Robust Decentralized Detection and Social Learning in Tandem Networks

0.24 MB·English

by  Jack Ho

Robust Decentralized Detection and Social Learning
in Tandem Networks
Jack Ho, Student Member, IEEE, Wee Peng Tay, Senior Member, IEEE, Tony Q. S. Quek, Senior Member, IEEE,
and Edwin K. P. Chong, Fellow, IEEE
Abstract—We study a tandem of agents who make decisions social learning is in the case of participatory sensing, where
5 about an underlying binary hypothesis, where the distribution inferenceabouta phenomenonofinterestismadethroughthe
of the agent observations under each hypothesis comes from
1 help of agents in the network [11], [18], [19]. For example,
an uncertaintyclass.Weinvestigatebothdecentralized detection
0 this can occur when users send a picture of litter in a park to
rules,whereagentscollaboratetominimizetheerrorprobability
2
of the final agent, and social learning rules, where each agent a social sensing platform [20], [21] or report congested road

n minimizes its own local minimax error probability. We then conditions. [22].
a extend our results to the infinite tandem network, and derive Ontheotherhand,iftheagents’decisionrulesaredesigned
J necessaryandsufficientconditionsontheuncertaintyclassesfor
tominimizetheerrorcriterionofthelastagentinthenetwork,
3 the minimax error probability to converge to zero when agents
or the asymptotic error probability in the case of an infinite
2 know their positions in the tandem. On the other hand, when
agents do not know their positions in the network, we study network, this is known as decentralized detection [8], [23],

] the cases where agents collaborate to minimize the asymptotic [24]. One major application of decentralized detection is in
T minimax error probability, and where agents seek to minimize sensor networks with a fusion center [25]–[27]. If the fusion
I their worst-case minimax error probability (over all possible
. center is able to relay information to the other agents, it will
s positions in the tandem). We show that asymptotic learning of
be able to select a set of globally optimal decision rules for
c thetruehypothesisisnolongerpossibleinthesecases,andderive
[ characterizations for the minimax error performance. everyagent.However,manypracticalnetworks,suchassocial

networks, do not have a fusion center. Furthermore, even for
1 IndexTerms—Sociallearning,decentralizeddetection,tandem
networks with fusion centers, the fusion center may not be
v networks, robust hypothesis testing
7 able to easily communicatewith the other agents. This is true
4 of the participatory sensing examples above.
8 I. INTRODUCTION Inthe aboveexamples,itisassumedthateachagentknows
5
In this paper, we formulate and study the robust social the distribution of its private observation, and that of its
0
learning problem in a tandem network. A tandem network predecessor, as well as its position in the network. However,
.
1
consists of agents connected in a serial fashion, where each in a real-life network, this is generally not the case. In this
0
agent receives an opinion about a binary hypothesis from a paper,weinvestigatewhathappenswhenoneorbothofthese
5
1 previousagent,andmakesadecisionaboutabinaryhypothesis assumptions do not hold.
: basedonthepreviousagent’sopinionanditsownobservation.
v
Despite the simple structure of the tandem network, studying
i A. Related Work
X it can lead to insights about more complicated network struc-
r tures such as those in social networks or Internet of Things Binary hypothesis testing in a tandem network model is
a
(IoT) networks. The tandem network approximates a single studied in [3], [5], which shows that learning the true hy-
information flow in a network, and it and its variants have pothesisasymptoticallyis possiblewith unboundedlikelihood
been widely studied in [1]–[10]. ratios, and not possible with bounded likelihood ratios when
Inourmodel,eachagent’sdecisionisbasedonalocalerror agents transmit only 1-bit messages. Decentralized detection
criterion, which it selfishly tries to optimize. This behavior is policies for tandem networks are also considered in [6], and
present in social networks, where users are mainly concerned conditions for the error probability approaching zero as the
withspreadingonlylocallyaccurateinformation.Inthispaper, number of agents grows are derived. This is a network where
we call this social learning [9]–[17]. One such application of each agent after the first receives exactly one decision from
its predecessor. The authors also study a sub-optimal scheme
Part of this paper was presented at the IEEE International Con- where each sensor “selfishly” tries to minimize its own error,
ference on Acoustics, Speech and Signal Processing, May 2013. This
as opposed to the error of the root agent. The reference [8]
research is supported in part by the Singapore Ministry of Educa-
tion Academic Research Fund Tier 2 grants MOE2013-T2-2-006 and shows that the rate of error decay is at most sub-exponential.
MOE2014-T2-1-028. J. Ho and W.P. Tay are with the School of Feedforward networks, in which an agent obtains informa-
Electrical and Electronic Engineering, Nanyang Technological University,
tion from a subset of previous agents not necessarily just
Singapore. E-mail: [email protected], [email protected].
T.Q.S. Quek is with the Information Systems Technology and Design Pil- the immediate predecessor, have been studied in [9], [10],
lar, Singapore University of Technology and Design, Singapore. E-mail: [17]. In [10], agents are able to access the decisions of their
[email protected] K.P. Chong is with the Department of
K most recent predecessors. It is demonstrated that almost
Electrical and Computer Engineering, Colorado State University, USA. E-
mail:[email protected]. sure learning is impossible for any value of K, and learning
2
in probability is possible for K ≥ 2. A new model where to the case where the contamination of the uncertainty
forward looking agents try to maximize the discounted sum classes are zero [6], [8], in which case asymptotic learn-
of the probability of a right decision is also considered. The ing happens if the log likelihood ratio is unbounded.
reference [9] studies the decentralized detection problem in a 3) When agents know their positions in the network, we
gametheoreticsetting,andexaminestheeffectofobtainingin- show that asymptotically learning the true hypothesis
formationfromdifferentsets ofpreviousagentsontherate of under social learning is achievable if and only if the log
errordecay.The reference[28] examinesthe asymptoticerror likelihoodratioofthenominaldistributionsisunbounded,
rate of feedforward topologies under two types of broadcast and there are two subsequences of agents, one corre-
errors, namely erasure and random flipping. spondingto eachhypothesis,such thatthe contamination
All the above works assume that agent’s observations are of the uncertainty class under that hypothesis converges
drawn from known distributions under each hypothesis. This to zero (Theorem 5).
assumption may not hold in practical networks like IoT net- 4) When agents do not know their positions in the tandem,
works,inwhichsensors’observationdistributionsmaychange weshowthatitisnotpossibletoasymptoticallylearnthe
overtime,orinsocialnetworks,inwhichagents’observations true hypothesis. We investigate the cases where agents
maybe affectedbythe agents’moodata particulartime. The collaborate to minimize the asymptotic minimax error
robust detection framework was first proposed by [29] for a probability, and where agents seek to minimize their
single agentto modelthe case wherethe observationdistribu- worst-case minimax error probability (over all possible
tions are not known exactly. A survey of results in this area positions in the tandem), and characterize the minimax
can be found in [30]. The underlyingprobability distributions error performance in these approaches (Theorems 6 and
governing the agent observations are assumed to belong to 7).
