A Probabilistic Approach to Inference with LimitedInformation in Sensor Networks

Rahul BiswasStanford University

Stanford, CA 94309 USA

rahul@cs.stanford.edu

Sebastian ThrunStanford University

Stanford, CA 94309 USA

thrun@cs.stanford.edu

Leonidas J. GuibasStanford University

Stanford, CA 94309 USA

guibas@cs.stanford.edu

ABSTRACTWe present a methodology for a sensor network to answerqueries with limited and stochastic information using prob-abilistic techniques. This capability is useful in that it al-lows sensor networks to answer queries effectively even whenpresent information is partially corrupt and when more in-formation is unavailable or too costly to obtain. We use aBayesian network to model the sensor network and MarkovChain Monte Carlo sampling to perform approximate infer-ence. We demonstrate our technique on the specific problemof determining whether a friendly agent is surrounded by en-emy agents and present simulation results for it.

1. INTRODUCTIONRecent advances in sensor and communication technolo-

gies have enabled the construction of miniature nodes thatcombine one or more sensors, a processor and memory, plusone or more wireless radio links [9, 12]. Large numbers ofthese sensors can then be jointly deployed to form whathas come to be known as a sensor network. Such net-works enable decentralized monitoring applications and canbe used to sense large-area phenomena that may be impos-sible to monitor in any other way. Sensor networks can beused in a variety of civilian and military contexts, includingfactory automation, inventory management, environmentaland habitat monitoring, health-care delivery, security andsurveillance, and battlefield awareness.

In some settings, sensors enjoy the use of external powersources and need not factor the energy costs of computation,sensing, and communication into their decisions. Informa-tion theory tells us that more information is always bet-ter [4] and any algorithm analyzing sensing results shouldbe able to perform at least the same and hopefully betterwith more information. In most actual deployment scenar-ios however, sensors must operate with limited power sourcesand carefully ration their energy utilization. In this world,computation, sensing, and communication comes at a costand algorithms designed for sensor networks must strike a

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good balance between increasing network longevity by de-creasing costs on the one hand and providing useful answersto queries on the other.

One aspect in which algorithms used for sensor networksmust differ from traditional algorithms is that they mustbe able to reason with incomplete and noisy information.Some areas of interest may lie outside sensor range. Anyinformation, if available, is not free but has a sensing costand comes at the expense of decreased network longevity.Also, real sensors do not always operate as they should andany algorithm that deals with sensor information must beable to handle partial corruption of the input data.

We present an algorithm that leverages both limited andstochastic sensor data to probabilistically answer queriesnormally requiring complete world state to answer correctly.By answering probabilistically, i.e. by assigning a probabil-ity to each possible answer, we provide the inquirer withmore useful information than only one answer whose cer-tainty is not known. This technique is applicable to a broadrange of queries and we demonstrate its effectiveness onan open problem in sensor networks, that of determiningwhether a friendly agent with known location is surroundedby enemy agents with unknown but stochastically sensablelocations [7].

We structure the probabilistic dependencies between sen-sor measurements, enemy agent locations, and the query ofinterest as a Bayesian network. We then enter the sensor ob-servations into the Bayesian network and use Markov ChainMonte Carlo (MCMC) methods to perform approximate in-ference on the Bayesian network and extract a probabilisticanswer to our original query.

Our experimental results demonstrate the effectiveness ofour inference algorithm on the problem of determining sur-roundedness on a wide variety of sensor measurements. Theinference algorithm consistently provides accurate estimatesof the posterior probability of being surrounded. It demon-strates the ability to incorporate both positive and negativeinformation, to use additional information to disambiguatealmost colinear points, to cope effectively with sensor failure,and provide useful prior probabilities of being surroundedwithout any sensor measurements at all.

2. PROBLEM DEFINITION

2.1 OverviewWe seek to determine whether an agent F is surrounded

by N enemy agents E1 . . . EN . Let the location of agentF be given as LF and the locations of agents E1 . . . EN be

given by LE1 . . . LEN respectively. Let the assertion D referto the condition that LF lies in the convex hull generatedby points LE1 . . . LEN , which we will henceforth refer to inaggregate as simply LE.

