Universal Scheme for Optimal Search and Stop

Transcription

1 Universal Scheme for Optimal Search and Stop Sirin Nitinawarat Qalcomm Technologies, Inc Morehose Drive San Diego, CA 92121, USA Vengopal V. Veeravalli Coordinated Science Laboratory Department of Electrical and Compter Engineering University of Illinois at Urbana-Champaign Urbana, IL, 61801, USA Abstract The problem of niversal search and stop sing an adaptive search policy is considered. When the target location is searched, the observation is distribted according to the target distribtion, otherwise it is distribted according to the absence distribtion. A niversal seqential scheme for search and stop is proposed sing only the knowledge of the absence distribtion, and its asymptotic performance is analyzed. The niversal test is shown to yield a vanishing error probability, and to achieve the optimal reliability when the target is present, niversally for every target distribtion. Conseqently, it is established that the knowledge of the target distribtion is only sefl for improving the reliability for detecting a missing target. It is also shown that a mltiplicative gain for the search reliability eqal to the nmber of searched locations is achieved by allowing adaptivity in the search. I. INTRODUCTION We stdy the problem of niversal search and stop sing an adaptive search policy. When the target location is searched, the observation is assmed to be distribted according to the target distribtion, otherwise it is distribted according to the absence distribtion. We assme that only the absence distribtion is known, and the target distribtion can be arbitrarily distinct from the absence distribtion. An adaptive search policy specifies the crrent search location based on the past observations and past search locations. At the stopping time, the target s location is detered or it is decided that it is missing. The overall goal is to achieve a certain level of accracy for the final decision sing the fewest nmber of observations. The reslts in this paper shold be regarded as a contribtion to the long-stdied area of search theory see, e.g., 1, 2, 3, 8, 10, 18, in particlar, searching for a stationary target in discrete time and space with a discrete search effort cf. 3Sbsection 4.2. Conceptally, a desirable goal of the search at each location shold be to detere if the target is there. To this end, a niversal seqential test for two hypotheses can be sed at each location to collect mltiple sbseqent observations that will eventally lead to a binary otcome that the target is there or not. To improve reliability for this binary decision at a particlar search location, one can se a test that takes more observations at that location. If we insist on sing the mentioned seqential binary test at each location as an inner test, then it is convenient to select the crrent search location based on the past binary otcomes of the sbseqent binary tests instead of all the past observational otcomes of all the searches, generally taken mltiple times at each of the locations. With this imposition, the search and stop problem can be conceptally redced to the problem of constrcting an oter test for the seqential design of sch inner experiments. This intitive decomposition leads to or proposed niversal seqential test for search and stop. Universal seqential testing for two hypotheses was first considered for certain parametric families of distribtions for continos observation spaces in 6, 7, 11, 17, the latest of which employed the concept of time-dependent thresholding. Here in Sbsection II-A, we look at a non-parametric family of distribtions for a finite observation space, for which we propose a niversal test sing a sitable time-dependent threshold and analyze its performance. Seqential design of experiments with a niform experimental cost was first considered in 5, 4 nder a certain positivity assmption for the model, which was sccessflly dispensed with later in 15, 14. A generalization of the model with a more complicated memory strctre for the experimental otcomes and with non-niform experimental cost was stdied in 16. We show that when the target is present, the proposed niversal test based on the aforementioned decomposition yields a vanishing error probability, and achieves the optimal reliability, in terms of a sitable exponent for the error probability, niversally for every target distribtion. Conseqently, we establish that the knowledge of the target distribtion is only sefl for improving the reliability for detecting a missing target. We also show that a mltiplicative gain for the search reliability eqal to the nmber of searched locations is achieved by allowing adaptivity in the search. We review the pertinent existing reslts on niversal seqential testing for two hypotheses and seqential design of experiments in Sbsections II-A and II-B, respectively. The general model for niversal search and stop is set p in Section III. We present the proposed seqential test for search and stop and state the main reslt pertaining to its performance in Section IV. De to space limitations, we omit all the proofs. II. PRELIMINARIES Throghot the paper, random variables rvs are denoted by capital letters, and their realizations are denoted by the corresponding lower-case letters. All rvs are assmed to take vales in finite sets, and all logarithms are the natral ones.