different uncertainty classes under different hypotheses, and In this paper, we consider only robust decentralized detec-
it is shown that under a minimax error criterion, the optimal tion and social learning in tandem networks, which are very
decision rule for the agent is a likelihood ratio test based on simple in structure. Social networks and IoT networks are
the pair of least favorable distributions(LFDs). Subsequently, much more complex in practice. Therefore, our results are
the work [31] investigates robust detection in a finite parallel limited,andcanonlybeappliedheuristicallytomorepractical
configuration, with and without a fusion center. The problem networks.Ouranalysisformsthefoundationforstudyingmore
of robustsequentialdetectionis studied in [32]. Robustsocial complex networks like trees and general loopy graphs, and
learning however has not been addressed in these works. In providesinsightsintodesigningoptimaldecisionrulesforsuch
addition, robust detection and learning have not been studied networks. For example, although using likelihood ratio tests
for the tandem network. based on LFDs at each agent is not known to be optimal for
loopy graphs, we expect this to produce reasonable results
in practice. Addressing the performance of social learning in
B. Our Contributions
more complex networks is part of future research.
Inthispaper,weconsiderrobustbinaryhypothesisdetection
The rest of this paper is organized as follows. In Section
and social learning in a tandem network in which the obser-
II, we introduce the robust decentralized detection and social
vationmodelsofagentsundereachhypothesisareuncertainty
learning problem in a tandem network. In Section III, we
classes of probability distributions. Our main goals are to
provide a characterization for the agents’ optimal decision
obtain the optimal agent policies under a minimax error cri-
rules in both decentralized detection and social learning in
terion, under both decentralized detection and social learning
a tandem network. We then derive necessary and sufficient
frameworks, and characterize the asymptotic minimax error
conditions for asymptotically learning the true hypothesis
probabilities under various scenarios. Our main contributions
undervarioussimplificationsinSectionIV. We alsostudythe
are the following:
case where agents do not know their positions in the tandem
1) For the tandem network, we show that the solutions in this section. In Section V, we illustrate some of our results
to the robust decentralized detection and social learning usinganumericalexample.Lastly,weconcludeinSectionVI.
problemsareequivalentto the respectivesolutionsof the
correspondingclassicalhypothesistestingproblemwhere II. PROBLEM FORMULATION
all the private observations are distributed according to WeconsideratandemnetworkconsistingofN agents,with
the LFDs (Theorem 1). Our proof can be extended to agent 1 being the first agent and N being the last (see Figure
generaltreetopologies,whichgeneralizesaresultin[31], 1). Consider a binaryhypothesistesting problemin whichthe
where a parallel topology is considered. true hypothesis H is H with prior probability π ∈ (0,1),
i i
2) We show that when the uncertainty classes for all agent for i = 0,1. Conditioned on H = H , each agent k in
i
observations are the same, and agents know their po- the network makes an observation Y , defined on a common
k
sitions in the tandem, asymptotically learning the true measurablespace(Y,A),andwithdistributionP belonging
i,k
hypothesis under both decentralized detection and social to an uncertainty class
learning frameworks is not possible if the contamination
P ={Q|Q=(1−ǫ )P∗+ǫ R,R∈R},
of both uncertainty classes are non-zero, even when the i,k i,k i i,k
log likelihood ratio of the nominal distributions is un- whereRisthesetofallprobabilitymeasureson(Y,A),P∗ ∈
i
bounded(Theorem3 and Theorem4). Thisis in contrast Risthenominalprobabilitydistribution,andǫ ∈[0,1)isa
i,k
3
In contrast to the decentralized detection problem in (1),
each agent myopically seeks to minimize its local maximum
probability of error.
For each agent k, let p be the density (with respect to
i,k
some commonmeasure)of P , andp∗ be the densityof P∗,
i,k i i
for i = 0,1. The least favorable distributions (LFDs) for two
givenuncertaintyclassesP andP isdefinedin[29]tobe
0,k 1,k
thepairofdistributions(Q ,Q )withdensities(q ,q )
0,k 1,k 0,k 1,k
Fig.1. Hypothesis testing inatandem network.
such that
(1−ǫ )p∗(y) for p∗(y)/p∗(y)<c′′
q (y)= 0,k 0 1 0
positiveconstantthatissufficientlysmallsothatP0,k andP1,k 0,k ((1−ǫ0,k)p∗1(y)/c′′ for p∗1(y)/p∗0(y)≥c′′
are disjoint. We assume that all distributionsin P and P
0,k 1,k
are absolutelycontinuouswith respectto oneanother,and the
distributionPj,k fromwhichtheobservationYk isdrawnfrom q (y)= (1−ǫ1,k)p∗1(y) for p∗1(y)/p∗0(y)>c′
is unknown. Furthermore, we assume that conditioned on the 1,k (c′(1−ǫ1,k)p∗0(y) for p∗1(y)/p∗0(y)≤c′,
truehypothesis,theobservationsofeachagentareindependent
where 0 ≤ c′ < c′′ ≤ ∞ are determined such that q and
from one another. The parameter ǫ is also known as the 0,k
i,k q are probability densities. Note that c′ = 0 if and only
contamination for the uncertainty class P . When ǫ = 0, 1,k
i,k i,k if ǫ = 0, and c′′ = ∞ if and only if ǫ = 0. Let b =
we recoverthe classical Bayesian hypothesistesting problem. 1,k 0,k k
(1−ǫ )/(1−ǫ ). We then have
Whileagentscanhavedifferentcontaminationvaluesǫ and 1,k 0,k
0,k
ǫ1,k, we assume thatthe nominaldistributionsP0∗ and P1∗ are bkc′ for p∗1(y)/p∗0(y)≤c′
φidkeF(nYotikrc,akUlk=f−or11)e,v∈.e.r{.y,0N,a1g,}eneatab.cohutatgheenhtykpomthaekseissHa,dwechiesrieonφkUiksa=n qq10,,kk((yy)) =bbkkc·′′pp∗0∗1((yy)) ffoorr cp′∗1(<y)pp/∗0∗1p((∗0yy())y<)≥c′′c′′. (4)
agent decision rule whose decision i corresponds to deciding
in favor of H , and U ≡ 0. For i = 0,1, let P(N) = P × In [29], it was shown that the LFDs of a pair of uncertainty
i 0 i i,1 classes are the two distributions that give the largest error
Pi,2×...×Pi,N. Similarly, define Pi(N) =Pi,1×Pi,2×...× probabilitywhenusingalikelihood-ratiotesttotellthemapart.