We assume that LF and N are known. We have no priorknowledge over the LE but form posteriors over them fromthe evidence gathered by the sensor measurements.

2.2 Sensor ModelSensors can detect the presence of enemy agents in a cir-

cular region surrounding the sensor. We have a total of M

sensors S1 . . . SM at our disposal. Let the sensor locationsbe given by LS1 . . . LSM . For a given sensor Si, let d̂i equal1 if there exists one or more enemy agents inside the circlewith its center collocated with the sensor and radius R, and0 otherwise. In other words, d̂i is a boolean variable that isthe response sensor Si should obtain if it is operating cor-rectly. Let di be the value sensor Si actually obtains andshares with other sensors. The variable di depends stochas-tically on d̂i as follows:

di =

�� � d̂i : with probability ζ

0 : with probability 1−ζ

2

1 : with probability 1−ζ

2.

(1)

Setting ζ to 1 is equivalent to having perfect sensors andsetting ζ to 0 is equivalent to having no sensors. Typically,ζ will be quite high since sensors usually tend to work cor-rectly.

2.3 Objective FunctionOur objective is to determine, given LF and a set of

P stochastic sensor measurements R1 . . . RP , the posteriorprobability of D. The set of sensor measurements may beempty.

We do not seek to address the issues of network controland communication in this paper. While these are impor-tant topics and are revisited in Section 6.2, our inference al-gorithm works independently of the control algorithm thatdecides who will sense and manages inter-sensor communi-cation.

3. PRIOR WORKThe traditional approach to solving this sort of problem

is to determine the values of each of the variables of inter-est and then use standard techniques to answer the querygiven the full world state. The potency of our approach liesin its ability to take partial world state and estimate thecomplete world state in expectation. By doing so, we canuse the same off-the-shelf techniques the other algorithmscan. In fact, in the example we have chosen to showcase,that of determining whether a friendly agent is surrounded,we use an unmodified traditional computational geometryalgorithm and simply plug it into our inference. This allowsus to probabilistically answer any other query just as easilyby just plugging in existing algorithms requiring full worldstate into the core of our MCMC sampler.

For the specific problem at hand, we can determine whetherthe friendly agent lies within the convex hull of enemy agentsby starting with the exact values of LE, computing theirconvex hull, and then determining whether LF lies withinthis convex hull. Many well-known computational geometry

Figure 1: Bayesian Network with 3 sensor measure-

ments and 4 enemy agents

algorithms exist that compute the convex hull of N pointsin the plane in Θ(N · logN) time [16].

In fact, convex hull inclusion can be tested directly inO(N) time. Let CH represent the convex hull of LE andlet CH1 . . . CHS represent the enemy agents, in order, lyingon this hull. S is less than or equal to N . LF lies withinCH if and only if either of the following conditions holds:

∀s ∈ S, CCW (CHs, CHs+1, LF ) (2)

or

∀s ∈ S, CW (CHs, CHs+1, LF ). (3)

Here, CHS+1 wraps around to CH1, CCW (a, b, c) refersto the assertion that point c is strictly counterclockwise toray ab, and CW (a, b, c) refers to the assertion that point c

is strictly clockwise to ray ab.Closest in spirit to our effort are techniques that try to

approximate a convex body using a number of probes [1],or those that try to maintain approximate convex hulls ina data stream of points, using bounded storage [8]. Theseworks both address similar problems but their methods can-not be applied to our problem at hand.

4. OUR APPROACHWe show in this section how the query, in this case, whether

the friendly agent is surrounded by enemy agents can be rep-resented by a variable in a Bayesian network. The proba-bilistic relationships between sensors, enemy agent locations,and the query are faithfully represented by the Bayesian net-work. Thus the problem of answering the query reduces tocomputing the posterior probability of the variable in thenetwork representing our query. We use MCMC samplingto perform this inference. We start with the formal defini-tion of our Bayesian network, identify why exact inference isnot possible, and then discuss our MCMC implementation.

4.1 Bayesian NetworkA Bayesian network [11] is characterized by its variables

and the prior and conditional probabilities relating the vari-ables to one another. We start with the variables and what

they represent and then move on to their conditional prob-abilities. Figure 1 shows a Bayesian Network for the case ofthree sensor measurements and four enemy agents. In prac-tice, the network structure will vary depending on the num-ber of sensor measurements and number of enemy agents.

