Randomized Sensor Selection in Sequential Hypothesis Testing

1 Randomized Sensor Select ion in Sequential Hypothesis T esting V aibhav Sri v astav a Kurt Plarre Francesc o Bullo Abstract —W e consider the pro blem of sensor selection for time-optimal detection of a hypothesis. W e consider a group of sensors transmitting their observ ation s to a fusi on center . Th e fusion center considers the output of only one randomly chosen sensor at the time, and perf orms a sequen tial h ypothesis test. W e consider the class o f sequen tial tests which are easy to i mplement, asymptotically optimal, a nd co mputationally amenable. F or three distinct perf ormance metrics, we show that, for a generic set of sensors and binary hypothesis, the fusion center needs to consider at most two sensors. W e also show that for the case of multiple hypothesis, the optimal policy n eeds at most as many sensors to be obser ved as the nu mber of underlying hypotheses. Index T erms —S ensor selection, decision making, S PR T , MSPRT , sequ ential hypothesis testing, linear-fra ctional pr ogram- ming. I . I N T R O D U C T I O N In today’ s information -rich world, different sourc es are best inform ers about different topics. If the topic under con - sideration is well known beforeh and, then on e cho oses the best source. Otherwise, it is not obviou s what sour ce or how m any sources one should ob serve. T his need to id entify sensors ( informatio n so urces) to be ob served in decision making pro blems is f ound in m any co mmon situations, e.g., when decid ing which news channel to follow . When a person decides what information source to follo w , she relies in general upon her experience, i.e., one knows th rough experience what combinatio n of news chann els to follow . In en gineering application s, a reliable decision on the un - derlying h ypothesis is mad e thr ough rep eated measuremen ts. Giv en infinitely many observations, decision m aking can b e perfor med accurately . Given a cost associated to each obser- vation, a well-known tr adeoff arises between accuracy and number of iterations. V ario us sequen tial hypoth esis tests hav e been propo sed to detect the underlyin g hyp othesis within a giv en degree of acc uracy . There exist two different classes of sequen tial tests. Th e first class includes sequen tial tests developed fro m the dy namic progra mming point of view . These tests are optimal and , in g eneral, difficult to imple- ment [ 5]. T he second class consists o f easily-imp lementable and asymp totically-optim al sequential tests; a wid ely-studied example is the Sequential Probab ility Ratio T est (SPR T) for binary hy pothesis testing and its extension , the Multi- hypoth esis Sequential Probability Ratio T est (MSPR T). This work has been supported in part by AFOSR MURI A ward F49620- 02-1-0325. V aibha v Sri v asta v a and Francesco Bullo are with the Cente r for Control, Dynami cal Systems, and Computat ion, Univ ersity of Califor nia, Santa Barbara, Santa Barbara , CA 93106, USA, { vaibhav,bullo } @engi neering.ucsb.edu Kurt Plarre is with the Departmen t of Computer Science, Univ ersity of Memphis, Memphis, T N 38152, USA, kplarre@m emphis.edu In this paper, we co nsider the prob lem of quickest decision making u sing sequential probability r atio tests. Recent ad - vances in cognitiv e p sycholog y [7] show that the perf ormance of a human performin g decision making tasks, such as ”two- alternative for ced choice tasks, ” is well mo deled by a drif t diffusion process, i.e., the continuo us-time version of SPR T . Roughly speaking, modelin g dec ision making as an SPR T process is somehow appr opriate even for situations in which a human is ma king th e decision. Sequential hy pothesis testing and quickest d etection pro b- lems h av e been vastly studied [17], [4]. The SPR T for binary decision makin g was introduc ed by W ald in [21], and was extended by Armitage to mu ltiple h ypothesis testing in [1]. The Armitage test, unlike th e SPR T , is no t necessarily op - timal [24]. V ariou s other tests for multiple hyp othesis test- ing h a ve been dev eloped throug hout the years; a survey is presented in [18]. D esigning hypothesis tests, i.e., choosing thresholds to decid e within a given expected numbe r of iterations, through any of the proced ures in [18] is infeasible as none of them provid es an y results on the expected sample size. A sequential test for multiple hypothesis testing was dev eloped in [5], [ 10], and [11], which provides with an asy mptotic expression for the expecte d sample size. T his sequential test is called the MSPR T and red uces to the SPR T in case of bin ary hypoth esis. Recent years have witnessed a significant interest in the problem of sensor selection f or optimal de tection and estima- tion. T ay et al [20] discu ss the p roblem of cen soring sensors for decentr alized binary detection. They assess the qu ality of sensor data b y the Neyman-Pearso n an d a Bayesian binary hypoth esis test and de cide o n which sensor s sh ould transmit their o bservation at that time instant. Gup ta e t al [13] fo cus o n stochastic sensor selection a nd minimize the error cov ariance of a process estimation prob lem. Isler et al [ 14] propose geometric sensor selection schemes f or erro r m inimization in target d etection. Debouk et al [9] for mulate a Markovian decision problem to ascer tain some pr operty in a dy namical system, an d cho ose sensors to min imize the associated cost. W ang et al [ 22] d esign en tropy-based sen sor selection algo - rithms for ta rget localizatio n. Joshi et al [15] present a con vex optimization -based heuristic to select multiple sensors fo r optimal parameter estimation. Bajovi ´ c et al [3] d iscuss sensor selection problem s for Neyman-Pear son binary hypo thesis testing in wireless senso r n etworks. A third and last set of references related to this p aper are tho se on linear-fraction al pr ogramm ing. V arious iterative and cu mbersome algorithms hav e