A Theoretical Framework for Estimating
Swarm Success Probability Using Scouts


Antons Rebguns, University of Arizona, USA
Diana Spears, Swarmotics LLC, USA
Richard Anderson-Sprecher, University of Wyoming, USA
Aleksey Kletsov, East Carolina University, USA



ABSTRACT
This paper presents a novel theoretical framework for swarms of agents. Before deploying an
entire swarm for a task, it is advantageous to predict whether a desired percentage of the swarm
will succeed. Our framework uses a small group of expendable “scout” agents to predict the
success probability of the entire swarm, thereby preventing many agent losses. The scouts apply
one of two formulas to predict – the standard Bernoulli trials formula or our new Bayesian
formula. For experimental evaluation, the framework is applied to simulated agents navigating
around obstacles to reach a goal location. Extensive experimental results compare the mean-
squared error of the predictions of both formulas with ground truth, under varying circumstances.
Results include the accuracy and robustness of the Bayesian approach. The framework also
yields an intriguing result, namely, that both formulas usually predict better in the presence of
(Lennard-Jones) inter-agent forces than when their independence assumptions hold.

Keywords: scouts; swarm of agents; success rate; sampling; Bayesian

INTRODUCTION
This paper presents a novel theoretical framework for swarm risk assessment. The framework is
applied to a scenario consisting of a swarm of agents that needs to travel from an initial location
to a goal location, while avoiding obstacles. Before deploying the entire swarm, we would like to
have a certain level of confidence that a desired portion of the swarm will successfully reach the
goal. If not, then perhaps the swarm should not be deployed. For example, for a swarm of
moving robots, the environment itself can pose a significant risk (rough terrain, sudden changes
in elevation that agents are not equipped to handle, water, etc.) and, as with any hardware, circuit
and mechanical failures can prevent agents from successfully reaching their destination. It is
alternatively plausible that the swarm consists of software agents trying to achieve a more
abstract goal, such as a successful transaction, while avoiding obstacles, such as provisions or
constraints. For simplicity, in our simulation agents are modeled as robots and obstacles are
modeled as physical objects.
The environment in which the agents are deployed is assumed to be static, though it may be
completely or partially unknown. This environment can be highly unstructured as well, as in
Mondada et al. (2005). Additionally, deployment of the entire swarm is potentially hazardous,
e.g., due to the possible loss or corruption of agents – for example, some of the obstacles might

contain explosives, agents could fall into inescapable holes as in Dorigo et al. (2006), or there
could be environmental hazards as in Tatomir and Rothkrantz (2006). In these and many similar
situations it is advantageous to do a preliminary phase of risk assessment before deploying the
full swarm. The information gained from this phase will help the practitioner decide what
deployment strategy to use, e.g. what starting location works best, how many agents to deploy in
order to ensure a desired success rate, and whether the task at hand is worth the risk of losing a
possibly large portion of the swarm.
The solution proposed here is the use of a group of expendable agent “scouts” to predict the
success probability for the swarm, during the risk assessment phase. For practical reasons, only a
few (less than 20) scouts are sent from the swarm, which may consist of hundreds of agents. A
human or artificial agent, called the “sender,” deploys the scouts. Then, an agent (e.g., a person,
an artificial scout, or a sensing device), called the “receiver,” counts the fraction of scouts that
arrive at the goal successfully. The sender and receiver must be able to communicate with each
other – to report the scout success rate, but no other agents require the capability of
communicating messages. Using the fraction of scouts that successfully reach the goal, we apply
a formula that predicts the probability that a desired percentage of the entire swarm will reach
the goal. Based on this probability, the sender can decide whether or not to deploy the full
swarm. Alternatively, based on this probability the sender can decide how many agents to deploy
in order to yield a high probability that a desired number of agents will reach the goal.
Our theoretical framework is based on two formulas that use agent scouts as “samples” for
making predictions regarding the success probability of a swarm. The first approach is the
standard Bernoulli trials formula, and the second is a novel Bayesian formula. We report
conclusions regarding the predictive accuracy of these formulas, based on an extensive set of
experiments during which parameters were varied methodically. Our measure of predictive
accuracy is the mean squared error (MSE) of each formula’s predictions versus “ground truth.”
Experimental conclusions include the value of a uniform prior for the Bayesian formula in
knowledge-lean situations, and the accuracy and robustness to changes in the environment of the
Bayesian approach. This paper also reports an intriguing result, namely, that both formulas
usually predict better in the presence of inter-agent forces (when a Lennard-Jones inter-agent
force law is used) than when their independence assumptions hold. Inter-agent forces are useful
for initiating and sustaining multi-agent formations while traveling to a target location. We
conclude that these formulas, and especially our Bayesian formula, provide extremely practical
solutions for solving “the swarm success rate prediction problem” in a variety of real-world
situations. Additionally, this paper provides conclusions which lead to advice on selecting the
values of controllable parameters in order to help the practitioner apply our framework.
The most notable contributions of this research are:
 A novel theoretical framework for swarm risk assessment, using very few scouts to
predict the swarm success probability, is presented. For the first time, scouts are used to
predict the probability that a given portion of a swarm will achieve its objective.
Although our framework has been applied to a particular navigation task in this paper,
there is nothing about our approach that depends on this specific task.
 A novel Bayesian formula for success probability calculation is introduced, along with its
mathematical derivation.
 A full factorial experimental design has been employed to evaluate and compare the
Bernoulli and Bayesian formulas.