4.1.1 Variable 1: Sensor MeasurementsLet there be P sensor measurement variables R1 . . . RP in

our Bayesian network. These variables are identical to thevariables mentioned in Section 2 and contain both the indexof the sensor it originated from and the stochastic value d

that the sensor perceived. These variables are observed inthe Bayesian network.

4.1.2 Variable 2: Enemy Agent LocationsLet there be N enemy agent location variables LE1 . . . LEN

in our Bayesian network. These variables are also identicalto the variables mentioned in Section 2. The variables arenot observed in the Bayesian network, i.e., their values arenot known to the inference algorithm. By doing inferencehowever, it is possible to form posterior probability distri-butions over them.

4.1.3 Prior Probability: Enemy Agent LocationsWe assume that the enemy agents are independently and

uniformly distributed across a bounded rectangular region.

4.1.4 Variable 3: Whether the Friendly Agent is Sur-rounded

Let there be a variable D in our Bayesian network. Likethe variables that came before it, this variable is identical tothe variable mentioned in Section 2. Indeed, the potency ofour Bayesian network lies in the simplicity of its conception.This variable is also not observed in the Bayesian networkbut it is possible to form a posterior probability distributionover it as well.

4.1.5 Conditional Probability 1: How Enemy AgentLocations Affect Sensor Measurements

Sensor measurements are stochastically determined by en-emy agent locations as presented in Section 2.

4.1.6 Conditional Probability 2: How Enemy AgentLocations Affect the Query

The value of D is deterministic given LE1 . . . LEN . Weuse the off-the-shelf convex hull generation and containmentalgorithm described in section 3 to establish this relationbetween D and LE1 . . . LEN .

4.2 Problems with Exact InferenceInference in Bayesian networks refers to the process of

discovering posterior probabilities of unobserved variablesgiven the values of observed variables. A family of tech-niques known as clique tree methods is used for this pur-pose.

Unfortunately, our Bayesian network has both continu-ous (LE) and discrete (R and D) variables. In fact, thisis the hardest class of Bayesian networks to perform infer-ence on [10]. Moreover, the nature of the problem preemptsthe use of multivariate Gaussians to approximate the LE

variables while maintaining the rich hypotheses inferencerequired to form accurate posteriors. Faced with such anetwork, clique tree methods would form an immense clique

containing each R variable along with each of the LE vari-ables. The inadequacy of multivariate Gaussians would forceus to perform discretization of the joint space. The expo-nential blowup that arises from this enormous state spacewould render the problem computationally intractable.

4.3 MCMC for Approximate InferenceFortunately, we can get an accurate estimate of the poste-

rior probability of interest, P (D|R1 . . . RP ). We do this withthe Metropolis algorithm, an instance of a MCMC methodthat performs approximate inference on Bayesian networks.It is equipped to handle the continuous nature of this dis-tribution as it samples evenly over the entire joint space.

We follow the treatment of Metropolis and use the termi-nology as presented in [5]. The function f(X) we want toestimate is the variable D and the space Xt we sample fromis the joint distribution over LE. Let us choose to draw T

samples using the Metropolis algorithm and let the hypothe-sis over LE while taking sample t be LEt. The starting stateoften systematically biases the initial samples so we discardthe first ι ·T samples and estimate our posterior probabilityof interest as the mean of the remaining samples:

P (D|R1 . . . RP ) =1

(1 − ι) · T

T�

t=ι·T

P (D|LEt) (4)

where ι is the fraction of samples discarded until the Markovchain converges.

There are four aspects to evaluating this expression. Thefirst three deal with the inner workings of Metropolis. First,where do we start the chain? Second, how does LEt tran-sition to LEt+1? Third, how do we decide whether to ac-cept the proposed transition? Fourth, how do we evaluateP (D|LEt)?

4.3.1 Starting the ChainWe place each of the enemy agents at a random location

uniformly distributed over the rectangular region where en-emy agents are known to be located.

4.3.2 Proposal DistributionOne of the enemy agents is chosen at random and is moved

to a random location uniformly chosen over the same rect-angular region as before. Thus, each component of LEt+1,that is, each enemy agent location, is conditionally indepen-dent, given LEt. This condition is required for Metropolisto work correctly.