2 For a finite set X, and a probability mass fnction pmf p on X we write X p to denote that the rv X is distribted according to p. The following technical facts will be sefl; their derivations can be fond in 9, Chapter 11. Consider random variables Y n = Y 1,..., Y n which are independent and identically distribted i.i.d. according to a pmf p on Y, i.e., Y i p, i = 1,..., n. Let y n = y 1,..., y n Y n be a seqence with an empirical distribtion γ = γ n on Y. It follows that the probability of sch seqence y n, nder the i.i.d. assmption according to the pmf p, is py n = e n Dγ p+hγ, II.1 where Dγ p and Hγ are the relative entropy of γ and p, and entropy of γ, defined as and Dγ p y Y γy log γy py, Hγ y Y γy log γy, respectively. Conseqently, it holds that for each y n, the pmf p that maximizes py n is p = γ, and the associated maximal probability of y n is γy n = e nhγ. II.2 Next, for each n 1, the nmber of all possible empirical distribtions from a seqence of length n in Y n is pper bonded by n + 1 Y. In particlar, sing this last fact, it can be shown that for any ɛ > 0, it holds that the probability of the i.i.d. seqence Y n nder p satisfies P D γ p ɛ n + 1 Y e nɛ. II.3 We now review the relevant preliary reslts on niversal seqential testing for two hypotheses, and model-based seqential design of experiments with varying experimental cost in Sbsections II-A and II-B, respectively. These reslts will be key to or proposed niversal test for search and stop. A. Universal Seqential Testing for Two Hypotheses Consider seqential testing between the nll hypothesis H 0 with i.i.d. observations Y k Y, k = 1, 2,..., according to a pmf π on Y, and the alternative hypothesis H 1 with i.i.d. Y k, k = 1, 2,..., according to a pmf µ π. We assme that only π is known, and nothing is known abot µ, i.e., it can be arbitrarily close to π. We frther assme that both µ and π have fll spport on Y. For a threshold parameter a > 1, we shall employ a seqential test defined in terms of the following Markov time: Ñ b arg n 1 nd γ π > II.4 where γ denotes the empirical distribtion of the observation seqence y 1,..., y n. The test stops at this time or a log a ρ1 for some ρ 1 > 1, depending on which one is smaller, i.e., it stops at time N b, where N b Ñ b, a log a ρ1. II.5 Correspondingly, the final decision is made according to δ b Y N b { 1 if Ñ = b a log a ρ1 0 if Ñ b II.6 > a log a ρ1. Lemma II.1. With µ and π having fll spport on Y, for every a > 1, the seqential test in II.4, II.5, II.6 yields that α a P 0 δ b Y N b = 1 1 a. II.7 In addition, for any ν < 1, µ π and every a > a ν, µ, π, the test also yields that c a E 1 N b E 1 Ñ b log a νd µ π, II.8 κ a E 0 N b a log a ρ1, II.9 β a P 1 δ b Y N b = 0 = P 1 Ñ b > a log a ρ1 b E 1 Ñ a log a ρ1 1 νd µ π a log a. ρ1 1 II.10 II.11 B. Seqential Design of Experiments with Varying Experimental Cost Now we trn or attention to another seqential decisionmaking problem a model-based one this time. Consider the problem of seqential design of experiments to facilitate the evental testing for H hypotheses. We assme a conditionally memoryless model for the otcome conditioned on the crrently chosen experiment. In particlar, nder the i- th hypothesis, i {1,..., H} = H, and conditioned on the crrent experiment t = U, at time t = 1, 2,..., the crrent otcome of the experiment, denoted by Z t, is assmed to be conditionally independent of all past otcomes and past experiments U t 1, Z t 1, and to be conditionally distribted according to a pmf p i on Z. There is a cost fnction c : H U R +, and the crrent experiment t is assmed to incr a cost of c i, t nder the i-th hypothesis. We assme that for every i = 1,..., H, U, z Z, p i z > 0, c i, > 0. A test consists of a adaptive policy φ that chooses each experiment as a sitable possibly randomized fnction of past experiments and their otcomes, log a + n Y log n + 1 a, stopping time τ, and a final decision rle δ that otpts a gess of a hypothesis in H. The goal is to design a test to optimize the tradeoff between the cost accmlated p to