been prop osed to optimize linear-fractional fun ctions [8], [2]. In particular, for the pr ob- lem of m inimizing the sum and the maxim um of linear- 2 fractional func tionals, some efficient iterati ve algorithm s ha ve been prop osed, includin g the algo rithms by F alk et a l [12] and by Benson [6]. In this p aper , we an alyze the prob lem o f time-optimal se- quential decision making in the pre sence of multiple switching sensors and d etermine a sensor selection strategy to achieve the same. W e consider a sensor network where all sensors are connected to a fusion cen ter . The f usion center, at each in stant, receives informa tion fr om only one sensor . Such a situation arises when we hav e interfering sensors (e.g., sonar sensors), a fusion center with limited attention or information processing capabilities, or sensors with sha red commu nication resources. The f usion center implemen ts a sequential hyp othesis test with the gath ered information. W e conside r tw o such tests, namely , the SPR T an d the MSPR T for binar y an d multiple hypoth esis, respectiv ely . First, we develop a version of the SPR T and the MSPR T where the sensor is randomly switched at each iteration , and dete rmine the expected time that these tests require to ob tain a decision within a given degree of accuracy . Secon d, we identify the set o f sensors that min imize the expected d ecision time. W e consider three different co st function s, namely , the conditioned decision time, the worst case decision time, and the average decision time. W e show that the expected decision time, conditioned on a gi ven hypo th- esis, using these sequen tial tests is a linea r-fractional func tion defined on th e p robability simplex. W e exploit the special structure of our do main (p robability simp lex), and the fact that ou r data is positive to tackle the prob lem of th e sum a nd the maximum of linear-fractional fu nctionals analy tically . Our approa ch provides insights into the behavior of these function s. The major contributions of this paper a re: i) W e de velop a version of the SPR T and the MSPR T where the sensor is selected random ly at each obser- vation. ii) W e determ ine th e a symptotic expre ssions for the thresh - olds an d the expected sample size for these sequen tial tests. iii) W e in corpor ate the processing time of the sen sors into these models to d etermine the e xpected decision time. iv) W e show that, to minimize the conditio ned expected decision time, th e o ptimal po licy req uires on ly on e sensor to be o bserved. v) W e sho w th at, for a generic set of sensors an d M underly ing h ypotheses, the o ptimal a vera ge decision time policy requires the fusion center to consider at m ost M sensors. vi) F or the binary h ypothesis case, we ide ntify the o ptimal set of sensors in the worst case and th e average decision time minimization problems. Moreover , we determine an optimal pro bability distribution for the sensor selection. vii) In the worst case and the average de cision time mini- mization problem s, we encoun ter th e p roblem of m in- imization of sum and max imum of linear-fractional function als. W e treat th ese p roblems analy tically , and provide insigh t into th eir o ptimal solutions. The remainder of the paper is organized in following way . In Section II, we presen t the problem setup. Some prelimina ries are pr esented in Section III. W e develop the switching- sensor version of the SPR T a nd the MSPR T pr ocedure s in Section IV. In Sectio n V, we fo rmulate the optimiza tion prob lems for time-optima l sensor selectio n, and determin e their solution. W e elucidate the results obtained throug h n umerical examples in Section VI. Ou r c oncludin g remarks are in Section VI I. I I . P RO B L E M S E T U P W e con sider a gro up of n > 1 agents (e.g., robots, senso rs, or cameras), wh ich take measureme nts a nd transmit th em to a fusion center . W e gene rically call these ag ents ”sensors. ” W e identify the fusion cen ter with a p erson super vising the agen ts, and call it ”th e su pervisor . ” Fig. 1. T he agents A transmit their obse rv ation to th e supervisor S , one at the time. The supervisor performs a sequential hypothesis test to decide on the underlyi ng hypoth esis. The g oal of the superv isor is to decide, ba sed on the measuremen ts it receives, which of the M ≥ 2 alternative hypoth eses or “states of natu re, ” H k , k ∈ { 0 , . . . , M − 1 } is correct. F or doing so, the superviso r uses sequential hypothesis tests, which we b riefly re view in the next section. W e assume that only on e sensor can transmit to the superv i- sor at each ( discrete) time instant. Equ i valently , the supe rvisor can process d ata from only one of the n a gents at ea ch time. Thus, at each time, the su pervisor must d ecide which sensor should tran smit its measurement. W e are interested in finding the optimal sensor(s), which must be observed in order to minimize the decision tim e. W e model the setup in the following w ay: i) Let s l ∈ { 1 , . . . , n } indicate wh ich sensor transmits its measuremen t at time instant l ∈ N . ii) Conditioned o n the h ypothesis H k , k ∈ { 0 , . . . , M − 1 } , the prob ability that the measuremen t at sensor s is y , is denoted by f k s ( y ) . iii) The prior pr obability of the hy pothesis H k , k ∈ { 0 , . . . , M − 1 } , being correct is π k . iv) The measure ment of sensor s at time l is y s l . W e assume that, conditione d on hypothesis H k , y s l is independ ent of y ( s ¯ l ) , for ( l, s l ) 6 = ( ¯ l , s ¯ l ) . v) The time it takes for s ensor s to t ransmit its measurement (or for th e supervisor to process it) is T s > 0 . vi) The superv isor chooses a sensor randomly at each tim e instant; th e prob ability to choose sensor s is stationa ry and giv en by q s . vii) The su pervisor uses the data collected to execute a sequential hy pothesis test with the desired pro bability of incorrect decision, conditioned on hypothesis H k , given by α k . 