# ELEC 303 – Random Signals (PowerPoint) by shuifanglj

VIEWS: 20 PAGES: 36

• pg 1
```									ELEC 303 – Random Signals

Lecture 17 – Hypothesis testing 2
Dr. Farinaz Koushanfar
ECE Dept., Rice University
Nov 2, 2009
outline
• Bayesian Hypothesis testing
• Likelihood Hypothesis testing
Four versions of MAP rule
•  discrete, X discrete
p  () p X| ( x | )

•  discrete, X continuous
p  ()f X| ( x | )

•  continuous, X discrete
f  () p X| ( x | )
•  continuous, X continuous
f  (  ) f X | ( x |  )
Example – spam filter
• Email may be spam or legitimate
• Parameter , taking values 1,2, corresponding to
spam/legitimate, prob p(1), P(2) given
• Let 1,…, n be a collection of special words,
whose appearance suggests a spam
• For each i, let Xi be the Bernoulli RV that denotes
the appearance of i in the message
• Assume that the conditional prob are known
• Use the MAP rule to decide if spam or not.
Bayesian Hypothesis testing
• Binary hypothesis: two cases
• Once the value x of X is observed, Use the Bayes
rule to calculate the posterior P|X(|x)
• Select the hypothesis with the larger posterior
• If gMAP(x) is the selected hypothesis, the correct
decision’s probability is       P(= gMAP(x)|X=x)
• If Si is set of all x in the MAP, the overall probability
of correct decision is
P(= gMAP(x))=i P(=i,XSi)
• The probability of error is: i P(i,XSi)
Multiple hypothesis
Example – biased coin, single toss
• Two biased coins, with head prob. p1 and p2
• Randomly select a coin and infer its identity
based on a single toss
• =1 (Hypothesis 1), =2 (Hypothesis 2)
• X=0 (tail), X=1(head)
• MAP compares P(1)PX|(x|1) ? P(2)PX|(x|2)
• Compare PX|(x|1) and PX|(x|2) (WHY?)
• E.g., p1=.46 and p2 =.52, and the outcome tail
Example – biased coin, multiple tosses
• Assume that we toss the selected coin n times
• Let X be the number of heads obtained
• ?
Example – signal detection and
matched filter
• A transmitter sending two messages =1,=2
• Massages expanded:
– If =1, S=(a1,a2,…,an), if =2, S=(b1,b2,…,bn)
• The receiver observes the signal with
corrupted noise: Xi=Si+Wi, i=1,…,n
• Assume WiN(0,1)
Likelihood Approach to Binary
Hypothesis Testing
BHT and Associated Error
Likelihood Approach to BHT (Cont’d)
Likelihood Approach to BHT (Cont’d)
Binary hypothesis testing
• H0: null hypothesis, H1: alternative hypothesis
• Observation vector X=(X1,…,Xn)
• The distribution of the elements of X depend
on the hypothesis
• P(XA;Hj) denotes the probability that X
belongs to a set A, when Hj is true
Rejection/acceptance
• A decision rule:
– A partition of the set of all possible values of the
observation vector in two subsets: “rejection
region” and “acceptance region”
• 2 possible errors for a rejection region:
– Type I error (false rejection): Reject H0, even
though H0 is true
– Type II error (false acceptance): Accept H0, even
though H0 is false
Probability of regions
• False rejection:
– Happens with probability
(R) = P(XR; H0)
• False acceptance:
– Happens with probability
(R) = P(XR; H1)
Analogy with Bayesian
• Assume that we have two hypothesis =0 and
=1, with priors p(0) and p(1)
• The overall probability of error is minimized using
the MAP rule:
–   Given observations x of X, =1 is true if
–   p(0) pX|(x|0) < p(1) pX|(x|1)
–   Define:         = p(0) / p(1)
–   L(x) = pX|(x|1) / pX|(x|0)
• =1 is true if the observed values of x satisfy the
inequality: L(x)> 
More on testing
• Motivated by the MAP rule, the rejection
region has the form R={x|L(x)>}
• The likelihood ratio test
– Discrete:
L(x)= pX(x;H1) / pX(x;H0)
– Continuous:
L(x) = fX(x;H1) / fX(x;H0)
Example
• Six sided die
• Two hypothesis

• Find the likelihood ratio test (LRT) and
probability of error
Error probabilities for LRT
• Choosing  trade-offs between the two error
types, as  increases, the rejection region
becomes smaller
– The false rejection probability (R) decreases
– The false acceptance probability (R) increases
LRT
• Start with a target value  for the false
rejection probability
• Choose a value  such that the false rejection
probability is equal to :
P(L(X) > ; H0) = 
• Once the value x of X is observed, reject H0 if
L(x) > 
• The choices for  are 0.1, 0.05, and 0.01
Requirements for LRT
• Ability to compute L(x) for observations X
• Compare the L(x) with the critical value 
• Either use the closed form for L(x) (or log L(x))
or use simulations to approximate
Example
• A camera checking a certain area
• Recording the detection signal
• X=W, and X=1+W depending on the presence
of the intruders (hypothesis H0 and H1)
• Assume W~N(0,)
• Find the LRT and acceptance/rejection region
Example
Example
Example
Example (Cont’d)
Example (Cont’d)
Error Probabilities
Example: Binary Channel
Example: Binary Channel
Example: Binary Channel
Example: More on BHT
Example: More on BHT
Example: More on BHT
Example: More on BHT

```
To top