3 the final decision, as measred by τ c i, U t, i H, and the accracy of the final decision, as measred by P max max P i δ Z τ i. The problem is model-based: all the i=1,...,h conditional distribtions p i, j H, U, and the cost fnction c are assmed to be known. For each hypothesis i H, let q i argmax q j i qd p i p j. II.12 qci, Then an asymptotically optimal test can be specified based on these distribtions as follows. At each time t 1, the ML estimate of the tre hypothesis î can be compted based on past experiments and their otcomes t 1, z t 1 sing the model p i, i H, U ties are broken arbitrarily. For b > 0, dring the sparse occasions t = e bk, k = 0, 1,..., the experiment is selected to explore all possible options in { U in a rond-robin } manner independently of î: for U = 1,..., U, t = k mod U + 1. II.13 At all other times, the crrent random experiment is selected as U t q. Denote the joint distribtion nder the i-th î hypothesis of all experiments and their otcomes p to time t indced by the control policy by p i z t, t. For a threshold a > 1, the test stops at time τ and decides in favor of the ML hypothesis according to the rle δ, where τ pî z t, t arg t j î p j z t, t > a, δ z τ, τ = î. II.14 Note that as the qi, i H, are, in general, not point-mass distribtions, in addition to the realization of all experimental otcomes z t, we also need to accont for the realization of the experiments t as well in the instantaneos compation of the ML hypothesis and the stopping criterion II.14. If the experiments have been chosen deteristically at all times, we can jst se the joint distribtions of all experimental otcomes p i z t, i H, t = 1, 2,... in these comptations. The reslting test is asymptotically optimal and its performance is characterized in Proposition II.1 as follows. Proposition II For b > 0, in II.13 chosen to be sfficiently small, and as a, the test in II.12, II.13, II.14 yields a vanishing error probability P max 0, and satisfies for each i = 1,..., H, that τ E i c i, U t = max q log P max j i qdp i p j qci, 1 + o1. 1 The reslt in 16 was proven for the model in which the cost fnction depends only on the experiment; however, the proof generalizes to the crrent setting when the cost fnction also depends on the hypothesis. In addition, the proposed test is asymptotically optimal in the sense that any seqence of tests φ, τ, δ that achieve P max 0 mst satisfy log P max E i τ c i, U t for every i = 1,..., H. max q j i qdp i p j qci, III. MODEL FOR SEARCH AND STOP 1 + o1, Consider searching for a single target located in one of the M locations. At each time k 1, if a location withot the target is searched, then the observation Y k Y is assmed to be conditionally independent of all past observations and past search locations, and to be conditionally distribted according to the absence distribtion π. The distribtion π represents pre noise, and we shall assme that this distribtion is known to the searcher. On the other hand, if the target location is searched, then the observation wold be conditionally distribted according to the target distribtion µ on Y and wold be conditionally independent of past observations and search locations. We assme that both µ and π have fll spport on Y. We also allow for the possibility of an absent target. In this latter case, the observations at all locations are distribted according to π. Denote the search location at time k 1 by U k M, which is allowed to be any fnction of all past observations Y k 1 = Y 1,..., Y k 1 and past search locations U k 1 = U 1,..., U k 1. It is interesting to note that the most basic search problem with an overlook probability α > 0 see, e.g., Chapters 4, 5 of 18 and Section 4.2 of 3 that is niform over all locations, corresponds to a special case of or general model wherein Y = {0, 1}, µ0 = α, π0 = 1. In contrast, or model allows for any general finite observation space Y, bt assmes that both µ and π have fll spports. The degeneracy in the model for the classic search problem as mentioned affords the constrction of a search plan that is more efficient than that for or model with the assmption of fll spport. The main concern for the classic search problem has been to come p with the search plan that is absoltely optimal non-asymptotically, whereas or main concern is to constrct a niversal test that is asymptotically efficient in the regime of vanishing error probability. We seek to design a niversal seqential test to search the target or to decide that it is missing. Precisely speaking, a test consists of a seqential search policy, a stopping rle and a final decision rle. The stopping rle defines a random stopping time, denoted by N, which is the nmber of searches taken ntil the final decision is made. At the stopping time, the final decision for the target location is made based on the decision rle δ : Y N M N {0, 1,..., M}, where the 0 otpt corresponds to the final decision for a missing target. The overall goal is to achieve a certain level of accracy for the final decision sing the fewest nmber of observations, niversally for all µ π.