3 viii) W e assum e th at th ere are no two sensors with id entical condition ed probability distribution f k s ( y ) and process- ing time T s . If there are such sensors, we club them together in a single nod e, and distribute th e probability assigned to that n ode equally am ong them. I I I . P R E L I M I N A R I E S A. Linear-fractional functio n Giv en p arameters A ∈ R q × p , B ∈ R q , c ∈ R p , and d ∈ R , the function g : { z ∈ R p | c T z + d > 0 } → R q , defined by g ( x ) = Ax + B c T x + d , is called a linear-fr actional function [8]. A linear-fractional function is q uasi-conve x a s well as q uasi-concave. In partic- ular , if q = 1 , then any scalar linea r-fractional fun ction g satisfies g ( tx + (1 − t ) y ) ≤ ma x { g ( x ) , g ( y ) } , g ( tx + (1 − t ) y ) ≥ min { g ( x ) , g ( y ) } , (1) for all t ∈ [0 , 1] and x, y ∈ { z ∈ R p | c T z + d > 0 } . B. K u llback-Leibler distance Giv en two prob ability distributions functions f 1 : R → R ≥ 0 and f 2 : R → R ≥ 0 , the Kullback -Leibler distance D : L 1 × L 1 → R is defined by D ( f 1 , f 2 ) = E  log f 1 ( X ) f 2 ( X )  = Z R f 1 ( x ) log f 1 ( x ) f 2 ( x ) dx. Further, D ( f 1 , f 2 ) ≥ 0 , and the equality h olds if and only if f 1 = f 2 almost e very where. C. Sequ ential Pr o bability Ratio T est The SPR T is a sequen tial binar y hy pothesis test tha t pr o- vides us with two thr esholds to decide on som e hypo thesis, opposed to c lassical hypothesis tests, where we have a single threshold. Con sider two hypoth esis H 0 and H 1 , with pr ior probab ilities π 0 and π 1 , respec ti vely . Gi ven th eir conditio nal probab ility distribution functions f ( y | H 0 ) = f 0 ( y ) and f ( y | H 1 ) = f 1 ( y ) , and repeated measurements { y 1 , y 2 , . . . } , with λ 0 defined by λ 0 = log π 1 π 0 , (2) the SPR T p rovides us with two constants η 0 and η 1 to decide on a hypothesis at each time instant l , in the following way: i) Compute th e log likelihood ratio: λ l := log f 1 ( y l ) f 0 ( y l ) , ii) Integrate evidence u p to time N , i.e. , Λ N := N P l =0 λ l , iii) Decide o nly if a threshold is crossed, i.e.,      Λ N > η 1 , say H 1 , Λ N < η 0 , say H 0 , Λ N ∈ ] η 0 , η 1 [ , continue sampling . Giv en the prob ability o f false alarm P ( H 1 | H 0 ) = α 0 and probab ility of missed d etection P ( H 0 | H 1 ) = α 1 , the W ald ’ s thresholds η 0 and η 1 are defined by η 0 = log α 1 1 − α 0 , and η 1 = log 1 − α 1 α 0 . (3) The expected sample size N , for decision u sing SPR T is asymptotically giv en by E [ N | H 0 ] → − (1 − α 0 ) η 0 + α 0 η 1 − λ 0 D ( f 0 , f 1 ) , and E [ N | H 1 ] → α 1 η 0 + (1 − α 1 ) η 1 − λ 0 D ( f 1 , f 0 ) , (4) as − η 0 , η 1 → ∞ . The as ymptotic expected samp le size expres- sions in equa tion (4) are vali d f or la rge thresh olds. The use of these asymptotic expressions as approximate expected sam ple size is a standar d ap proxim ation in the inf ormation theory lit- erature, and is known as W ald’ s approx imation [4 ], [ 17], [ 19]. For g i ven error pro babilities, the SPR T is the op timal se- quential binary hyp othesis test, if the sample size is considered as the cost fu nction [19]. D. Multi-hypo thesis Sequ ential Pr obability Ratio T est The MSPR T for multiple hypoth esis testing was introduced in [5], and was fur ther gene ralized in [10] an d [11]. It is described as follows. Gi ven M hy potheses with th eir prio r probab ilities π k , k ∈ { 0 , . . . , M − 1 } , the posterior probability after N observations y l , l ∈ { 1 , . . . , N } is given by p k N = P ( H = H k | y 1 , . . . , y N ) = π k N Π l =1 f k ( y l ) M − 1 P j =1 π j  n Π l =1 f j ( y l )  , where f k is th e pro bability density fun ction of the ob servation of the sensor , condition ed on hypothe sis k . Before we state the MSPR T , for a gi ven N , we define ¯ k by ¯ k = argmax j ∈{ 0 ,...,M − 1 } π j N Π l =1 f j ( y l ) . The MSPR T at each sampling iteration l is defined as ( p k l > 1 1+ η k , for at least one k , say H ¯ k , otherwise, continue sampling, where the thresholds η k , for giv en freque ntist e rror pro babili- ties (accep t a given hyp othesis wro ngly) α k , k ∈ { 0 , . . . , M − 1 } , are gi ven by η k = α k π k γ k , (5) where γ k ∈ ]0 , 1 [ is a constant function of f k (see [5]). It can be shown [5] that the expec ted sample size of the MSPR T , cond itioned on a hypo thesis, satisfies E [ N | H k ] → − lo g η k δ k , as max k ∈{ 0 ,... ,M − 1 } η k → 0 + , where δ k = min {D ( f k , f j ) | j ∈ { 0 , . . . , M − 1 } \ { k }} . The MSPR T is an easily-implem entable hypo thesis test and is sho wn to be asymp totically o ptimal in [5 ], [10]. 4 I V . S E Q U E N T I A L H Y P O T H E S I S T E S T S W I T H S W I T C H I N G S E N S O R S A. SPRT with switc hing sensors Consider the case whe n the fusion cen ter collects data fro m n sensors. At each iter ation the f usion center loo ks at one sensor chosen rando mly with pr obability q s , s ∈ { 1 , . . . , n } . The fusion center perfor ms SPR T with the collected data. W e d efine this pr ocedure as SPR T with switching senso rs. If we assume th at sensor s l is obser ved at iteratio n l , and th e observed value is y s l , then SPR T with switching sensors is described as following, with the threshold s η 0 and η 1 defined in equation (3), an d λ 0 defined in equation ( 2): i) Compute lo g likelihood r atio: λ l := lo g f 1 s l ( y s l ) f 0 s l ( y s l ) , ii) Integrate evidence u p to time N , i.e. , Λ N := N P l =0 λ l , iii) Decide o nly if a threshold is crossed, i.e.,      Λ N > η 1 , say H 1 , Λ N < η 0 , say H 0 , Λ N ∈ ] η 0 , η 1 [ , continue sampling . Lemma 1 (Expected sa mple size): For the SPR T with switching sensors de scribed above, the expe cted samp le size condition ed on a hypothesis is asymp totically g i ven by: E [ N | H 0 ] → − (1 − α 0 ) η 0 + α 0 η 1 − λ 0 n P s =1 q s D ( f 0 s , f 1 s ) , and E [ N | H 1 ] → α 1 η 0 + (1 − α 1 ) η 1 − λ 0 n P s =1 q s D ( f 1 s , f 0 s ) , (6) as − η 0 , η 1 → ∞ . Pr oof: Similar to the proof