4.3.3 Determining Proposal AcceptanceWe use the Metropolis proposal acceptance criterion:

α(LEt, LE

t+1) = min(1,P (LEt+1, R1 . . . RP )

P (LEt, R1 . . . RP )). (5)

The proposal is accepted with probability α(LEt, LEt+1)as given above. This is easily done by generating a ran-dom number U uniformly drawn from between 0 and 1 andaccepting the proposal if U ≤ α(LEt, LEt+1).

We can simplify the expression

P (LEt+1, R1 . . . RP )

P (LEt, R1 . . . RP )(6)

substantially. We discover from the Bayesian network thatthe expression P (LEt, R1 . . . RP ) decomposes as follows:

N�

i=1

P (LEti )

P�

j=1

P (Rj |LEt) (7)

where LEti refers to the hypothesized location of enemy

agent i when Metropolis takes sample t.Similarly, the expression P (LEt+1, R1 . . . RP ) decomposes

as:

N�

i=1

P (LEt+1

i )

P�

j=1

P (Rj |LEt+1). (8)

Thus, we can rewrite 6 as:

� N

i=1P (LEt+1

i )� P

j=1P (Rj |LEt+1)

� N

i=1P (LEt

i )� P

j=1P (Rj |LEt)

. (9)

If we recall that LEti is uniformly distributed and that

each of the LE terms cancel out in both the numerator anddenominator, this expression simplifies to:

� P

j=1P (Rj |LEt+1)

� P

j=1P (Rj |LEt)

. (10)

Our Bayesian network does not directly allow us to sim-plify this expression further but since sensors are not af-fected, even stochastically, by enemy agents outside theirsensing range, we can exploit context-specific independenceto reduce this expression further. Specifically, only one ofthe LEi variables has changed value from step t to step t+1.Only sensors for which this enemy agent is within range willbe affected by this transition. Let there be K such sensorsand let Kk refer to the index of the k-th such sensor.

The sensor measurements that are not affected by thechange in LEi cancel out in the numerator and denominator,leaving us with:

K�

k=1

P (RKk|LEt+1)

P (RKk|LEt)

. (11)

Computing this expression is extremely fast and is furthersped up in our implementation by discretizing the rectangu-lar region and associating with each grid cell a list of sensormeasurements that affect that cell.

The conditional probabilities of the sensor measurementsgiven the enemy agent locations can be computed directlyfrom the expressions of the conditional probability. Thisformulation of the problem, which relies only on the con-ditional probabilities as they are stated, i.e. the forwardmodel, and not on evidential reasoning make computationof likelihoods and thus inference easier to implement, faster,and more accurate [17].

4.3.4 Computing P (D|LEt)

This conditional probability can be computed directly fromthe definition as presented in 4.1.6. The locations of the en-emy agents are given by LEt. The expression will yield aprobability that is either 0 or 1 but not in between.

4.4 Alternate QueriesUp to this point, we have explored our algorithm for the

specific query of determining whether a friendly agent issurrounded by enemy agents. This problem dealt with de-termining properties of the convex hull of a set of stochasti-cally sensable points. While we present the scenario in 2D,it generalizes directly to higher dimensional spaces.

We briefly note two other scenarios where we can use ouralgorithm to discover useful geometric properties of a setof points. The first scenario explores discovering both theextent and area of a set of stochastically sensable points.The second deals with finding a point maximally distantin expectation from all other points. These scenarios werechosen to explore different areas of computational geometrybut our algorithm works on any query where small changesin the input induce small changes in the output.

4.4.1 Extent and Area of a Point SetThe problem of finding the farthest point along a cer-

tain direction in a point set is more generally known as theextremal polytope query and can be solved efficiently byKirkpatrick’s algorithm [16]. Let us define a new variable Bthat contains both the minimal and maximal point in eachdirection. We replace the last step of our algorithm, comput-ing P (D|LEt), with computing P (B|LEt). This conditionalprobability is deterministic, as its predecessor was. The ex-pected value of the extents are the mean of the componentsof the individual samples. The expected value of the areais the mean of the areas stemming from the individual sam-ples and will typically not equal the area stemming from themean extents. Using only discovered points in this scenariowill tend to underestimate the extent of the point set.