4 A. Fndamental Performance Limit When both µ and π are known, the search and stop problem falls nder the mbrella of seqential design of experiments with a niform experimental cost 5. In particlar, there are M + 1 hypotheses: 0, 1,..., M, where the nll 0-th hypothesis corresponds to the possibility of a missing target. Each i-th hypothesis, i = 1,..., M, corresponds to a possible location of the present target. The experiment set corresponds to U = M; and the model p i y, i = 0,..., M, for seqential design of experiments can be identified as p i = µ, = i, p i = π, i, i = 1,..., M, p 0 = π, = 1,..., M. III.1 Then in this idealistic sitation when the probabilistic model both µ and π is known, by particlarizing the characterization of the asymptotically optimal performance in Proposition II.1 to or search and stop problem sing III.1 and ci, = 1, for each i = 0,..., M, for each U, we get that as the error probability P max = max i=0,...,m P i δ Y N, U N i is driven to zero, the optimal asymptotes of E i N, i = 0,..., M, can be characterized as follows. Proposition III.1. There exists a seqence of tests to search the target that satisfy P max 0 and yield log P max Dπ µ 1 + o1, i = 0, M E i N = log P max Dµ π 1 + o1, i = 1,..., M. III.2 Frthermore, the asymptotic performance in III.2 each term in the denoators is optimal for every i = 0,..., M simltaneosly. Of corse, the asymptotic performance in Proposition III.1 is idealistic, as it reqires the knowledge of µ with π being already known. When µ is not known, since µ can be arbitrarily close to π, this asymptotic performance cannot be achieved niversally. Nevertheless, or main contribtion Theorem IV.1 described below shows that one can design a niversal test withot the knowledge of µ that drives the error probability to zero and achieves the optimal exponent of D µ π nder all the non-nll hypotheses niversally for any µ π. IV. PROPOSED UNIVERSAL SCHEME FOR SEARCH AND STOP AND ITS PERFORMANCE A. Motivation Intitively speaking, a desirable goal of the search at each location shold be to detere if the target is there. To this end, the niversal seqential test for two hypotheses in Sbsection II-A can be sed at each location to collect mltiple sbseqent observations that will eventally lead to a binary otcome say 1 if it is gessed that the target is there, and 0 otherwise. To improve reliability for this binary decision at a particlar search location, one can increase the threshold a in II.4, II.5, II.6 with the cost of taking more observations at that location. If we se the mentioned seqential binary test at each location as the inner test, then it is convenient to select the crrent search location based on the past binary otcomes of the sbseqent binary tests instead of the past Y-ary otcomes of every search, generally taken mltiple times at each of the locations. With this imposition, the search and stop problem can be redced to a problem of constrcting an oter test for the seqential design of sch inner experiments, each of which has a binary otcome. Mathematically speaking, we have redced the original problem of seqential design of Y-ary-otpt experiments specified by the conditional distribtions as in III.1 to one of seqential design of binary-otpt experiments specified as µ b 0 = 1 µ b 1 = β a, π b 1 = 1 π b 0 = α a, IV.1 and p i = µ b, = i, p i = π b, i, i = 1,..., M, p 0 = π b, = 1,..., M IV.2 where α a, β a are as defined in II.7, II.10. On the other hand, each binary-otpt experiment will not have the same cost as for the original Y-ary-otpt experiment. In particlar, the cost of each binary-otpt experiment can be specified as ci, = c a, = i, ci, = κ a, i, i = 1,..., M, c0, = κ a, = 1,..., M, IV.3 where c a, κ a are