of Th eorem 3.2 in [23]. The expected sample size con verges to th e values in equ a- tion (6) f or large threshold s. From equation (3), it f ollows that large thr esholds correspo nd to small error pr obabilities. In th e remainder of the paper, we assume that the error probab ilities are cho sen sma ll enough, so th at the above asymptotic expression fo r sample size is close to the actual expected samp le size. Lemma 2 (Expected d ecision time): Giv en th e processing time of the sensors T s , s ∈ { 1 , . . . , n } , the expected decision time o f th e SPR T with switchin g senso rs T d , co nditioned on the hypoth esis H k , k ∈ { 0 , 1 } , is E [ T d | H k ] = n P s =1 q s T s n P s =1 q s I k s = q · T q · I k , for each k ∈ { 0 , 1 } , (7) where T , I k ∈ R n > 0 , a re constant vectors for each k ∈ { 0 , 1 } . Pr oof: The decision time using SPR T with switching sensors is the sum of sen sor’ s p rocessing time at each iter - ation. W e observe that the nu mber of iteration s in SPR T and the processing time o f sensors are in depende nt. Hen ce, the expected value o f the d ecision time T d is E [ T d | H k ] = E [ N | H k ] E [ T ] , f or each k ∈ { 0 , 1 } . (8) By the d efinition of expected value, E [ T ] = n X s =1 q s T s . (9) From equations (6), ( 8), an d (9) it fo llows that E [ T d | H k ] = n P s =1 q s T s n P s =1 q s I k s = q · T q · I k , for each k ∈ { 0 , 1 } , where I k s ∈ R > 0 is a con stant, for each k ∈ { 0 , 1 } , and s ∈ { 1 , . . . , n } . B. MSPRT with switc hing sensors W e call th e MSPR T with the data collected from n sen sors while o bserving on ly one sensor at a time as th e MSPR T with switching sensor s. The on e senso r to be o bserved at each time is determined throu gh a randomized policy , and the pr obability of ch oosing sensor s is stationary and given b y q s . Assume that the sensor s l ∈ { 1 , . . . , n } is cho sen at time instant l , and the prior p robabilities of the h ypothesis are given by π k , k ∈ { 0 , . . . , M − 1 } , then th e p osterior probab ility after N observations y l , l ∈ { 1 , . . . , N } is g i ven b y p k N = P ( H k | y 1 , . . . , y N ) = π k N Π l =1 f k s l ( y l ) M − 1 P j =0 π j  N Π l =1 f j s l ( y l )  , Before we state th e MSPR T with switchin g sensors, for a giv en N , we d efine ˜ k by ˜ k = argmax k ∈{ 0 ,... ,M − 1 } π k N Π l =1 f k s l ( y l ) . For the thre sholds η k , k ∈ { 0 , . . . , M − 1 } , defined in equation ( 5), th e MSPR T with switching sensors at each sampling iteration N is defin ed by ( p k n > 1 1+ η k , f or at least on e k , say H ˜ k , otherwise, continue sampling. Before we state the results on asymptotic sam ple size and expected dec ision time , we intro duce the f ollowing notation. For a g i ven hy pothesis H k , and a sen sor s , we define j ∗ ( s,k ) by j ∗ ( s,k ) = argmin j ∈{ 0 ,... ,M − 1 } j 6 = k D ( f k s , f j s ) . W e also define E D : ∆ n − 1 × ( L 1 ) n × ( L 1 ) n → R by E D ( q , f k , f j ∗ k ) = n X s =1 q s D ( f k s , f j ∗ ( s,k ) s ) , where ∆ n − 1 represents the probability simplex in R n . 5 Lemma 3 (Expected sa mple size): Giv en thresholds η j , j ∈ { 0 , . . . , M − 1 } , the sam ple size N r equired for decision satisfies E [ N | H k ] − lo g η k → 1 E D ( q , f k , f j ∗ k ) , as ma x j ∈{ 0 ,...,M − 1 } η j → 0 . Pr oof: The proof f ollows f rom T heorem 5.1 of [5] an d the observation that lim N →∞ 1 N N X l =1 log f k s l ( X l ) f j s l ( X l ) = n X s =1 q s D ( f k s , f j s ) . Lemma 4 (Expected d ecision time): Giv en th e processing time of th e sen sors T s , s ∈ { 1 , . . . , n } , the expected deci- sion time T d condition ed on the hypothesis H k , fo r each k ∈ { 0 , . . . , M − 1 } , is g iv en by E [ T d | H k ] = − lo g η k E D ( q , f k , f j ∗ k ) n X s =1 q s T s = q · T q · ¯ I k , (10) where T , ¯ I k ∈ R n > 0 are constants. Pr oof: Similar to the proof of Le mma 2. V . O P T I M A L S E N S O R S E L E C T I O N In this sectio n we con sider sensor selection pro blems with the aim to min imize the expected d ecision time of a sequ ential hypoth esis test with switching sensors. As exemplified in Lemma 4, th e prob lem features multiple conditio ned decision times and, therefor e, multiple distinct cost functions are of interest. In Scen ario I below , we aim to minimize the decision time conditio ned upon one specific hypo thesis being true; in Scenarios II an d III we will consid er worst-case and average decision times. In all three scenarios the decision variables take v alues in the pro bability simplex. Minimizing d ecision time co nditioned upon a specific hy- pothesis m ay be of interest wh en fast reaction is req uired in response to the specific h ypothesis being indeed true . For example, in chan ge detection problems one aims to quickly detect a ch ange in a stochastic process; th e CUSUM algorithm (also refer red to as Page’ s test) [16] is widely used in such problem s. It is kn own [4] th at, with fixed thr eshold, th e CUSUM algo rithm for quickest chang e detec tion is eq uiv alen t to an SPR T on the observations taken af ter the ch ange h as occurre d. W e c onsider the minimiza tion pr oblem for a single condition ed d ecision time in Scenario I below and we sh ow that, in this case, observin g th e best sensor each time is the optimal strategy . In g eneral, no specific hypo thesis might play a special role in the problem and, th erefore, it is of interest to simultane- ously min imize multiple decision times over the probability simplex. Th is is a mu lti-objective optimization p roblem, and may h av e Pareto-optimal solutions. W e ta ckle this p roblem by co nstructing a sing le aggr egate o bjective func tion. In the binary hyp othesis case, we construct two sing le agg regate objective fu nctions as the maximum and the a verage of the two condition ed d ecision times. These two functions are discussed in Scen ario II and Scena rio III respecti vely . In the mu ltiple hypoth esis setting , we consider the single ag gregate objectiv e