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Course by: Clayton Scott. E-mail the author

# The Bayes Risk Criterion in Hypothesis Testing

Module by: Clayton Scott, Robert Nowak. E-mail the authors

The design of a hypothesis test/detector often involves constructing the solution to an optimization problem. The optimality criteria used fall into two classes: Bayesian and frequent.

In the Bayesian setup, it is assumed that the a priori probability of each hypothesis occuring ( π i π i ) is known. A cost C ij C ij is assigned to each possible outcome: C ij =Pr say H i when H j true C ij say H i when H j true The optimal test/detector is the one that minimizes the Bayes risk, which is defined to be the expected cost of an experiment: C-= i , j C ij π i Pr say H i when H j true C i j C ij π i say H i when H j true

In the event that we have a binary problem, and both hypotheses are simple, the decision rule that minimizes the Bayes risk can be constructed explicitly. Let us assume that the data is continuous (i.e., it has a density) under each hypothesis: H 0 : x f 0 x H 0 : x f 0 x H 1 : x f 1 x H 1 : x f 1 x Let R 0 R 0 and R 1 R 1 denote the decision regions corresponding to the optimal test. Clearly, the optimal test is specified once we specify R 0 R 0 and R 1 = R 0 R 1 R 0 .

The Bayes risk may be written

C - = ij =01 C i j π i f j xd x = C 00 π 0 f 0 x+ C 01 π 1 f 1 xd x + C 10 π 0 f 0 x+ C 11 π 1 f 1 xd x C - i j 0 1 C i j π i x R i f j x x R 0 C 00 π 0 f 0 x C 01 π 1 f 1 x x R 1 C 10 π 0 f 0 x C 11 π 1 f 1 x
(1)
Recall that R 0 R 0 and R 1 R 1 partition the input space: they are disjoint and their union is the full input space. Thus, every possible input xx belongs to precisely one of these regions. In order to minimize the Bayes risk, a measurement xx should belong to the decision region R i R i for which the corresponding integrand in the preceding equation is smaller. Therefore, the Bayes risk is minimized by assigning xx to R 0 R 0 whenever π 0 C 00 f 0 x+ π 1 C 01 f 1 x< π 0 C 10 f 0 x+ π 1 C 11 f 1 x π 0 C 00 f 0 x π 1 C 01 f 1 x π 0 C 10 f 0 x π 1 C 11 f 1 x and assigning xx to R 1 R 1 whenever this inequality is reversed. The resulting rule may be expressed concisely as Λx f 1 x f 0 x H 0 H 1 π 0 ( C 10 C 00 ) π 1 ( C 01 C 11 )η Λ x f 1 x f 0 x H 0 H 1 π 0 C 10 C 00 π 1 C 01 C 11 η Here, Λx Λ x is called the likelihood ratio, ηη is called the threshold, and the overall decision rule is called the Likelihood Ratio Test (LRT). The expression on the right is called a threshold.

## Example 1

An instructor in a course in detection theory wants to determine if a particular student studied for his last test. The observed quantity is the student's grade, which we denote by r r. Failure may not indicate studiousness: conscientious students may fail the test. Define the models as

• 0 0 : did not study
• 1 1 : did study
The conditional densities of the grade are shown in Figure 1. Based on knowledge of student behavior, the instructor assigns a priori probabilities of π 0 =14 π 0 1 4 and π 1 =34 π 1 3 4 . The costs C i j C i j are chosen to reflect the instructor's sensitivity to student feelings: C 01 =1= C 10 C 01 1 C 10 (an erroneous decision either way is given the same cost) and C 00 =0= C 11 C 00 0 C 11 . The likelihood ratio is plotted in Figure 1 and the threshold value η η, which is computed from the a priori probabilities and the costs to be 13 1 3 , is indicated. The calculations of this comparison can be simplified in an obvious way. r50 0 1 13 r 50 0 1 1 3 or r 0 1 503=16.7 r 0 1 50 3 16.7 The multiplication by the factor of 50 is a simple illustration of the reduction of the likelihood ratio to a sufficient statistic. Based on the assigned costs and a priori probabilities, the optimum decision rule says the instructor must assume that the student did not study if the student's grade is less than 16.7; if greater, the student is assumed to have studied despite receiving an abysmally low grade such as 20. Note that as the densities given by each model overlap entirely: the possibility of making the wrong interpretation always haunts the instructor. However, no other procedure will be better!

A special case of the minimum Bayes risk rule, the minimum probability of error rule, is used extensively in practice, and is discussed at length in another module.

## Problems

### Exercise 1

Denote α=Pr declare H 1 when H 0 true α declare H 1 when H 0 true and β=Pr declare H 1 when H 1 true β declare H 1 when H 1 true . Express the Bayes risk C - C - in terms of α α and ββ, C i j C i j , and π i π i . Argue that the optimal decision rule is not altered by setting C 00 = C 11 =0 C 00 C 11 0 .

### Exercise 2

Suppose we observe x x such that Λx=η Λ x η . Argue that it doesn't matter whether we assign x x to R 0 R 0 or R 1 R 1 .

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