as defined in II.8 and II.9, respectively. There is still a large gap in trning the motivation described above into a working test for search and stop. To this end, there are two major challenges. First, the optimal test for seqential design of experiments in II.12, II.13, II.14, achieving the performance stated Proposition III.1, reqires precise knowledge of the model. In contrast, the indced model for seqential design of binary-otpt experiments in IV.1, IV.2 is a complicated fnction of the inner threshold a for the seqential binary test in II.4, II.5, II.6. Only an estimate of this tre indced model is available throgh the bonds for α a, β a stated in II.7, II.11 of Lemma II.1. Second, as the threshold a for the optimal test in II.12, II.13, II.14 increases, the model for seqential design of experiments in Proposition III.1 remains fixed. In contrast, in or proposed test, the oter threshold for the test for seqential design of binary-otpt experiments increases together with the inner threshold a, the latter of which deteres the indced model in IV.1, IV.2. Conseqently, the analysis leading to Proposition III.1 does not apply to or proposed test. Or main technical contribtion are precisely, first, to overcome these challenges throgh the proposed test described below in Sbsection IV-B, employing an oter threshold, which is an appropriate fnction of the inter threshold, and, second, to provide the analysis for its performance, stated in Theorem IV.1 below.

5 B. Proposed Universal Test As mentioned in the previos sbsection, the tre indced model for seqential design of the binary-otpt experiments in IV.1, IV.2 is a complicated fnction of the inner threshold a and is not available to s. Nevertheless, Lemma II.1 yields that for a sfficiently large as a fnction of ν, in II.8 and µ, π α a, β a 1 a. IV.4 Or idea wold be to se a mismatched model defined in terms of µ b, π b, where µ b 0 = 1 µ b 1 = π b 1 = 1 π b 0 = 1 a IV.5 to perform the seqential design of the binary-otpt experiments. Specifically, instead of IV.1, IV.2, consider the following mismatched model for seqential design of binaryotpt experiments p i = µ b, = i, p i = π b, i, i = 1,..., M, p 0 = π b, = 1,..., M. IV.6 Heristically speaking, by IV.4, this mismatched model is more noisy than the tre model for large a; hence, the test designed based on this mismatched model shold be conservative enogh to work well for the tre model as well. This intition will be proven to be correct. With the mismatched model specified in IV.6, we can now describe or niversal test as follows. At each time t 1, we compte the estimate of the tre hypothesis î based on past searched locations and their binary otcomes t 1, z t 1 sing the mismatched model p i, i = 0,..., M, M in IV.6. Denote Ni, 1, Ni, 0, i M, as the nmber of times the i-th location were searched and the seqential binary test in II.4, II.5, II.6 decides that the target is there, and that the target is not there, respectively. By the reciprocity of µ b and π b, in IV.5, the comptation of this estimate can be simplified as î = argmax i M 0 Ni, 1 Ni, 0 if max Ni, 1 Ni, 0 > 0; i M if max i M Ni, 1 Ni, 0 0. IV.7 The estimation in IV.7 is qite intitive, as the difference between the nmbers of searched-and-fond and searchedand-not-fond at the i-th location: Ni, 1 Ni, 0, i M, shold approximate the likelihood that the target is there. When all these nmbers are negative, it is most likely that the target is missing. For b > 0, dring the sparse occasions t = e bk, k = 0, 1,..., the experiment is selected to explore all locations in a rond-robin manner as t = k mod M + 1 IV.8 independently of î. At all the other times, if î 0, we shall search at the i-th location, i.e., t = î, if