4.4.2 Maximally Distant PointThe problem of finding a point that is maximally distant

from its nearest neighbor stems from the Voronoi diagramof a point set and can be computed as described in [6]. Letus define a new variable E that uses a discretized grid torepresent the distance of each cell to its nearest neighbor inthe point set. By replacing the last step with computingP (E|LEt) (again deterministic), we calculate the expecteddistance from each cell and choose the one with the highestexpected distance. Using only discovered points in this sce-nario will tend to ignore the effects of undiscovered pointson the solution.

5. EXPERIMENTAL RESULTSWe show in this section some examples of the sort of re-

sults the inference algorithm produces. First, we have somespecific networks and show the posterior probabilities pro-duced conditioned on different sets of sensor measurements.Second, we show how the average posterior probabilitiesvary as different parameters of the sensor network varies.Third, we examine how many iterations MCMC requires toconverge.

5.1 Specific Inference ExamplesIn Figure 2, we see a sensor network. Here, LF corre-

sponds to the friendly agent and is designated by a plus sign,LEi corresponds to the ith enemy agent and is designatedby a circle, and LSj the jth sensor and is designated by anasterisk. The circle surrounding a sensor shows the extentof its sensing range. The labels and circles for only a subset

+LE 0

LE 1

LE 2

LE 3LE 4

LF

LS 2

LS 37

LS 41

LS 56

LS 68

LS 36

Figure 2: An Example Sensor Network

of sensors are drawn to reduce clutter in the diagram. Thefollowing table provides the posterior probabilities resultingfrom inference when different sets of sensors are chosen tosense. None of the sensors failed during the simulation runsbut the algorithm runs with ζ set to 0.01.

Posterior Probabilities for Figure 2

Sensors Sensing Posterior Probability

56, 68 44none 512, 41 8041 8141, 56, 68 8636, 37, 41, 56, 68 93

We analyze our results from a geometric perspective. It isimportant to note that our inference algorithm approachesthe problem in a fundamentally different way than the dis-cussion that follows.

Let us start with the prior probability that the agent issurrounded without any sensor measurements. In this in-stance, the entire field is 100 by 100 and the friendly agentis at 58, 40. Ignoring collinearity, which is of measure 0 any-way, the friendly agent is only surrounded if all of the enemyagents do not lie to one side of it. Let us analyze the casethat all the enemy agents are to the north of the friendlyagent. The probability of any one of them being there is60% and the probability of all five of them being there isapproximately 8%.

The alternative scenario is that each of the agents is to thesouth. The probability of any one of them being there is 40%and the probability of all five of them being there is approx-imately 1%. Thus the probability of not being surroundedbecause all the agents lie to one side of the east-west merid-ian is approximately 9%. We need to extend this case to

each line passing through LF but without double counting.For example, the probability of all of the enemies lying toone side of the north-south meridian is also approximately9% because of the approximate symmetry of the situation.We do not seek to compute the prior via this method butonly to show where the prior comes from and demonstratethe difficulty of computing it.

We next consider the situation where sensors 56 and 68have sensed and found enemy agents but the posterior hasdecreased to 44%. This may seem counterintuitive thatwe have useful information and our posterior has droppedbut with a fixed number of enemies, finding two of themin inessential locations decreases the probability of enemieselsewhere. Specifically, the probability of enemy agents tothe south and east of the friendly agent, where they werealready less likely to be, is further reduced. Contrast thiswith the situation where only sensor 41 has sensed. Findingan enemy where it was unlikely but more potentially usefulincreases the posterior probability substantially, in this caseto 81%.

Let us next consider the case where sensors 2 and 41 haveboth sensed. The posterior probability is lower than the sce-nario where only 41 has sensed. Intuitively, the sensor is al-most collinear to the previously detected sensor and friendlyagent and the discovery of this enemy is neither immenselyhelpful nor it useless. It does however reduce the number ofunknown enemies that might have been used to expand theconvex hull in a useful direction.