Pi,N. In the decentralized detection problem, our aim is to In the rest of this paper, for any random variable Y with
find a sequence of decision rules φ(N) = (φ1,φ2,...,φN) to distributions drawn from a given pair of uncertainty classes,
minimize the maximum probability of error given by we let l∗(Y) be the likelihood ratio q (Y)/q (Y), where
1 0
q and q are the respective densities of the LFDs of the
0 1
PNDD(φ(N)) given uncertainty classes. In addition, we use l∗(Y = y) to
= sup P (φ(N),P(N),P(N)), (1) denote the realization of l∗(Y) when Y = y. A sequence
e,N 0 1
(P0(N),P1(N))∈P0(N)×P1(N) x1,x2,...,xn is denoted as (xi)ni=1.
where
III. ROBUST LEARNING IN A TANDEMNETWORK
P (φ(N),P(N),P(N))=π P (φ(N),P(N)) When there is only a single agent, the minimax error
e,N 0 1 0 F,N 0 (2)
inf PSL(φ) is achieved by setting φ to be a likelihood ratio
+π P (φ(N),P(N)). φ 1
1 M,N 1 test using the LFDs (Q ,Q ) [29]. A similar result is
0,1 1,1
proven in [31] for a parallel network configuration. In this
In (2), P and P are the false alarm and missed
F,N M,N
section, we show the same result for the tandem network.We
detection probabilities of agent N respectively, given the
decision rules φ(N) and the agents’ observation distributions dothisbyfirstshowingthatforanysequenceofagentdecision
P(N), i=0,1. rules that consists of likelihood ratio tests between LFDs, the
i error probabilities (1) and (3) are maximized when all the
In the social learning problem, the first agent chooses its
private agent observations are drawn from the corresponding
decision rule φ to minimize
1
LFDs. We then show that when agents’ private observations
sup Pe,1(φ1,P0,1,P1,1). aredrawnfromtherespectiveLFDs,thenlikelihoodratiotests
(P0,1,P1,1)∈P0,1×P1,1 usingLFDsminimizetheerrorprobabilities.We firststatethe
Each other agent k, given the decision rules φ(k−1) of the following lemma proven in [29].
previous agents 1,...,k−1, is able to derive the false alarm Lemma 1. Supposethatthe LFDsfor(P ,P ) are(Q ,Q ).
0 1 0 1
and miss detection probabilities of agent k−1. It then seeks Then, for any random variable Y with distributions P ∈P ,
i i
to find φ to minimize
k where i=0,1, we have
PSL(φ |φ(k−1)) P0(l∗(Y)>t)≤Q0(l∗(Y)>t)
k k
= sup P (φ(k),P(k),P(k)). (3) ≤Q1(l∗(Y)>t)
e,k 0 1 ≤P (l∗(Y)>t).
(P(k),P(k))∈P(k)×P(k) 1
0 1 0 1
4
We can now present our first result. For all k ≥ 1, let From Lemma 3, the product of l∗(Y ) and l∗(U ) is
k k−1
(Q ,Q )betheLFDsfor(P ,P ),andQ(N) =Q × stochasticallylargerunderQ(k) thanunderanyotherdistribu-
0,k 1,k 0,k 1.k i i,1 0
Qi,2×...×Qi,N for i=0,1. tion P0(k) as well. Therefore, we have
Theorem1. Letφ(N) beanysequenceofmonotonelikelihood
ratio tests based on the LFDs (Q(N),Q(N)) for a tandem Q(0k)(Uk =1)=Q(0k)(l∗(Uk−1,Yk)>tk)
0 1
network. Then for all (P0(N),P1(N))∈P0(N)×P1(N), we have ≥P0(k)(l∗(Uk−1,Yk)>tk)
P (φ(N),Q(N))≥P (φ(N),P(N)), (5) =P0(Uk =1).
F,N 0 F,N 0
and The proof for the missed detection probability inequality (6)
P (φ(N),Q(N))≥P (φ(N),P(N)). (6) is similar, and the induction is complete. The theorem is now
M,N 1 M,N 1
proved.
Proof: We proceed by mathematical induction on N.
Theorem 2. Let φ(N) be an optimal sequence of decision
FromLemma1,thetheoremholdsforN =1.Wenowassume ∗
rules such that
that it holds for N < i. We also make use of the following
two lemmas, the first of which is proved in Appendix A-A,
φ(N) =argminP (φ(N),Q(N),Q(N)).
while the second is shown in [31]. ∗ e,N 0 1
φ(N)
Lemma 2. ForanyN,Q(N)(U =1)≥Q(N)(U =1)and
Q1(N)(UN =0)≤Q(0N)(U1N =N0). 0 N Then,φ(∗N) minimizesPNDD(·)in(1).Similarly,foreachk ≥1,
define ψ recursively as
∗,k
Lemma 3. Let Z and Z be non-negative, independent
1 2
random variables. If for k =1,2, we have ψ =argminP ((ψ(k−1),ψ ),Q(k),Q(k)),
∗,k e,k ∗ k 0 1
F(Z >t) ≥G(Z >t), ∀t≥0, (7) ψk
k k
then where ψ∗,0 is ignored. Then, ψ∗,k minimizes PkSL(·|ψ∗(k−1))
F(Z Z >t) ≥G(Z Z >t), ∀t≥0. in (3) for all k ≥1.
1 2 1 2
If(7) holds,we say thatZ is stochasticallylargerunderF Proof: In [23], it was shown that φ(∗N) is a sequence of
than G. Since the observationk of agent k is independentfrom likelihood ratio tests based on (Q(0N),Q(1N)). Hence, for any
the decision it receives, for i=0,1 we have sequence of decision rules φ(N), we have from Theorem 1,
Qi(k)(l∗(Yk)>t)=Qi,k(l∗(Yk)>t). sup Pe,N(φ(∗N),P0(N),P1(N))
(P(N),P(N))∈P(N)×P(N)
FromLemma1,l∗(Y )isstochasticallylargerunderQ(k) than 0 1 0 1
k 0 (N) (N) (N)
underanyotherdistributionP(k) ∈Pk.Toshowthesamefor =Pe,N(φ∗ ,Q0 ,Q1 )
0 0
l∗(U ), we obtain from Lemma 2 that
k−1 ≤P (φ(N),Q(N),Q(N)) (8)
e,N 0 1
Q(k−1)(U =1)
l∗(U =1)= 1 k−1
k−1 Q0(k−1)(Uk−1 =1) ≤ sup Pe,N(φ(N),P0(N),P1(N))
≥1 (P0(N),P1(N))∈P0(N)×P1(N)
1−Q(k−1)(U =1)
≥ 1 k−1 where (8) follows from the theorem assumption. Therefore,
1−Q0(k−1)(Uk−1 =1) the minimax error in the decentralized detection problem is
≥l∗(U =0). equal to the minimum error when all the distributions of the
k−1
privateobservationsareequaltotheLFDs.Asimilarargument
For any l∗(Uk−1 =0)<t<l∗(Uk−1 =1), holdsforthesociallearningprobleminthesecondpartofthis
theorem. The proof is now complete.