function constructed as the average of the conditioned decision times. An analytica l treatment of this function for M > 2 , is difficult. W e determine the optimal num ber of sensors to be observed, and direct th e in terested reader to some iterative algorithm s to solve su ch optimiz ation problem s. This case is also considered u nder Scenario III. Before we pose the problem of op timal sensor selectio n, we introdu ce the following notation. W e d enote the probab ility simplex in R n by ∆ n − 1 , and the vertices of the pr obability simplex ∆ n − 1 by e i , i ∈ { 1 , . . . , n } . W e refer to the line joining any two vertices of th e simp lex as an edge . Finally , we define g k : ∆ n − 1 → R , k ∈ { 0 , . . . , M − 1 } , by g k ( q ) = q · T q · I k . A. Scena rio I (Optimization of conditione d decision time): W e con sider the case when the su pervisor is trying to detect a particu lar hyp othesis, irrespective of the pre sent hy - pothesis. T he correspondin g optimization pr oblem for a fixed k ∈ { 0 , . . . , M − 1 } is p osed in the fo llowing way: minimize g k ( q ) subject to q ∈ ∆ n − 1 . (11) The solu tion to this minimization problem is given in the following theor em. Theor em 1 (Optimization of condition ed decision time): The solution to the minimization problem (11) is q ∗ = e s ∗ , where s ∗ is gi ven by s ∗ = ar gmin s ∈{ 1 ,...,n } T s I k s , and the m inimum objectiv e func tion is E [ T ∗ d | H k ] = T s ∗ I k s ∗ . (12) Pr oof: W e notice th at o bjective fu nction is a linear- fractional function. In the following argume nt, we show that the minima occurs at on e o f the vertices of the simplex. W e first notice th at the probability simplex is the conve x hull of th e vertices, i.e., any p oint ˜ q in the p robability simplex can be wr itten as ˜ q = n X s =1 α s e s , n X s =1 α s = 1 , and α s ≥ 0 . W e inv oke equation (1), and observe th at for some β ∈ [0 , 1] and for any s, r ∈ { 1 , . . . , n } g k ( β e s + (1 − β ) e r ) ≥ min { g k ( e s ) , g k ( e r ) } , (13) which can be easily gen eralized to g k ( ˜ q ) ≥ min l ∈{ 1 ,...,n } g k ( e s ) , (14) for any po int ˜ q in th e probab ility simplex ∆ n − 1 . Hence, minima will occur at one of the vertices e s ∗ , where s ∗ is giv en by s ∗ = argmin s ∈{ 1 ,...,n } g k ( e s ) = a rgmin s ∈{ 1 ,...,n } T s I k s . 6 B. Scena rio II (Op timization of the worst case decision time): For the bin ary hypothesis testing, we consider the multi- objective optimizatio n pro blem of min imizing both decision times simultaneo usly . W e construct single aggregate objecti ve function by co nsidering the maximum of the two o bjectiv e function s. This tur ns out to be a worst case analysis, and the optimization pro blem for this case is posed in the fo llowing way: minimize max  g 0 ( q ) , g 1 ( q )  , subject to q ∈ ∆ n − 1 . (15) Before we move on to the solu tion of above minimization problem , we state the following results. Lemma 5 (Monotonicity of con ditioned d ecision times): The function s g k , k ∈ { 0 , . . . , M − 1 } are mono tone on the probability simplex ∆ n − 1 , in the sense that giv en two points q 1 , q 2 ∈ ∆ n − 1 , the function g k is monoton ically non-in creasing o r monoton ically no n-decreasin g alon g the line joining q 1 and q 2 . Pr oof: Consider p robability vectors q 1 , q 2 ∈ ∆ n − 1 . Any point q on line joining q 1 and q 2 can be w ritten as q ( t ) = tq 1 + (1 − t ) q 2 , t ∈ ]0 , 1 [ . W e no te th at g k ( q ( t )) is given by: g k ( q ( t )) = t ( q 1 · T ) + (1 − t )( q 2 · T ) t ( q 1 · I k ) + (1 − t )( q 2 · I k ) . The deriv ati ve of g k along the line join ing q 1 and q 2 is giv en by d dt g k ( q ( t )) =  g k ( q 1 ) − g k ( q 2 )  × ( q 1 · I k )( q 2 · I k ) ( t ( q 1 · I k ) + (1 − t )( q 2 · I k )) 2 . W e note that the sign of the der i vati ve o f g k along the line joining two po ints q 1 , q 2 is fixed by the choice of q 1 and q 2 . Hen ce, the f unction g k is mon otone over the line jo ining q 1 and q 2 . Moreover, note tha t if g k ( q 1 ) 6 = g k ( q 2 ) , then g k is strictly mo notone. Otherwise, g k is co nstant over the line joining q 1 and q 2 . Lemma 6 (Location o f min-max): Define g : ∆ n − 1 → R ≥ 0 by g = max { g 0 , g 1 } . A minimum o f g lies at the intersection of the grap hs of g 0 and g 1 , or at so me vertex of the prob ability simplex ∆ n − 1 . Pr oof: Case 1: The graphs of g 0 and g 1 do not intersect at any po int in the simplex ∆ n − 1 . In this case, one o f the functions g 0 and g 1 is an upper bound to th e other fun ction at ev ery point in the probab ility simplex ∆ n − 1 . Hence, g = g k , for some k ∈ { 0 , 1 } , at every point in the p robability simplex ∆ n − 1 . From The orem 1, we know th at the minim a of g k on the pro bability simplex ∆ n − 1 lie at some vertex o f the probability simplex ∆ n − 1 . Case 2: The graphs of g 0 and g 1 intersect at a set Q in the probab ility simplex ∆ n − 1 , and let ¯ q b e so me point in th e set Q . Suppose, a minimum of g occurs at some p oint q ∗ ∈ relint (∆ n − 1 ) , and q ∗ / ∈ Q , where relint ( · ) denotes the r elati ve interior . With out loss o f gene rality , we can a ssume that g 0 ( q ∗ ) > g 1 ( q ∗ ) . Also, g 0 ( ¯ q ) = g 1 ( ¯ q ) , and g 0 ( q ∗ ) < g 0 ( ¯ q ) by assumption. W e inv oke Lemma 5, an d notice th at g 0 and g 1 can intersect at most once on a line. Moreover , we no te th at g 0 ( q ∗ ) > g 1 ( q ∗ ) , hen ce, along th e half -line fro m ¯ q through q ∗ , g 0 > g 1 , that is, g = g 0 . Since g 0 ( q ∗ ) < g 0 ( ¯ q ) , g is decreasin g along this h alf-line. Hence, g sho uld ach iev e its m inimum at th e bound ary of the simplex ∆ n − 1 , which contradicts that q ∗ is in the relative in terior o f the simplex ∆ n − 1 . In summ ary , if a m inimum of g lies in the relative interior of the pro bability simplex ∆ n − 1 , then it lies at the intersectio n of the grap hs of g 0 and g 1 . The same argument can be applied recursiv ely to show that if a minimu m lies at some p oint q † on th e bounda ry , then