The next scenario we consider, where sensors 41, 56, and68 have sensed, is perhaps the most intuitive. We can seethat the discovered enemies form a triangle that just enclosesthe friendly agent. The sensors have less information thanwe do though – they only know that enemy agents lie withintheir sensing radius. It may be that both enemy agents lienorth of the sensors that have located them. If this were the

+LF

LE 0LE 1 LE 2

LE 3

LE 4

LS 8

LS 35

LS 36

LS 46

LS 50

LS 58

LS 75

LS 86

Figure 3: Another Example Sensor Network

case, the friendly agent would not longer be surrounded.The last scenario, where sensors 36, 37, 41, 56, and 68 have

sensed, shows the powerful effect of negative information.The addition of results from sensors 36 and 37 demonstratethat the enemy agent that sensor 41 found is not in thesensable region of either sensor 36 and 37 and therefore mustlie farther south, where it is more likely to participate insurrounding the friendly agent.

Another example sensor network and inference results fol-low. In this case, none of the sensors failed during the sim-ulation runs but the algorithm now runs with ζ set to 0.2.We will see how this difference affects inference.

Posterior Probabilities for Figure 3

Sensors Sensing Posterior Probabilitynone 6735, 50, 58 708, 36, 58 748, 36, 46 758, 35, 50, 58 788, 35, 36, 46, 50, 58, 75, 86 91

There is much to be learned from these results but we fo-cus on the three essential points that were not demonstratedby the previous example. First, the prior probability of be-ing surrounded is much higher. This is due solely to the factthat the friendly agent is closer to the center of the regionwhere enemy agents are distributed.

Second, even when the sensor evidence supports the as-sertion that the friendly agent is surrounded, as in the casewhere sensors 8, 35, 50, and 58 have sensed, the posteriorprobability is not as high as one may expect. The sensornetwork realizes due to the relatively high value of ζ that itis likely that one or more of these sensor measurements areinaccurate and considers these possibilities in its inference.

Third, when there is evidence from multiple sensors sup-porting the presence of an enemy agent, as in the case wheresensors 8, 35, 36, 46, 50, 58, 75, and 86 have all sensed andeach enemy agent is covered by at least two sensors, theposterior probability increases substantially.

5.2 Effects of Changing Network ParametersWe now move on to aggregate results of how different

parameters of the scenario affect the posterior probability ofbeing able to detect that the friendly agent is surrounded.It is possible to perform such analysis for any query and canprovide valuable insights into how the parameters of thesensor network help or hinder in answering specific types ofqueries.

Figure 4 shows the aggregate of two sets of ground truthscenarios. The Surrounded case shows examples where theagent is actually surrounded. The Not Surrounded caseshows examples where the agent is actually not surrounded.The distinction is determined by the simulator and not throughsensor measurement. The chart thus shows how the aver-age posterior of the inference algorithm across a variety ofscenarios evolves. The number of enemies is set to four andthe sensing radius is set to 12. As we see, more informa-tion brings the posterior probability closer and closer to theground truth.

Figure 5 shows the same scenario but with the numberof enemies varying now. The number of sensors is set totwenty-five and the sensing radius remains at twelve. Wesee that it is easier to locate enemies and believe that one issurrounded when there are more enemies present but thatthe effect is not strong.

Figure 6 shows the same scenario but with the range ofthe sensor varying now. The number of sensors is set totwenty-five and the number of enemies is set at four. Whileit is easier to locate enemies with increased sensing range,

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Sensors

Pro

babi

lity

Sur

roun

ded

Posterior Probability Evolution as Number of Sensors Varies

SurroundedNot Surrounded

Figure 4: Posterior Probability Evolution as Num-

ber of Sensors Varies

the uncertainity associated with such information makes thisinformation less useful. These contrary forces appear to bal-ance out and varying the sensor range appears to neitherhelp nor hurt inference.

5.3 MCMC ConvergenceAlthough MCMC converges in the limit to the exact pos-

terior, it is useful to know how many iterations are requiredin practice to obtain accurate results. It is unfortunatelyonly feasible to compute analytical bounds for MCMC al-gorithms for only the simplest distributions though and theconditions required by most bounding theorems are not sat-isfied by general distributions. However, it often suffices inpractice to answer this question heuristically through ex-perimentation [15]. Figure 7 shows the standard deviationof the MCMC posterior estimate errors before convergenceoccurs. Regardless of the number of queries, we find thatthe posterior estimate errors level off after approximately15,000 steps of MCMC, a computation that takes only afew hundred milliseconds to perform.