Q(k)(l∗(U )>t)=Q(k)(U =1)
0 k−1 0 k−1 We remark that Theorem 2 can be extended to generaltree
≥P(k)(U =1) topologies. This is because in such a topology, the decisions
0 k−1
received by each agent are mutually independent. Hence,
=P(k)(l∗(U )>t),
0 k−1
m
where the inequality follows from the induction hypothesis. l∗(Y ,U ,...,U )=l∗(Y ) l∗(U ),
Note that l∗(U ) only takes the two values l∗(U = 0) k k1 km k kj
k−1 k−1 j=1
and l∗(U = 1), so for any t not in between these values, Y
k−1
the equality Q0(k)(l∗(Uk−1) > t) = P0(k)(l∗(Uk−1) > t) where k1,...,km are the agents that agent k is receiving
is trivially true. Therefore, l∗(Uk−1) is stochastically larger decisions from. The rest of the proof then proceeds similarly
under Q(k). as that of Theorems 1 and 2.
0
5
IV. ASYMPTOTICDETECTIONAND SOCIAL LEARNING sufficiency of the given condition then follows immediately.
Toshowthatitisalsonecessary,observethatifǫ >0orǫ >
Theorem 2 shows that there is no loss in optimality in 0 1
0, then the log-likelihood ratio of Q versus Q is bounded
both the decentralized detection and social learning problems 1 0
from either above or below respectively, and Theorem 2 and
if agents in a tandem network are restricted to monotone
Proposition3 of [6] implies that PDD(φ(N)) is boundedaway
likelihood ratio tests based on the LFDs. It however does not N
from zero. The proof is now complete.
tell us the minimum minimax error probability achievable.
Fromthetheoremsabove,itcanbeseenthatforasymptotic
In this section, we study the minimax error probability in
learningtooccurinthesociallearningcase,itisnecessarythat
long tandems under various technical assumptions in order to
both the distributionsbe uncontaminated.In the decentralized
simplify the problem. In particular, we investigate the condi-
detectioncase,itisonlynecessarythatoneofthedistributions
tions under which the minimax error probability convergesto
be uncontaminated.
zero as the numberof agentsincreases.1 We first consider the
case where every agent has identical uncertainty classes, and
B. Varying uncertainty classes for social learning
provide necessary and sufficient conditions for the minimax
error probability to approach zero under both decentralized In this subsection, we relax the assumption of identical
detectionandsociallearning.We willthenproceedtoanalyze uncertainty classes for all agents in the previous subsection,
sociallearninginlongtandemswherethecontaminationofthe and study the effect of varying contamination values on the
uncertainty class can differ. Finally, we study the achievable asymptotic error probability in the social learning framework.
asymptotic minimax error probability when agents do not We make the following assumptions.
know their own positions in the tandem.
Assumption 2. We have
(i) the log-likelihood ratio of P∗ versus P∗ is unbounded;
1 0
A. Identical Uncertainty Classes and
(ii) each agent k ≥ 1 knows its own contamination values
In this subsection, we make the following assumption that
ǫ , for i = 0,1, as well as those of its predecessors,
the uncertainty classes of every agent are identical. i,k
and its position in the tandem network.
Assumption 1. For all k ≥ 1, we have ǫ = ǫ , ǫ =
0,k 0 1,k
Assumption 2(i) is necessary because otherwise learning
ǫ , and the LFDs of each agent’s uncertainty classes are
1
the true hypothesis is not possible under a social learning
(Q ,Q ).
0 1
framework even if all the contamination values are zero,
The following two results give necessary and sufficient as shown in Theorem 4. We will show that under these
conditions for the minimax error probability to approach assumptions,learningthetruehypothesishappensifthereexist
zero under the decentralized detection and social learning infinite subsequences (ǫ ) and (ǫ ) (which may
0,kn n≥1 1,jn n≥1
frameworks, respectively. potentially be distinct) that converge to zero as n increases.
We first observe that under the social learning framework,
Theorem 3 (Decentralized detection). Suppose that Assump-
agents minimize their local maximum error probability. This
tion 1 holds, and that the decision rules for every agent
implies that regardless of the values of ǫ and ǫ , the
are chosen so as to minimize PDD(φ(N)) in (1). Then 0,k 1,k
N minimaxerrorprobabilityofeachagentk isnon-increasingin
PDD(φ(N)) → 0 as N → ∞ if and only if at least one of
N k.Thisisbecauseanyagentcansimplypassonthedecisionof
the following is true:
thepreviousagentifnootherdecisionruleleadstoadecrease
1) ǫ = 0 and the log-likelihood ratio of P∗ versus P∗ is
0 1 0 in minimax error probability.
unbounded from above, For ease of notation, we let Q = P (φ(k),Q(k)) and
2) ǫ1 = 0 and the log-likelihood ratio of P1∗ versus P0∗ is Q = P (φ(k),Q(k)) be theFL,kFD faFls,ek alarm a0nd miss
unbounded from below. M,k M,k 0
detection probabilities of agent k respectively, where φ(k) is
Proof: See Appendix A-B. the sequence of optimal social learning decision rules that
minimizesPSL(φ |φ(k−1)) in(3). From[6], itcanbeshown
Theorem 4 (Social learning). Suppose that Assumption 1 k k
that the decision rule used by agent k is of the form
holds, and that the decision rule for each agent is chosen
sequentially so as to minimize PSL(φ | φ(k−1)) in (3). Then 0 if l∗(Y )< π0QF,k−1
PankSdL(tφhke|loφg(-kl−ik1e)l)ih→ood0raastioko→f P∞1k∗ viferaksnudsoPn0∗lyisifuǫn0b=ounǫ1de=d.0 uk =u1 iiff l∗(πY0kkQ)F≥,k−ππ101π((111−−Q≤QQMFMl,k∗,−,k(k−1Y−11)))< π0(1−QF,k−1).