either g 0 ( q † ) = g 1 ( q † ) or the m inimum lies at th e vertex. In the following a rguments, let Q b e the set of points in the simplex ∆ n − 1 , where g 0 = g 1 , that is, Q = { q ∈ ∆ n − 1 | q · ( I 0 − I 1 ) = 0 } . (16) Also notice that the set Q is no n empty if and only if I 0 − I 1 has at lea st o ne non- negativ e and one n on-positive entry . If the set Q is empty , then it fo llows from Lemma 6 that the solutio n of optimization prob lem in equation (15) lies at som e vertex of the probab ility simp lex ∆ n − 1 . Now we consider the case when Q is non emp ty . W e assume that the sensors have been re- ordered such that the entries in I 0 − I 1 are in ascending o rder . W e fu rther assume that, for I 0 − I 1 , the first m entries, m < n , are non positi ve, and the remaining e ntries are positi ve. Lemma 7 (Intersection polytop e): If the set Q defined in equation (1 6) is non e mpty , then th e po lytope generated b y the points in th e set Q has vertices gi ven by: ˜ Q = { ˜ q sr | s ∈ { 1 , . . . , m } and r ∈ { m + 1 , . . . , n }} , where for each i ∈ { 1 , . . . , n } ˜ q sr i =      ( I 0 r − I 1 r ) ( I 0 r − I 1 r ) − ( I 0 s − I 1 s ) , if i = s, 1 − ˜ q sr s , if i = r, 0 , otherwise . (17) Pr oof: Any q ∈ Q satisfies the following constrain ts n X s =1 q s = 1 , q s ∈ [0 , 1] , (18) n X s =1 q s ( I 0 s − I 1 s ) = 0 , (19) Eliminating q n , using equ ation (18) and eq uation (1 9), we get: n − 1 X s =1 β s q s = 1 , wher e β s = ( I 0 n − I 1 n ) − ( I 0 s − I 1 s ) ( I 0 n − I 1 n ) . (20) The equ ation (2 0) defines a hy perplane, whose extrem e points in R n − 1 ≥ 0 are gi ven by ˜ q sn = 1 β s e s , i ∈ { 1 , . . . , n − 1 } . Note that fo r s ∈ { 1 , . . . , m } , ˜ q sn ∈ ∆ n − 1 . Hence , these points define some vertices of the polytope generated by points in the set Q . Also note that the other vertices of the polyto pe 7 can be determined b y the in tersection of each p air o f lines throug h ˜ q sn and ˜ q r n , and e s and e r , for s ∈ { 1 , . . . , m } , and r ∈ { m + 1 , . . . , n − 1 } . In particular, the se vertices are g iv en by ˜ q sr defined in equation ( 17). Hence, a ll the vertices o f the polyto pes are defined by ˜ q sr , s ∈ { 1 , . . . , m } , r ∈ { m + 1 , . . . , n } . Th erefore, the set of vertices of th e polygo n gen erated by the points in the set Q is ˜ Q . Before we state th e solution to the optimization pro blem (15), we define th e f ollowing: ( s ∗ , r ∗ ) ∈ argmin r ∈{ m +1 ,...,n } s ∈{ 1 ,...,m } ( I 0 r − I 1 r ) T s − ( I 0 s − I 1 s ) T r I 1 s I 0 r − I 0 s I 1 r , and g two-sensors ( s ∗ , r ∗ ) = ( I 0 r ∗ − I 1 r ∗ ) T s ∗ − ( I 0 s ∗ − I 1 s ∗ ) T r ∗ I 1 s ∗ I 0 r ∗ − I 0 s ∗ I 1 r ∗ . W e also define w ∗ = argmin w ∈ { 1 ,...,n } max  T w I 0 w , T w I 1 w  , and g one-sensor ( w ∗ ) = max  T w ∗ I 0 w ∗ , T w ∗ I 1 w ∗  . Theor em 2 ( W orst case optimization): For the optimization problem (15), an o ptimal probability vector is gi ven by: q ∗ = ( e w ∗ , if g one-sensor ( w ∗ ) ≤ g two-sensors ( s ∗ , r ∗ ) , ˜ q s ∗ r ∗ , if g one-sensor ( w ∗ ) > g two-sensors ( s ∗ , r ∗ ) , and the m inimum v alue of the fu nction is given by: min { g one-sensor ( w ∗ ) , g two-sensors ( s ∗ , r ∗ ) } . Pr oof: W e in voke L emma 6, an d note that a minimum should lie at some vertex of the simp lex ∆ n − 1 , o r a t some point in the set Q . Note that g 0 = g 1 on th e set Q , hence the problem o f minim izing max { g 0 , g 1 } red uces to m inimizing g 0 on the set Q . From Theo rem 1, we know that g 0 achieves the m inima at some extreme po int of the feasib le region. Fro m Lemma 7, we kn ow that the vertices of the p olytope gen erated by p oints in set Q are giv en by set ˜ Q . W e fu rther note that g two-sensors ( s, r ) and g one-sensor ( w ) are the value of objective function a t the points in the set ˜ Q and the vertices of th e probab ility simplex ∆ n − 1 respectively , whic h comp letes the proof . C. Scen ario III (Optimization of the avera ge decisio n time) : For th e multi-ob jectiv e optimization pr oblem of minim izing all the decision times simultan eously o n the simplex, we for- mulate the sing le aggregate objectiv e fu nction as the average of th ese d ecision times. The r esulting o ptimization p roblem, for M ≥ 2 , is posed in the following way: minimize 1 M ( g 0 ( q ) + . . . + g M − 1 ( q )) , subject to q ∈ ∆ n − 1 . (21) In th e f ollowing discu ssion we assume n > M , unless otherwise stated. W e analyze the op timization problem in equation (21) a s follows: Lemma 8 (Non-vanishing Jacobian): The ob jectiv e func- tion in optimization proble m in equation (21) has no critical point on ∆ n − 1 if the vectors T , I 0 , . . . , I M − 1 ∈ R n > 0 are linearly independen t. Pr oof: The Jaco bian of the objective function in the optimization problem in eq uation (21) is 1 M ∂ ∂ q M − 1 X k =0 g k = Γ ψ ( q ) , where Γ = 1 M  T − I 0 . . . − I M − 1  ∈ R n × ( M +1) , and ψ : ∆ n − 1 → R M +1 is defined by ψ ( q ) =  M − 1 P k =0 1 q · I k q · T ( q · I 0 ) 2 . . . q · T ( q · I M − 1 ) 2  T . For n > M , if the vecto rs T , I 0 , . . . , I M − 1 are linear ly indepen dent, then Γ is fu ll rank. Fur ther , th e entries o f ψ are non-zer o on the pr obability simplex ∆ n − 1 . Hence, the Jacobian d oes n ot vanish anywher e on the pro bability simplex ∆ n − 1 . Lemma 9 (Case of Ind ependen t Information) : For M = 2 , if I 0 and I 1 are linearly ind ependen t, and T = α 0 I 0 + α 1 I 1 , for some α 0 , α 1 ∈ R , then the f ollowing statements hold: i) if α 0 and α 1 have o pposite signs, then g 0 + g 1 has n o critical point o n the simplex ∆ n − 1 , and ii) for α 0 , α 1 > 0 , g 0 + g 1 has a critical point on the simplex ∆ n − 1 if and only if there exists v ∈ ∆ n − 1 perpen dicular to the vector √ α 0 I 0 − √ α 1 I 1 . Pr oof: W e no tice th at the Jaco bian of g 0 + g 1 satisfies ( q · I 0 ) 2 ( q · I 1 ) 2 ∂ ∂ q ( g 0 + g 1 ) = T  ( q · I 0 )( q · I 1 ) 2 + ( q · I 1 )( q · I 0 ) 2  − I 0 ( q · T )( q · I 1 ) 2 − I 1 ( q · T )( q · I 0 ) 2 . (22) Substituting T = α 0 I 0 + α 1 I 1 , e quation (22) becomes ( q · I 0 ) 2 ( q · I 1 ) 2 ∂ ∂ q ( g 0 + g 1 ) =  α 0 ( q · I 0 ) 2 − α 1 ( q · I 1 ) 2   ( q · I 1 ) I 0 − ( q · I 0 ) I 1  . Since I 0 , a nd I 1 are linearly independen t, we h av e ∂ ∂ q ( g 0 + g 1 ) = 0 ⇐ ⇒ α 0 ( q · I 0 ) 2 − α 1 ( q · I 1 ) 2 = 0 . Hence, g 0 + g 1 has a cr itical p oint on th e sim plex ∆ n − 1 if and only if α 0 ( q · I 0 ) 2 = α 1 ( q · I 1 ) 2 . (23) Notice that, if α 0 , and α 1 have opp osite signs, then equation (23) ca n not be satisfied for any q ∈ ∆ n − 1 , and hen ce, g 0 + g 1 has no critical po int on the simp lex ∆ n − 1 . If α 0 , α 1 > 0 , then equation (23) leads to q · ( √ α 0 I 0 − √ α 1 I 1 ) = 0 . Therefo re, g 0 + g 1 has a cr itical point on the simplex ∆ n − 1 if and only if there exists v ∈ ∆ n − 1 perpen dicular to the vector √ α 0 I 0 − √ α 1 I 1 . 