î 0. IV.9 If î = 0, we search among all locations with eqal freqency, namely, t = i t mod M + 1, IV.10 where i t was the search location at the last time t < t sch that î = 0. Denote the joint mismatched distribtion nder the i-th hypothesis of all binary searched otcomes p to time t indced by the above control policy by p i z t. The test stops at time τ and decides in favor of the crrent estimate of the hypothesis as: τ arg t δ z τ = î, j=0,...,m, j î pî z t p j z t > e aρ 2 log a ρ 1, IV.11 where a is the inner threshold for the binary test in II.4, II.5, II.6 and some ρ 2 > 1. As clarified at the end of the paragraph preceding Proposition II.1, since the search policy in IV.8, IV.9, IV.10 specifies the search location as a deteristic fnction of the crrent estimate of the hypothesis at all times, it sffices to work with the joint distribtion p i z t instead of p i z t, t, i.e., t can be written as a deteristic fnction of z t. Using IV.5, IV.6, we can simplify IV.11 as where τ arg t: î 0 τ 0 arg t: î=0 τ = τ, τ 0, IV.12 Nî, 1 Nî, 0, Nî, 1 Nî, 0 Nj, 1 Nj, 0 j î > aρ2 log a ρ1 ; log a 1 Ni, 0 Ni, 1 > i M IV.13 aρ2 log a ρ1 log a 1 IV.14 Of corse, the Markov time in IV.13 is reached first when δ z τ = î 0, whereas the other Markov time in IV.14 is reached first when δ z τ = î = 0. Note that the total nmber of Y-ary-otpt observations N sed to prodce the search reslt is related to the stopping time τ above as τ N = Nt b,.

6 where each Nt b, t = 1,..., τ, is the nmber of observations taken at each location ntil the seqential test in II.4, II.5, II.6 prodces a binary reslt Z t. Conseqently, we get from sccessive ses of the property of conditional expectation and IV.3 that nder the tre hypothesis i = 0,..., M, it holds that τ 1 E i N = E i N b t + E i N b τ Y τ 1 τ 1 = E i Nt b + E i N τ b U τ τ 1 = E i Nt b + c i, U τ τ = E i c i, U t. C. Performance of Proposed Test N b t Theorem IV.1. For any ν < 1 in II.8 and for b > 0 sed in IV.8 chosen to be sfficiently small, as a, the test in IV.7, IV.8, IV.9, IV.10, IV.11 yields a vanishing error probability P max 0 and also satisfies E i N = E i τ c i, U t i = 1,..., M, niversally for every µ π. log P max 1 + o1, νd µ π IV.15 Remark IV.1. Compared to the idealistically optimal performance when µ is known in Proposition III.1, it is interesting to note that or niversal test is niversally asymptotically optimal, except only when the target is missing. In other words, the knowledge of the target distribtion is only sefl in improving reliability for detecting the missing target. This conseqence of or reslt is directly relevant in practical settings, wherein the knowledge of the target distribtion µ wold be lacking before the target is fond. D. Comparison with Universal Non-Adaptive Scheme for Search and Stop Or main reslt in Theorem IV.1 illstrates that one can constrct a test with adaptive search policy, sing only the knowledge of π, that yields a vanishing error probability and achieves the exponent of D µ π niversally for every µ π when the target is present. A natral qestion that arises is how mch can be gained by employing sch an adaptive search policy beyond a non-adaptive one. A nonadaptive search policy φ has to specify the seqence of search locations at the otset and cannot adapt to the otcomes of the instantaneos searches. By the symmetry of the problem, there is no reason for a non-adaptive search policy to favor any location. Conseqently, the only non-adaptive search policy that shold be considered in the niversal setting is the one that search all locations with eqal freqency: k = k mod M + 1, k 0. IV.16 We denote this non-adaptive search policy by φ. With this search policy, an efficient niversal test has been