( P F
δ˜ ′ ) (^) −
(^) P F (^) ( δ˜ ) )
η ( α (^) −
(^) P F (^) ( δ˜ ) )
Since
F (^) ( δ˜ ′ ) =
(^) α
Since
F (^) ( δ˜ ) (^) ≤
α
Thus,
D ( δ˜ ′ ) ≥
(^) P D ( δ˜ ) as required, and any size
α
decision rule of the
form of (17) is a Neyman-Pearson rule.
ECE642: Detection and Estimation Theory
Proof of the Neyman-Pearson Lemma: (ii) Existence (ctd...)
Let
η (^) =
(^) η
0 be the smallest number such that
0 (^) ( p 1 ( Y (^) ) > η p 0 ( Y
α
Figure 1:
Note that
(^) P
0 (^) ( p 1 ( Y (^) ) (^) >
(^) η
p 0 ( Y (^) ))
increases as
η (^) decreases.
ECE642: Detection and Estimation Theory
Proof of The Neyman-Pearson Lemma: (iii) Uniqueness (ctd...)
Suppose that
δ˜ ′ is an
α -level Neyman-Pearson rule of the form of
(17) with
F (^) ( δ˜ ′ ) =
α , and let
δ˜ ′′ be any other
α -level
Neyman-Pearson rule.
detectors, they must haveThen by the definition of the Neyman-Pearson optimality of the two
D ( δ˜ ′ ) =
D ( δ˜ ′′ )
Then, from (10) (letting
δ˜ ′′ be the test
δ˜ in (10)):
α (^) −
(^) P F (^) ( δ˜ ′′ ) )
≥
(^0)
since
F (^) ( δ˜ ′′ ) (^) ≤
α
F (^) ( δ˜ ′′ )
α
ECE642: Detection and Estimation Theory
Proof of The Neyman-Pearson Lemma: (iii) Uniqueness (ctd...)
to (8), we getBy starting with (14) and (15), and then working backward from (10)
D ( δ˜ ′ ) (^) −
(^) P D ( δ˜ ′′ ) =
(^) η
( α (^) −
(^) P F (^) ( δ˜ ′′ ) )
D ( δ˜ ′ ) (^) −
(^) P D ( δ˜ ′′ ) (^) −
(^) η
( α (^) −
(^) P F (^) ( δ˜ ′′ ) )
=
(^0)
Z Γ δ ˜ ′ p 1 dμ
Z Γ δ ˜ ′′ p 1 dμ
(^) η
( Z Γ δ ˜ ′ p 0 dμ
Z Γ δ ˜ ′′ p 0 dμ
Z Γ ( δ˜ ′ ( y ) (^) −
(^) δ˜ ′′ ( y ) ) ( p 1 ( y )
(^) η
p 0 ( y ))
(^) μ ( dy
Since the integrand is non-negative (note that
δ˜ ′′ also can take values
only in
), (16) implies that, the integrand is equal to zero except
possibly on a set of zero probability under
0 and
1
Thus
δ˜ ′ and
δ˜ ′′ differ only on the set
y (^) ∈
Γ | p 1 ( y
η p 0 ( y ) }
, which