8 Lemma 10 (Optimal nu mber of sensors ): For n > M , if each ( M + 1 ) × ( M + 1) subm atrix of the matrix Γ =  T − I 0 . . . − I M − 1  ∈ R n × ( M +1) is full rank, th en the following statements hold: i) e very so lution of the optimization prob lem (21) lies on the probability simplex ∆ M − 1 ⊂ ∆ n − 1 ; and ii) e very tim e-optimal policy requ ires a t most M sensors to be observed. Pr oof: From Lemma 8, we kn ow that if T , I 0 , . . . , I M − 1 are linearly independe nt, then the Jacobian of the objectiv e function in equ ation (21) does not vanish anywh ere o n the simplex ∆ n − 1 . Hence, a minimum lies at some simplex ∆ n − 2 , which is the boun dary of the simp lex ∆ n − 1 . Notice th at, if n > M and the conditio n in th e lemma holds, then the projections of T , I 0 , . . . , I M − 1 on the simplex ∆ n − 2 are also linearly indepen dent, and the argume nt repeats. Hen ce, a min imum lies at some simplex ∆ M − 1 , whic h implies that optim al policy requires at m ost M sensors to b e o bserved. Lemma 11 (Optimization o n an ed ge): Gi ven two vertices e s and e r , s 6 = r , of the proba bility simplex ∆ n − 1 , then for the ob jectiv e function in the problem (21) with M = 2 , the following statements hold: i) if g 0 ( e s ) < g 0 ( e r ) , and g 1 ( e s ) < g 1 ( e r ) , then the minima, along the edge join ing e s and e r , lies at e s , and optimal v alue is given by 1 2 ( g 0 ( e s ) + g 1 ( e s )) ; and ii) if g 0 ( e s ) > g 0 ( e r ) , and g 1 ( e s ) < g 1 ( e r ) , or vice versa, then the m inima, alo ng th e edge join ing e s and e r , lies at the p oint q ∗ = (1 − t ∗ ) e s + t ∗ e r , where t ∗ = 1 1 + µ ∈ ]0 , 1 [ , µ = I 0 r p T s I 1 r − T r I 1 s − I 1 r p T r I 0 s − T s I 0 r I 1 s p T r I 0 s − T s I 0 r − I 0 s p T s I 1 r − T r I 1 s > 0 , and the o ptimal v alue is given b y 1 2 ( g 0 ( q ∗ ) + g 1 ( q ∗ )) = 1 2 s T s I 1 r − T r I 1 s I 0 s I 1 r − I 0 r I 1 s + s T r I 0 s − T s I 0 r I 0 s I 1 r − I 0 r I 1 s ! 2 . Pr oof: W e ob serve from Lemma 5 th at both g 0 , and g 1 are m onoton ically non-incr easing or n on-dec reasing along a ny line. He nce, if g 0 ( e s ) < g 0 ( e r ) , and g 1 ( e s ) < g 1 ( e r ) , then the minima should lie at e s . This co ncludes the proof of the first statement. W e now establish th e secon d statement. W e note that any point o n the line segment conne cting e s and e r can be written as q ( t ) = (1 − t ) e s + te r . The v alue of g 0 + g 1 at q is g 0 ( q ( t )) + g 1 ( q ( t )) = (1 − t ) T s + tT r (1 − t ) I 0 s + tI 0 r + (1 − t ) T s + tT r (1 − t ) I 1 s + tI 1 r . Differentiating with respect to t , we get g 0 ′ ( q ( t )) + g 1 ′ ( q ( t )) = I 0 s T r − T s I 0 r ( I 0 s + t ( I 0 r − I 0 s )) 2 + I 1 s T r − T s I 1 r ( I 1 s + t ( I 1 r − I 1 s )) 2 . (24) Notice that the tw o term s in equation ( 24) have op posite sign. Setting the deriv ati ve to zero, an d choosin g the value of t in [0 , 1] , we get t ∗ = 1 1+ µ , where µ is as defined in the statement o f the theo rem. The optimal value of the fu nction can be o btained, by substituting t = t ∗ in th e expression for 1 2 ( g 0 ( q ( t )) + g 1 ( q ( t ))) . Theor em 3 (Optimization of average decision time): For the optimization problem in equatio n (2 1) with M = 2 , the following statements hold: i) If I 0 , I 1 are lin early dependent, then the so lution lies at some vertex of the simplex ∆ n − 1 . ii) If I 0 and I 1 are linearly ind ependen t, and T = α 0 I 0 + α 1 I 1 , α 0 , α 1 ∈ R , then the following statements hold: a) If α 0 and α 1 have opp osite signs, then th e optimal solution lies at so me e dge of th e simplex ∆ n − 1 . b) If α 0 , α 1 > 0 , then the optimal so lution may lie in the interior of th e simplex ∆ n − 1 . iii) If every 3 × 3 sub -matrix of the matrix  T I 0 I 1  ∈ R n × 3 is full rank, then a m inimum lies at an edge of the simplex ∆ n − 1 . Pr oof: W e start by establishing the first statement. Since, I 0 and I 1 are linea rly depen dent, ther e exists a γ > 0 such that I 0 = γ I 1 . For I 0 = γ I 1 , we have g 0 + g 1 = (1 + γ ) g 0 . Hence, the minima o f g 0 + g 1 lies at the same point where g 0 achieves th e min ima. From Theorem 1 , it fo llows th at g 0 achieves the minima at som e vertex of the simplex ∆ n − 1 . T o prove the seco nd statement, we n ote tha t fro m Lemma 9, it follows that if α 0 , and α 1 have o pposite signs, then the Jacobian of g 0 + g 1 does not vanish a nywhere on the simplex ∆ n − 1 . Hen ce, the m inima lies at the b oundar y of th e simplex. Notice that the bound ary , of the simplex ∆ n − 1 , are n simplices ∆ n − 2 . Notice that the argumen t repe ats till n > 2 . Hen ce, the optima lie on one of the  n 2  simplices ∆ 1 , which are the ed ges of the origin al simplex. Moreover , we note that from Lemma 9 , it follows that if α 0 , α 1 > 0 , the n we can not guarantee the number of optimal set of sen sors. This concludes the p roof of the second statement. T o prove the last statement, we note that it follows im- mediately fr om Lemma 10 that a solution of the optimization problem in equation (21) would lie at some simplex ∆ 1 , which is an edge of the origin al simp lex. Note that, we h av e shown that, for M = 2 and a gen eric set of sensor s, the solution of the optimization p roblem in equation (21) lies at an edge of the simplex ∆ n − 1 . The optimal value of the objective function on a gi ven edge was determined in Lemma 1 1. Hence, an o ptimal solution of th is p roblem can be determined by a comparison o f the op timal values at each edge. For the mu ltiple hypo thesis case, we have determined the time- optimal nu mber of the sen sors to be observed in Lemma 10. In order to identify these sensor s, one need s to solve the optimizatio n prob lem in equation (21). W e notice that th e objective function in this optimizatio n problem is no n- conv ex, and is hard to tackle analytically for M > 2 . Interested reader may refer to some efficient iterative algorithms in linear-fractional prog ramming literature (e.g., [6]) to solve these problems. 