Proof: It was demonstrated in Proposition 3 of [6] that k−1 π1(1−QM,k−1) k π1QM,k−1
using social learning decision rules, when ǫ0 = ǫ1 = 0, the  (9)
error probability in a tandem network where all agents have Theorem 5. Suppose that Assumption 2 holds, and that each
thesameobservationdistributionswillconvergetozeroifand agent k in a tandem network chooses decision rule φk to
only if the log-likelihood ratios between the two probability minimizePkSL(φk |φ(k−1))in(3).ThenPkSL(φk |φ(k−1))→0
distributions is unbounded from both above and below. The as k → ∞ if and only if there exist infinite subsequences
ǫ →0 and ǫ →0 as n→∞.
1In the case where the agents’ contamination values for their uncertainty 0,kn 1,jn
classes arezero,[9]calls thisasymptotic learning. Proof: See Appendix A-C.
6
We observe that agents in a tandem network in the social deterministic decision rules may yield a lower asymptotic
learningframeworkhavetotalerrorprobabilitiesatleastthatof maximumerrorprobabilitythaneitherofthetwodeterministic
agentsadoptingdecentralizeddetectionrules.Hence,Theorem rules.Thus,weintroducetherandomizedlikelihoodratiotest:
5 also provides a sufficient condition for the minimax error
0 if l∗(Y )<t
probability (1) under a decentralized detection framework to k 1
converge to zero. A if l∗(Yk)=t1
C. Unknown agent positions uk =uBk−if1li∗f(Yt1k)<=l∗t(0Yk)<t0 (11)
knIonwainsgochioawl nemtwanoyrkh,oupsesrsinhfaovrme atotiomnakheasthbeeirendepcirsoipoangsanteodt where A is equal to10ifwli∗th(Ypkr)ob>abt0il,ity p and equal to uk−1
from a source node. We model this in a tandem network by
with probability 1−p, B is equal to u with probability
k−1
assuming that each agent has no knowledge of its position in
q and equal to 1 with probability q, and t , t , p and q are
1 0
the network. We make the following assumption, in addition
constants to be determined.
to Assumption 1, in this subsection.
We will now show how to obtain the minimax asymptotic
Assumption 3. Every agent k > 1 uses the same decision error probability. To do so, we start with the following two
rule. lemmas.
Exceptfor agent1 (which knowsits positionin the tandem Lemma 4. Suppose that Assumptions 1 and 3 hold. Then,
because it does not receive any preceding messages), every lim P ((φ ,φ),Qk,Qk)
e,k 1 0 1
otheragentinthetandemdoesnotknowitsownposition,and k→∞
π Q (φ(Y ,0)=1)
has access to exactly the same information when it comes to 0 0 1
=
choosingadecisionrule,whichisbasedsolelyonthenominal Q0(φ(Y1,0)=1)+Q0(φ(Y1,1)=0)
distributions, P0∗ and P1∗, as well as the contaminationvalues + π1Q1(φ(Y1,1)=0) .
ǫ and ǫ . Therefore, it is natural to make Assumption 3. Q (φ(Y ,1)=0)+Q (φ(Y ,0)=1)
0 1 1 1 1 1
BecauseofAssumption3,anysequenceofk agentdecision
Proof: We have the two following recurrence relations:
ruleshastheformφ(k) =(φ ,φk−1),whereφ isthedecision
1 1
rule used by the first agent, and φ is the decision rule used PF,k((φ1,φ),Qk0)
by every other agent with φk−1 = (φ,...,φ) consisting of =Qk(U =1)
0 k
k−1copiesof φ. Forsimplicity,andby abusingnotation,we =Qk−1(U =0)·Q (U =1|U =0)
0 k−1 0 k k−1
replace φ(k) in our notations by (φ1,φ) in the sequel. +Qk−1(U =1)·Q (U =1|U =0)
In the following, we consider two different scenarios. 0 k−1 0 k k−1
=(1−Qk−1(U =1))·Q (φ(Y ,0)=1)
1) Minimizing asymptotic error: We consider the case 0 k−1 0 k
+Qk−1(U =1)·Q (φ(Y ,1)=1)
where agents are collaborating to minimize the asymptotic 0 k−1 0 k
error. This mightoccur when there is a chain of agents trying
=P ((φ ,φ),Qk−1)
torelaysomeinformationto afusioncenter,buteachagentis F,k−1 1 0
·[Q (φ(Y ,1)=1)−Q (φ(Y ,0)=1)] (12)
unsure of how many other agentsthere are between itself and 0 1 0 1
the fusion center. For a given decision rule φ, the asymptotic +Q0(φ(Y1,0)=1),
maximum error probability is given by
and
P∞DD(φ)=kl→im∞PkDD(φ1,φ), (10) PM,k((φ1,φ),Qk0)
=Qk(U =0)
where we have implicitly assumed that PDD does not depend 1 k
onφ1.Thisassumptionisvalid,asshown∞inthenexttheorem, =Q1k−1(Uk−1 =0)·Q1(Uk =0|Uk−1 =0)
whichalsoprovidesacharacterizationfortheoptimalφwhich +Q1k−1(Uk−1 =1)·Q1(Uk =0|Uk−1 =1)
obtains the asymptotic minimax error probability, defined as =Qk−1(U =0)·Q (φ(Y ,0)=0)
1 k−1 1 k
infP∞DD(φ). +(1−Q1k−1(Uk−1 =0))·Q0(φ(Yk,1)=0)
φ
In the previous sections, we had no need to consider ran- =PM,k−1((φ1,φ),Q1k−1)
domized decision rules. This is because under the decentral- ·[Q (φ(Y ,1)=1)−Q (φ(Y ,0)=1)] (13)
1 1 1 1
ized detection framework, the final error probability under a +Q (φ(Y ,0)=0),
1 1
randomized sequence of decision rules is no less than the
minimum final error probability of each of the respective The first recurrence relation converges linearly to
deterministic sequences of decision rules. Similarly, under
Q (φ(Y ,0)=1)
the social learning framework, the error probability of an 0 1 , (14)
Q (φ(Y ,0)=1)+Q (φ(Y ,1)=0)
0 1 0 1
agent using a randomized decision rule is no less than the
and the second recurrence relation convergeslinearly to
minimum error probability under each of the deterministic
rules.However,thispropertydoesnotholdforthe asymptotic π Q (φ(Y ,1)=0)
1 1 1 . (15)
maximum error probability. A randomized version of two Q (φ(Y ,1)=0)+Q (φ(Y ,0)=1)
1 1 1 1
7
Hence, the proof is complete. Theorem 6. Suppose that Assumptions 1 and 3 hold, and let
Next, we will show that if the observation distributions are φ be the randomized likelihood ratio test such that
∗
alldrawnfromtheLFDs,thenthedecisionrulethatminimizes
φ =argmin lim P ((φ ,φ),Qk,Qk).
that asymptotic error probability is a randomized likelihood ∗ e,k 1 0 1
φ k→∞
ratio test between Q and Q .
