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Detection & EstimationLecture 9
Sequential Tests
Xiliang Luo
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decay rate
decay rate
Repeated Observations
ln
exp exp
For any ∈ , , both false alarm and miss rates decay exponentially!
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Sequential Hypothesis Testing
• For binary hypothesis testing, we can take a series of measurements. After completing measurements, we decide whether to take one more observation or to make a decision just with these measurements
• Two decisions to make:• Stopping rule:
• Terminal decision rule:
, … ,1, 0,
, … , 0or1
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only when , … , 1
Sequential Testing• Assume zero cost for correct decision
• False alarm cost:
• Missing cost:
• Each additional sample cost:
• Let denote the stopping time, we have
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Overall Bayes Risk:
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Sequential Testing
• Optimal Bayesian sequential decision rule ,is the one minimizing the overall cost
• Consider the set of sequential decision rules taking at least one sample
• the min Bayes risk/cost strategy is:
5First sample analysis
Sequential Bayesian Test
• When 0or1, the best decision will make no error, so
• , is linear in
• We have the following fact• is concave in
• For zero sample case, either decide 1 or decide 0:
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Sequential Bayesian Test
• Zero sample risk:
• Bayes risk over all strategies satisfies:
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more likely
more likely
Sequential Bayesian Test
1 2
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• After getting one more sample, we have the conditional prior:
• Like the 0th step, with this conditional prior, we can have the “no‐sampling” cost:
• Meanwhile, addition cost for strategies with at least one more sample is !
• Back to previous figure!
Sequential Bayesian Test
,
1 , . .
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Optimal Bayes Decision Rule
Reasonable assumption:
:
the cost of deciding when is true
Likelihood Ratio Test (LRT)
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Optimal Bayes Decision Rule
Reasonable assumption:
:
the cost of deciding when is true
Likelihood Ratio Test (LRT)
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Equivalently, we have:
In case of zero sample, the rule is:
Sequential Bayesian Test
• We have the following recursion rule:
• Then, we can also get:
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Sequential Bayesian Test• Assuming n samples have been obtained:
, , … , , the posterior probability becomes:
• Optimal Bayesian rule:
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Sequential Bayesian Test• The threshold rule: translates to LRT:
Optimal Bayesian rule: Sequential Probability Ratio Test!14
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Sequential Probability Ratio Test
• SPRT with lower and upper thresholds and such that: 0 1 ∞:
• Select H0 whenever:
• Select H1 whenever:
• Take another sample if:
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• Convenient to express the test in logarithmic form:
• Select H0: • Λ
• Select H1:• Λ
• Keep sampling: • Λ ∈ ,
SPRT
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SPRT
• Random walk is the sum of IID random variables
• Adding zero‐mean fluctuations to the mean trajectories
• ⋅ under
• ⋅ under
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SPRT
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SPRT• Stopping time of SPRT:
• A random process , 0 is a martingale if it satisfies the following two properties:
• 1. ∞• 2. , 0
• A nonnegative random variable is a stopping time adapted to the martingale if
• Event can be expressed in terms of the values , 0,1, … ,
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SPRT‐Wald’s Identity
• Wald’s Identity• Let ; 1 be a sequence of i.i.d. random variables whose generating function : is defined over , , 0 and let Λ ⋯ .
• Let be the martingale defined as .
• Let N be an adapted stopping time such that:
Then:
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Λ : sum of LLR at the time of stopping
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SPRT‐Wald’s Identity
• When ln , its generating functions are under and 1 under :
• With SPRT, we have
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1⋅
1
⋅
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SPRT‐Wald’s Identity
• Under , with SPRT, we have
| 1
expΛ Λ , expΛ Λ , 1
1 zero‐overshoot approximation
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SPRT
• Under , at 1, using the Wald’s identity:
exp Λ 1
exp Λ Λ , exp Λ Λ , 1
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Under zero‐overshoot approx., falsealarm and miss prob can be expressedin terms of the thresholds A and B.
Conversely, given desired false alarmand miss prob, we can set the thresholdsaccordingly! 23
1⋅
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SPRT• Take differentiation of the following wrt :
• we have:
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Under :
Under :
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SPRT• Due to the fact that:
• We have:
• Typically, , ≪ 1, we have:
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Sequential vs Fixed‐size Tests
• SPRT:
• Fixed‐size tests with 0:
ln0
ln,
, min∈ ,
lnChernoff distance
Chernoff distance is alwaysless than the KL divergence!
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Optimality of SPRT
• Property: Let , denote an SPRT with thresholds , . Let , be an arbitrary sequential decision
rule such that
Then
Among all sequential decision rules achieving certain false alarm and miss prob, SPRT requires on average the smallest number of samples!
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