6. FUTURE WORKWhile we feel that we have made significant progress with

the methods presented in this paper, there are two majordirections we plan to proceed with this work. First, we wantto use this algorithm with a set of actual sensors. Second, wewant the sensor network to guide its own sensing activitiesand direct sensing in such a way to minimize both sensingand communication cost. We discuss these ideas in turn.

6.1 Moving from Simulation to Actual SensorsWe plan to use this algorithm to help mobile robots play-

ing laser tag with one another [14] determine where theirenemy agents may or may not be. For example, if a robotknew that a certain corridor was most likely free of enemyagents, it could focus its searching efforts elsewhere.

Specifically, we plan to use vibration sensors that will de-tect the absence or presence of enemy robots and route this

3 4 5 6 7 8 9 10 110

0.1

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0.3

0.4

0.5

0.6

0.7

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0.9

1

Number of Enemies

Pro

babi

lity

Sur

roun

ded

Posterior Probability Evolution as Number of Enemies Varies

SurroundedNot Surrounded

Figure 5: Posterior Probability Evolution as Num-

ber of Enemies Varies

information to the friendly agents to determine where en-emies may or may not be. These sensors will be subjectto noise as they will sometimes miss a robot going by ormistake a person walking by for an enemy robot.

6.2 Intelligent Sensor SelectionWhile our algorithm can effectively leverage very limited

information, to run on an autonomous sensor network, itneeds to be paired with a decentralized algorithm that runson each sensor node that decides when to sense and alsowhen and what type of information to communicate to itsneighbors.

Such an algorithm will need to make tradeoffs betweenenergy conservation and answering queries more effectively.One way to address this problem is a cost-based schemethat casts utility and battery depletion into a single cur-rency [3]. While the cost of sensing, asking another nodeto sense, or sending information to another node will falldirectly out of a coherent energy-aware sensor model, thevalue of such information is intricately tied to the inferencealgorithm. Specifically, to prevent bias, the value of sensingshould be the expected reduction in entropy stemming fromthe acquisition of this information.

Constructing such an algorithm requires answering severaldifficult questions. First, how can sensors with only partialworld state display utility-maximizing emergent behavior?Second, what is the tradeoff between expending more energyand acquiring a more accurate answer? Third, how cansensors distribute work in a way that takes into account theeffects of remaining power at the granularity of individualnodes?

7. CONCLUSIONWe have presented an inference algorithm for a sensor

network to effectively answer generalized queries. Usingprobabilistic techniques, the algorithm is able to answerthese queries effectively with only partial information over

0 5 10 15 20 250

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Sensing Range

Pro

babi

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roun

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Posterior Probability Evolution as Sensor Range Varies

SurroundedNot Surrounded

Figure 6: Posterior Probability Evolution as Sensing

Range Varies

the world state. Moreover, it can reason coherently despitethe stochastic nature of the sensor data. The results showndemonstrate that the inference algorithm is highly accuratein the scenario presented and is able to handle a wide rangeof types of sensor information.

8. ACKNOWLEDGEMENTSThis research is supported in part by DARPA’s MARS

Program (Contract number N66001-01-C-6018), DARPA’sSensor Information Technology Program (Contract numberF30602-00-C-0139), the National Science Foundation (CA-REER grant number IIS-9876136 and regular grant numberIIS-9877033), Intel Corporation, the Stanford NetworkingResearch Center, ONR MURI grant N00014-02-1-0720, anda grant from the DARPA Software for Distributed Roboticsprogram under a subcontract from SRI, all of which is grate-fully acknowledged. The views and conclusions contained inthis document are those of the authors and should not beinterpreted as necessarily representing official policies or en-dorsements, either expressed or implied, of the United StatesGovernment or any of the sponsoring institutions.

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0 5 10 15 20 250.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055Posterior Probability Convergence as MCMC Progresses

Number of MCMC Steps, in Thousands

Sta

ndar

d D

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tion

of P

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0 queries2 queries4 queries6 queries8 queries

Figure 7: Posterior Probability Convergence as

MCMC Converges

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