0 1 Then,
Lemma 5. Suppose that Assumptions 1 and 3 hold. For any
infPDD(φ)= lim P ((φ ,φ ),Qk,Qk).
φ , let ∞ e,k 1 ∗ 0 1
1 φ k→∞
φ∗ =argmin lim Pe,k((φ1,φ),Qk0,Qk1). Proof: From Theorem 1, we have
φ k→∞
Then there is no loss in optimality if φ∗ is restricted to be a kl→im∞Pe,k((φ1,φ∗),Qk0,Qk1)
randomized likelihood ratio test between Q and Q .
0 1 = lim sup P ((φ ,φ ),P(k),P(k))
e,k 1 ∗ 0 1
Proof: Assume that φ∗ is not a randomized likelihood k→∞(P0(k),P1(k))∈P0k×P1k
ratio test between Q0 and Q1. For any agent k ≥2, consider =PDD(φ ).
a randomizedlikelihood ratio test φ′ in the form presented in ∞ ∗
(11), where t ,t ,p,q are chosen such that By the theorem assumption, for any decision rule φ we have
0 1
Q0(φ′(Yk,0)=1)=Q0(l∗(Yk)≥t0) lim Pe,k((φ1,φ∗),Qk0,Qk1)≤ lim Pe,k((φ1,φ),Qk0,Qk1).
k→∞ k→∞
=Q (φ (Y ,0)=1)
0 ∗ k
Hence, for any decision rule φ, we have
Q (φ′(Y ,1)=0)=Q (l∗(Y )<t )
1 k 1 k 1 PDD(φ)= lim sup P ((φ ,φ),P(k),P(k))
=Q1(φ∗(Yk,1)=0). ∞ k→∞P(k)∈Pk,P(k)∈Pk e,k 1 0 1
0 0 1 1
From the Neyman-Pearson lemma, we then have ≥ lim P ((φ ,φ),Qk,Qk)
e,k 1 0 1
k→∞
Q1(φ′(Yk,0)=0)=Q1(l∗(Yk)<t0) ≥ lim P ((φ ,φ ),Qk,Qk)
e,k 1 ∗ 0 1
≤Q (φ (Y ,0)=0) k→∞
1 ∗ k
=PDD(φ ).
∞ ∗
Q (φ′(Y ,1)=1)=Q (l∗(Y )<t )
0 k 0 k 1 The proof is now complete.
≤Q (φ (Y ,1)=1).
0 ∗ k Theorem 6 states that each agent should find the decision
rule to optimize the asymptotic minimax error as if its ob-
Hence, from (12) and (13), it is clear that for any φ and any
1
servations were distributed according the the LFDs of the
k ≥1, we have
uncertainty class. This is consistent with our results when
P ((φ ,φ ),Qk,Qk)≥P ((φ ,φ′),Qk,Qk),
F,k 1 ∗ 0 1 F,k 1 0 1 agents do know their positions (Theorem 1). However, the
exact threshold values for φ are difficult to compute in
and ∗
general, but can be found using numerical methods. Together
P ((φ ,φ ),Qk,Qk)≥P ((φ ,φ′),Qk,Qk),
M,k 1 ∗ 0 1 M,k 1 0 1 with Lemma 4, since the uncertainty classes P and P
0,k 1,k
and hence are disjoint for all k, Theorem 6 shows that asymptotically
learningthe true hypothesisis impossible when agentsdo not
P ((φ ,φ ),Qk,Qk)≥P ((φ ,φ′),Qk,Qk).
e,k 1 ∗ 0 1 e,k 1 0 1 know their own positions and also providesan expression for
By choosing suitable values of t ,t ,p,q, we can set the asymptotic minimax error.
0 1
Q (φ(Y ,0)=1)andQ (φ(Y ,1)=1)toanyvaluebetween 2) Minimizing errorofcurrentagent: We nowassumethat
0 k 0 k
0 and 1. Similarly, we can also set Q (φ(Y ,1)=0) and every agent is acting to minimize its local minimax error
1 k
Q (φ(Y ,0)=0) to any value between 0 and 1. Hence, probability, and that each agent past the first does not know
1 k
which position it is in. This is true in general for most
min lim P ((φ ,φ),Qk,Qk)
e,k 1 0 1 social networks, where it is difficult to find the root of any
φ k→∞
information spread. Hence, users typically would not know
is attainable, and the decision rule used to attain it can be
how many hops information has been propagated from its
assumed to be in the form of a randomized likelihood ratio
source.
test.
We find the decision rule to minimize the maximum error
The proof of Lemma 5 shows that the asymptotic minimax
probability by allowing each agent to consider the maximum
error probability is attainable even when the likelihood ratio
errorforeverypossiblepositionitmightbein,thenfindingthe
of Y is not continuous. To avoid cumbersome notation, for
k decision rule that minimizes this value. Like in the previous
the rest of the paper, we will assume that the likelihood ratio
subsection, we also assume that Assumptions 1 and 3 are in
of Y is continuous. It is easy to extend our results if this is
k effect. More specifically, for each agent k ≥ 2, we wish to
not the case.
find φ that minimizes
We can now prove the following theorem, which provides
an expression for the minimax asymptotic error. PSL (φ)=supPSL(φ|(φ ,φ)), (16)
max k 1
k≥2
8
where φ is the optimal decision rule that minimizes PSL(·). We nowpresenta numericalexampleusingthe exponential
1 1
We will show that for the optimal decision rule φ, the distribution. First, we fix P∗ as an exponential distribution
0
maximumerrorprobabilityoccurseitherinthesecondposition with mean 1 and let P∗ be an exponential distribution with
1
or at the asymptotic limit, as defined in (10). mean 2. We set ǫ = ǫ = 0.01, and π = π = 0.5. Then,
0 1 0 1
the optimal decision rule φ for agent 1 has the form:
Theorem 7. Suppose that Assumptions 1 and 3 hold. Then 1
φ∗ =argminφPmSLax(φ) is a randomized likelihood ratio test 0 if l∗(Y1)<1
between the LFDs Q0 and Q1, and u1 =(1 if l∗(Y1)≥1.
PSL (φ )=max PSL(φ |φ ),PDD(φ ) ,
max ∗ 2 ∗ 1 ∞ ∗ Note that for this case, if t 6= c′ and t 6= c′′, then it does
1 0
where P∞DD(φ∗) is as defin(cid:8)ed in (10). (cid:9) not matter what p and q are. This is because Q0(l∗(Yk)=x)
and Q (l∗(Y ) = x) are both zero unless x = c′ or x = c′′
Proof: See Appendix A-D. 1 k
when the nominal distributions are both continuous. We plot
Findingan analyticalformfor φ in Theorem7 is difficult.
