4.6 Sequential hypothesis testing  (Page 2/3)

Rather explicitly attempting to relate thresholds to performance probabilities, we obtain simpler results by usingbounds and approximations. Note that the expression for $P_D$ may be written as $$P_D=\int_{{}_1} \frac{p(r, {}_1, r)}{p(r, {}_0, r)}p(r, {}_0, r)\,d r=\int_{{}_1} {}_L(r)p(r, {}_0, r)\,d r$$ In the decision region ${}_1$ , ${}_L(r)\ge {}_1$ ; thus, a lower bound on the detection probability can be established by substituting this inequality into theintegral. $$P_D\ge {}_1\int_{{}_1} p(r, {}_0, r)\,d r$$ The integral is the false-alarm probability $P_F$ of the test when it terminates. In this way, we find that $\frac{P_D}{P_F}\ge {}_1$ . Using similar arguments on the miss probability, we obtain a similar bound on the threshold ${}_0$ . These inequalities are summarized as $${}_0\ge \frac{1-P_D}{1-P_F}$$ and $${}_1\le \frac{P_D}{P_F}$$ These bounds, which relate the thresholds in the sequential likelihood ratio test with the false-alarm and detectionprobabilities, are general, applying even when sequential tests are not being used. In the usual likelihood ratio test,there is a single threshold $$ ; these bounds apply to it as well, implying that in a likelihood ratio test the errorprobabilities will always satisfy

Only with difficulty can we solve the inequality constraints on the sequential test's thresholds in the general case.Surprisingly, by approximating the inequality constraints by equality constraints we can obtain a result having pragmatically important properties. As anapproximation, we thus turn to solving for ${}_0$ and ${}_1$ under the conditions $${}_0=\frac{1-}{1-}$$ and $${}_1=\frac{}{}$$ In this way, the threshold values are explicitly specified in terms of the desired performance probabilities. We use thecriterion values for the false-alarm and detection probabilities because when we use these equalities, the test'sresulting performance probabilities $P_F$ and $P_D$ usually do not satisfy the design criteria. For example, equating ${}_1$ to a value potentially larger than its desired value might result in a smaller detection probability and a largerfalse-alarm rate. We will want to understand how much actual performance departs from what we want.

The relationships derived above between the performance levels and the thresholds apply no matter how the thresholds arechosen. $$\frac{1-}{1-}\ge \frac{1-P_D}{1-P_F}$$ $$\frac{}{}\le \frac{P_D}{P_F}$$ From these inequalities, two important results follow: $$P_F\le \frac{}{}$$ $$1-P_D\le \frac{1-}{1-}$$ and $$P_F+1-P_D\le +1-$$ The first result follows directly from the threshold bounds. To derive the second result, we must work a little harder.Multiplying the first inequality by $(1-)(1-P_F)$ yields $(1-)(1-P_F)\ge (1-P_D)(1-)$ . Considering the reciprocal of the second inequality and multiplying it by $P_D$ yields $P_D\ge P_F$ . Adding the two inequalities yields the second result.

The first set of inequalities suggest that the false-alarm and miss (which equals $1-P_D$ ) probabilities will increase only slightly from their specified values: the denominators on the right sidesare very close to unity in the interesting cases (e.g., small error probabilities like 0.01). The second inequalitysuggests that the sum of the false-alarm and miss probabilities obtained in practice will be less than the sumof the specified error probabilities. Taking these results together, one of two situations will occur when we approximatethe inequality criterion by equality: either the false alarm probability will decrease and the detection probabilityincrease (a most pleasing but unlikely circumstance) or one of the error probabilities will increase while the other decreases. The false-alarm and miss probabilities cannot both increase. Furthermore, whichever one increases, the first inequalities suggest that the incremental change will be small. Our ad hoc approximation to the thresholds does indeed yield a level of performance close to that specified