constrcted in 13, which we now describe. For each time k = lm, l = 1, 2,..., let γ i, i = 1,..., M denote the empirical distribtion of the observations when the i-th location is searched, namely, γ i = y i, y M+i,..., y l 1M+i, i = 1,..., M. Next, denote the estimate of the target location î as î = argmax D γ i π. i M IV.17 With the non-adaptive search policy φ, consider the stopping rle defined in terms of the following Markov time: N M arg D γî π max D γ j π j î. l 1 > log a + M Y log l + 1 IV.18 The test stops at time N, where N N, a log a. IV.19 Correspondingly, the final decision is made according to { δ Y N î if N a log a = 0 if N IV.20 > a log a. The performance of this test with the non-adaptive search scheme follows from the reslt in 13. Proposition IV With the non-adaptive search policy φ in IV.16, the test in IV.17, IV.18, IV.19, IV.20 yields a vanishing error probability P max 0 and also satisfies E i N log P max 1 + o1, i = 1,..., M, Dµ π M niversally for every µ π. In smmary, adaptivity offers a mltiplicative gain of M for search reliability beyond non-adaptive searching. This gain increases with the size of the area to be searched. ACKNOWLEDGEMENTS This work was spported by the Air Force Office of Scientific Research AFOSR nder the Grant FA throgh the University of Illinois at Urbana-Champaign and by the U.S. Defense Threat Redction Agency throgh sbcontract at the University of Illinois from prime award HDTRA REFERENCES 1 R. Ahlswede and I. Wegener, Search Problems, Chichester, Wiley, S. Alpern and S. Gal, The Theory of Search Games and Rendezvos, Boston, MA, Klwer Academic Pblishers, S.J. Benkoski, M.G. Monticino and J.R. Weisinger, A srvey of the search theory literatre, Naval Res. Logistics, vol. 38, pp , 1991.

7 4 S.A. Bessler, Theory and applications of the seqential design of experiments, k-actions and infinitely many experiments, Tech. Report, vol. 55, Dept. of Statistics, Stanford University, pt. I-Theory, H. Chernoff, Seqential design of experiments, Ann. Math. Statist., vol. 30, pp , H. Chernoff, Seqential tests for the mean of a normal distribtion, In Proceedings of the Forth Berkeley Symposim on Mathematical Statistics and Probability, vol. 1, pp Berkeley, CA, Univ. of Calif. Press, H. Chernoff, Seqential tests for the mean of a normal distribtion, Ann. Math. Statist., vol. 36, pt. III Small t, pp , D.V. Chdnovsky and G.V. Chdnovsky, Search Theory, Some Recent Developments, New York, Marchel Dekker, Inc., T.M. Cover and J.A. Thomas, Elements of Information Theory, 2nd edn., Hoboken, NJ, Wiley, B.O. Koopman, Search and Screening, Elmsford, NY, Pergamon Press, Inc., T.L. Lai, Nearly optimal seqential tests of composite hypotheses, Ann. Statist., vol. 16, pp , Y. Li, S. Nitinawarat and V.V. Veeravalli, Universal otlier hypothesis testing, IEEE Trans. Inf. Theory, vol. 60, pp , Y., Li, S. Nitinawarat and V.V. Veeravalli, Universal seqential otlier hypothesis testing, In Proceedings of the Asilomar Conference on Signals, Systems, and Compters, Pacific Grove, California, IEEE, To appear. Available at 14 M. Naghshvar and T. Javidi, Active seqential hypothesis testing, Ann. Statist., vol. 41, pp , S. Nitinawarat, G. Atia and V.V. Veeravalli, Controlled sensing for mltihypothesis testing, IEEE Trans. Atom. Contr., vol. 58, pp , S. Nitinawarat and V.V. Veeravalli, Controlled sensing for seqential mltihypothesis testing with controlled Markovian observations and non-niform control cost, Seqential Analysis, To appear. Available at 17 G. Schwarz, Asymptotic shapes of Bayes seqential testing regions, Ann. Math. Statist., vol. 33, pp , L.D. Stone, Theory of Optimal Search, 2nd edn., Topics in Operations Research Series, Hanover, MD, INFORMS, 2007.