∗
PSL(φ | (φ ,φ)) against k when φ = φ and φ , which are
However, as φ∗ is known to be a randomized likelihood ratio k 1 A B
randomized likelihood ratio tests of the form (11). Both rules
testofQ andQ ,thiscanbedonenumericallybyminimizing
max{PS0L(φ | φ1 ),PDD(φ )} with respect to the thresholds have t1 = c′ and p = 1, but φA has t0 = 5 and φB has
2 ∗ 1 ∞ ∗
t =1.1.
for φ . 0
∗
V. NUMERICAL RESULTS 0.4
Inthissection,weprovidenumericalresultstoillustratepart φA
of our theoretical contributionsin Section IV-C, which shows
φ
B
that even if an agent k’s position is unknown, where k ≥ 2, 0.38
itsoptimaldecisionruleφ canbechosentobearandomized
∗
likelihood ratio test between the LFDs Q and Q . This is
0 1
truewhethertheagentiscollaboratingwithotherstominimize SLk0.36
P
the asymptotic error (decentralized detection) or is trying to
minimize its own error probability (social learning). We have
shownthisinTheorems6and7respectively.Theformofthis 0.34
randomized likelihood ratio test φ is given in (11). Now, we
∗
can rewrite this in terms of an optimzation problem.
First, given the nominal distributions P0∗ and P1∗, as well 0.32
as the contamination values ǫ and ǫ , we can use a binary 5 10 15 20 25
0 1
searchtocomputec′ andc′′. FromthedefinitionoftheLFDs, Position k
weknowthattherangeofthresholdswehavetooptimizeover
1−ǫ Fig.2. Comparisonofdecision rules.
is boundedbetween bc′ and bc′′, whereb= 1. Then,we
1−ǫ
0
can derive the LFDs Q0 and Q1 and obtain Figure 2 shows that the total error probabilityis decreasing
Q (φ (Y,1)=1)=Q (l∗(Y)>t ) over k for φA. Hence, the maximum error probability occurs
0 ∗ 0 1
when k =2. For φ , the maximum occurs at the asymptotic
+(1−p)Q (l∗(Y)=t ), B
0 1 limit k →∞. This is in line with our conclusion in Theorem
and 7.
Q0(φ∗(Y,0)=1)=Q0(l∗(Y)>t0) Next, we let P0∗ be an exponential distribution with mean
+(1−q)Q (l∗(Y)=t ). 1 and P1∗ be an exponential distribution with variable mean.
0 0
We let ǫ =ǫ =0.01. We denote
0 1
Similarly, we have
φSL =argminPSL (φ),
Q (φ (Y,1)=0)=Q (l∗(Y)<t )+pQ (l∗(Y)=t ) ∗ φ max
1 ∗ 1 1 1 1
and
and
φDD =argminPDD(φ),
Q1(φ∗(Y,0)=0)=Q1(l∗(Y)<t0)+qQ1(l∗(Y)=t0). ∗ φ ∞
In the case where agents collaborate to minimize the where PSL (φ) and PDD(φ) are defined as in (10) and (16)
max ∞
asymptotic maximumerror PDD, we minimize the expression respectively.
∞
in Lemma 4 to obtain the optimal decision rule φ . From Figure 3 shows that as the mean of P∗ increases, both
∗ 1
Theorem6, since φ is a randomizedlikelihood ratio test, we PSL (φSL) and PDD(φDD) decrease. This is intuitive as the
∗ max ∗ ∞ ∗
can perform the optimization over t , t , p and q in (11). Kullback-Leibler divergence of P∗ from P∗ increases as the
1 0 1 0
Similarly, in the case where agents minimize their local mean of P∗ increases. As the nominal distributions become
1
maximumerrorprobabilityPSL ,weminimizetheexpression easier to differentiate, the asymptotic error probability de-
max
in Theorem 7 over t , t , p and q. creases.
1 0
9
the uncertainty classes have sizes bounded away from one,

φSL (φSL) evenwhentheloglikelihoodratioofthenominaldistributions
0.4 max ∗ of the uncertainty classes is unbounded. To achieve asymp-
φDD(φDD)
∞ ∗ totic learning of the true hypothesis in social learning, we
0.35
require the additional condition that the uncertainty classes’
0.3 contamination values decay over the tandem network. In the
case where agents do not know their positions in the tandem,
0.25
asymptoticlearningofthetruehypothesisbecomesimpossible
0.2 even if contamination values are zero. We characterized the
minimaxerrorperformanceinthiscase,whichprovidedaway
0.15
to determine the optimal decision rules for the agents.
In this work,we haverestricted ourattention to the tandem
0.1
network. It would be of interest to extend some of our results
0.05 to tree networks, and even to loopy general graphs. Another
1 2 3 4 5 6 7 8
Mean of P∗ future research direction would be to consider the robust
1
detectionproblemwith morethan two hypotheses.A possible
Fig.3. Errorprobability versusmeanofP1∗. approach to this problem could be to focus on the LFDs on
eachpossiblepairofuncertaintyclass.Lastly,fortheproblem
where each agent does not know its position in the network,
0.5 wecouldinsteadconsidereachagenthavingpartialknowledge
ofhispositioninthenetwork,andfindconditionsunderwhich
0.45 learning the true hypothesis asymptotically is possible.
0.4 APPENDIX A
PROOFS OF MAIN RESULTS
A. Proof of Lemma 2
0.35
We will prove this lemma using mathematicalinduction on
N. The likelihood ratio test for agent k ≥1 is of the form
0.3
PSLPSP
1 if l∗(Y ,U )>t ,
k k−1 k
PDDPDD U =
i
0.25 (0 otherwise,

0.02 0.04 0.06 0.08 0.1 0.12 0.14
Contamination (ǫ) where tk is some chosen threshold. From Lemma 1, we have
Q(1)(U =1)=Q(1)(l∗(Y )≥t )
Fig.4. Errorprobability versuscontamination value. 1 1 1 1 1
≥Q(1)(l∗(Y )≥t )
0 1 1
=Q(1)(U =1),
Figure 4 shows that both PSL (φSL) and PDD(φDD) in- 0 1
max ∗ ∞ ∗
creases as ǫ = ǫ increases. For this graph, P∗ and P∗ so that the lemma holds for N =1.