9 V I . N U M E R I C A L E X A M P L E S W e consider four sensors c onnected to a fusion center . W e assume th at the sensors take binary measurements. The probab ilities of their measurement being zero, u nder two hypoth eses, and their processing times a re given in the T able I. T ABL E I C O N D I T I O NA L P RO B A B I L I T I E S O F M E A S U R E M E N T B E I N G Z E RO Sensor Probabil ity(0) Processing Time Hypothesis 0 Hypothesis 1 1 0.4076 0.5313 0.6881 2 0.8200 0.3251 3.1960 3 0.7184 0.1056 5.3086 4 0.9686 0.6110 6.5445 W e perfor med Monte-Carlo simulations with the m is- detection an d false-alarm probabilities fixed at 10 − 3 , and computed expected decision times. In T able II, the numerical expected decision times are comp ared with the d ecision times obtained an alytically in equation ( 7). Th e difference in the numerical and the an alytical d ecision times is explained b y the W ald’ s asymptotic appro ximations. T ABL E II D E C I S I O N T I M E S F O R VAR I O U S S E N S O R S E L E C T I O N P RO BA B I L I T IE S Sensor selection Expected Decision Time probabil ity Hypothesis 0 Hypothesis 1 Analyti cal Numerical Analytic al Numerical [1,0,0,0] 154.39 156.96 152.82 157.26 [0.3768,0,0,0.6232] 124.35 129.36 66.99 76.24 [0.25,0.25,0.25,0.25] 55.04 59.62 50.43 55.73 In th e T able III, we co mpare optimal policies in each Scenario I, II, and II I with the policy when each sensor is chosen unifor mly . It is o bserved that the optimal policy improves the expected decision time significantly over th e unifor m policy . T ABL E III C O M PA R I S O N O F T H E U N I F O R M A N D O P T I M A L P O L I C I E S Scenari o Uniform polic y Optimal polic y Objecti ve Optimal probabili ty Optimal objecti ve functio n v ector functio n I 59.62 [0,0, 1,0] 41.48 II 55.73 [0,0, 1,0] 45.24 III 57.68 [0,0.4788,0.5212,0] 4 3.22 W e performed another set of simulations fo r the mu lti- hypoth esis case. W e consid ered a ternar y d etection pro blem, where the underlyin g signal x = 0 , 1 , 2 need s to be detected from the available noisy data. W e con sidered a set of fo ur sensors a nd their cond itional prob ability distribution is g i ven in T ables IV an d V. The processing time o f the sensors were chosen to be the same as in T able I. The set o f optimal sensor s were determin ed for this set of data. Monte-Carlo simulation s were pe rformed with th e thresholds η k , k ∈ { 0 , . . . , M − 1 } set at 10 − 6 . A comp arison of the uniform sensor selection policy and an optim al sensor T ABL E IV C O N D I T I O NA L P RO BA B I L I T I E S O F M E A S U R E M E N T B E I N G Z E RO Sensor Probabil ity(0) Hypothesis 0 Hypothesis 1 Hypothe sis 2 1 0.4218 0.2106 0.2769 2 0.9157 0.0415 0.3025 3 0.7922 0.1814 0.0971 4 0.9595 0.0193 0.0061 T ABL E V C O N D I T I O NA L P RO BA B I L I T IE S O F M E A S U R E M E N T B E I N G O N E Sensor Probabil ity(1) Hypothesis 0 Hypothesis 1 Hypothe sis 2 1 0.1991 0.6787 0.2207 2 0.0813 0.7577 0.0462 3 0.0313 0.7431 0.0449 4 0.0027 0.5884 0.1705 selection policy is presented in T able VI. Again, the significant difference b etween the average decision time in the u niform and the o ptimal policy is evident. T ABL E VI C O M PA R I S O N S O F T H E U N I F O R M A N D A N O P T I M A L P O L I C Y Polic y Select ion Probabil ity A ver age Dec ision Time q ∗ Analyti cal Numeric al Optimal [0, 0.9876, 0, 0.0124] 38.57 42.76 Uniform [0.25, 0.25, 0.25, 0.25] 54.72 54.14 W e note that the op timal results, we obtained, may only be sub-optima l because of the asymptotic a pproxim ations in equation s ( 3) and (5). W e fu rther note that, for small error pr obabilities and large sample sizes, these a symptotic approx imations y ield fairly accu rate results [5], and in fact, this is the regime in which it is o f interest to minimize the expected d ecision time. Th erefore, for all practical pu rposes the obtained op timal s cheme is very close to the actual optimal scheme. V I I . C O N C L U S I O N S In this paper, we considered the problem of sequential decision making. W e developed versions SPR T an d MSPR T where the sensor switches a t each observation. W e used these sequential proced ures to decide reliably . W e found out the set of optimal sensors to decide in minimum time. For the binary hypoth esis case, thr ee performan ce metrics were considered and it was foun d that for a generic set o f sensors at most two sensors are optimal. Further, it was shown that for M underly ing hypoth eses, and a generic s et of sensors, an optimal policy requires at most M sensors to be observed. A procedure for identification of the o ptimal sensor was developed. In the binary h ypothesis case, th e c omputation al complexity of the proced ure for th e three scena rios, namely , the con ditioned decision time, th e worst case decisio n time, and th e average decision time, was O ( n ) , O ( n 2 ) , and O ( n 2 ) , respectively . Many furth er extensions to the results p resented here ar e possible. First, the time-optimal sch eme may not be en ergy optimal. 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