0 1 0 1
are kept constant as exponential distributions with means 1 Assumethatthe lemmaistrueforN =k−1.Eachagent’s
and 2 respectively. As expected, as uncertainty increases, observation is independent of the previous agent’s decision.
so does the asymptotic error. When the uncertainty is large Hence, we have
enough, both PmSLax(φS∗L) and P∞DD(φD∗D) converge towards l∗(Yk,Uk−1)=l∗(Yk)l∗(Uk−1),
0.5. Furthermore, as uncertainty increases, the gap between
PSL (φSL) and PDD(φDD) decreases. In a network with a lot and the likelihood ratio test can be rewritten in the form
max ∗ ∞ ∗
of uncertainty, there is not much incentive in trying to get 1 if l∗(Y )≥t0,
agents to collaborate, as agents selfishly trying to minimize k k
U = U if t1 ≤l∗(Y )<t0
their own errorprobabilityleadsto similar errorperformance. k  k−1 k k k
0 if t1k >l∗(Yk),
VI. CONCLUSION where tik =tk/l∗(Uk−1 =i). We obtain
We have shown that in a tandem network where agents’
Q(k)(U =1)
observationdistributionsarenotknownexactly,andbelongto 1 k
uncertainty classes, the minimax error probability is obtained =Q1,k(Uk =1|Uk−1 =1)Q1(k−1)(Uk−1 =1)
by assuming that each observation is distributed according to +Q (U =1|U =0)Q(k−1)(U =0)
1,k k k−1 1 k−1
the LFDs of the uncertainty classes. In the case where agents
=Q (l∗(Y )≥t1)Q(k−1)(U =1)
know their positions in the tandem network, asymptotically 1,k k k 1 k−1
learning the true hypothesis is in general impossible when +Q (l∗(Y )≥t0)(1−Q(k−1)(U =1)).
1,k k k 1 k−1
10
From the induction hypothesis, we then have For 1<k <N∗,
Q1,k(l∗(Yk)≥t1k)Q1(k−1)(Uk−1 =1) U = 0 for l∗(Yk)<t and Uk−1 =0
+Q1,k(l∗(Yk)≥t0k)(1−Q1(k−1)(Uk−1 =1)) k (1 otherwise.
≥Q1,k(l∗(Yk)≥t1k)Q0(k−1)(Uk−1 =1) For k ≥N∗,
+Q (l∗(Y )≥t0)(1−Q(k−1)(U =1)) Uk =Uk−1.
1,k k k 0 k−1
≥Q0,k(l∗(Yk)≥t1k)Q0(k−1)(Uk−1 =1) Note that Φδ is a sequence of likelihood ratio tests between
Q and Q . It was shown in [6] that by choosing a suitable t
+Q0,k(l∗(Yk)≥t0k)(1−Q0(k−1)(Uk−1 =1)) an1d N∗, w0e can get an arbitrarily small error rate if P(N) =
=Q0(k)(Uk =1), QN1 . To see this, consider a point on the ROC curve1of an
agent using the likelihood ratio test between Q and Q with
wherethesecondinequalityfollowsfromLemma1.Theproof 0 1
its tangent to the ROC curve having slope t. This point is
of the lemma is now complete.
(Q (l∗(Y) ≥ t),Q (l∗(Y) ≥ t)). As P∗ = Q , the initial
0 1 1 1
slope of the ROC curve is ∞, and so such a point always
B. Proof of Theorem 3 exists for any t. From the concavity of the ROC curve, we
We consider three separate cases, dependingon whether ǫ have
0 Q (l∗(Y)≥t)
or ǫ1 is nonzero or not. Q (l∗(Y)≥t)< 1 .
0
Case 1: ǫ =ǫ =0. t
0 1
This reduces to P and P (and hence P(N) and P(N)) The asymptotic miss detection probability using the decision
0 1 0 1
being known exactly, and Proposition 1 in [5] has shown that rules outlined above is thus
tthheelmogax-liimkeulmihoeordroorfrPate∗ aisndboPu∗ndisedboaubnodveedzferroomifeaitnhderoanblyovief Nli→m∞PM,N(Φδ,QN1 )=Q1(UN∗ =0)
0 1
or below. =(1−Q (l∗(Y)≥t))N∗.
1
Case 2: Either ǫ =0 or ǫ =0, but not both.
0 1
Similarly, the asymptotic false alarm probability is
Inthiscase,oneoftheP reducestothenominalprobability
i
dDiestfirnibeu(tiQon,QPi∗.)=Wi(tPho∗u,tQlo)ssasotfhegeLnFeDraslitoyf, Plet athnidsPbe. PFo0r. Nli→m∞PF,N(Φδ,QN0 )=1−(1−Q0(l∗(Y)≥t))N∗
0 1 0 1 0 1
i=0,1, let the probabilitydensity of Pi∗ be p∗i. Since ǫ0 =0 <1−(1− Q1(l∗(Y)≥t))N∗.
and ǫ 6= 0, we have c′′ = ∞ and c′ > 0. Note that for t
1
p∗1(x)/p∗0(x) > c′, we have l∗(x) = bp∗1(x)/p∗0(x), where Choose an arbitrary δ > 0. To get the asymptotic miss
b = 1−ǫ1. Hence, if p∗1(x)/p∗0(x) is bounded from above, detection probability smaller than δ, we have
then for p∗(x)/p∗(x)>c′ we have
1 0 (1−Q (l∗(Y)≥t))N∗ <δ
0<bc′ 1
≤l∗(x) and so
log(δ)
p∗(x) N∗ > . (17)
=b 1 log(1−Q (l∗(Y)≥t))
p∗(x) 1
0
<∞. Similarly,togettheasymptoticfalsealarmprobabilitysmaller
than δ, we have
From Theorem 1, we have
log(1−δ)
φi(nNf)(P(N),P(N)s)u∈pP(N)×P(N)Pe,N(φ(N),P0(N),P1(N)) log(1− Q1(l∗(tY)≥t)) >N∗. (18)
0 1 0 1
≥ inf P (φ(N),QN,QN), We now make use of the following lemma.
e,N 0 1
φ(N)
Lemma A.1. For any probability distribution Q ∈R,
which is bounded above zero as N → ∞ since log(l∗(x)) 1
is bounded from both above and below (similar to case 1). log(1−Q1(l∗(Y)≥t))
lim =∞.
Hence we will assume that the log-likelihood of P∗ and P∗ t→∞ log(1− Q1(l∗(Y)≥t))
0 1 t
is unboundedfromabove.Then,the log-likelihoodof Q and
0
Proof: Let g(x) = log(1−x), a concave function. For
Q is bounded from below but not from above as well.
1
0<x<1, we have g(x)<0.
Using the scheme proposed in [6], we can show that we
For any fixed t>0, by Jensen’s Inequality,
can make the maximum error arbitrarily small as the number
of agents tends to infinity. We denote this scheme as Φ . The 1 t−1 x
δ g(x)+ g(0)≤g( ).
decision rules of Φδ are as follows: t t t
For a given N∗ and a threshold t,
As g(0)=0, we have
0 for l∗(Y1)<t g(x)
U1 =(1 for l∗(Y1)≥t. g(x) ≥t.
t