RELATED WORK
Our framework is applicable to a wide range of tasks, including search-and-rescue, navigation,
surveillance, optimization, and so on. It assumes that each agent within the swarm will succeed
or fail at the task, and that this binary outcome can be determined. This research instantiates our
framework by applying it to a simulation in which the agents’ task is to navigate through a field
of obstacles to get to a goal location.
For our control strategy, we chose physicomimetics, also called artificial physics (AP),
invented by Spears and Gordon (1999). This was a serendipitous choice. Alternative control
strategies that could have been employed include behavior-based, rule-based or biomimetic. As
will be seen in the experimental results below, there appears to be a favorable relationship
between our scouts approach and physicomimetics control (using a Lennard-Jones control law).
Our scouts framework is related to statistical sampling. The process of sampling from a
population to infer conclusions about the general population has a long and very important
history in the field of statistics, starting in the early 1800s with census taking (Wright & Farmer,
2000). The Bernoulli formula applied in this work is a fundamental formula from the literature
on statistical sampling (Zwillinger, 2002). Statistical sampling has permeated a wide array of
social and scientific fields, including artificial intelligence (AI). For example, search and
evolutionary algorithms depend fundamentally on stochastic sampling (Russell & Norvig, 2003).
Also, inductive inference within machine learning uses sampling theory (Hastie et al., 2001;
Kearns & Vazirani, 1994). Furthermore, our Bayesian formula is related to Bayesian statistical
approaches in AI, including Bayesian inference/updating (Russell & Norvig, 2003).
There has been little prior research on agent scouts. Most studies are found in natural science
– because scouts are a popular mechanism for assisting biological swarms. One noteworthy
example is that of Greene and Gordon (2007), who published a study of red harvest ants that use
a group of “patrollers” as scouts before sending out the full swarm of ants. These patrollers
establish a path for foraging. Our scouts also act as “patrollers,” but they quantify the probability
of success for the swarm rather than setting the path. Future work will include merging the use
of scouts for predicting the success rate and for subsequently guiding the swarm.
Scouts are particularly relevant for search-and-rescue applications (Rybski et al., 2001), for
remote reconnaissance, and within the context of swarms they are relevant for distributed sensor
networks (Tikanmäki et al., 2006) and swarm “nest assessment” (Şahin & Franks, 2002). Like
this previous work, our approach also uses scouts for assessing the safety of the environment for
the swarm. We have not, however, been able to find any prior research on the topic of sending
out a small set of scout agents specifically for the purpose of predicting the probability of swarm
success. The ability to predict the swarm success rate is fundamental for tasks in which the loss
of a substantial fraction of the swarm would be costly or might prevent completion of the task.

PREDICTING SWARM SUCCESS RATE WITH SCOUTS
Our objective is to predict the probability that y or more out of a total of n agents will
successfully navigate through an obstacle field and get to a goal location within a time limit
(Rebguns et al., 2008). We predict this probability by sending out a sample of k scout agents
(which, we assume, are not part of the n swarm agents, although modifying our formulas if they
are a subset of the swarm is straightforward). For practical reasons, we assume that k ≪ n.

The Formalized Prediction Problem
Formally, the problem being addressed in this paper is the following. Assume that k scouts are
used to find an estimate
ˆ Successesp k
of p, where Successes is the number of scouts
successfully reaching the goal, and p is the true probability of one agent succeeding in the time
limit. We use p as a measure of environmental difficulty, which, in the simplest case is related to
the number of obstacles to be avoided by the agents. Given pˆ as input, we want our approach
(using one of the formulas below) to output the following probability:
ˆ( | ) P Y y p
where Y is a random variable and y is a problem-dependent lower bound parameter whose value
is given by the system user.

Bernoulli Trials Formula
The process of predicting the swarm success rate can be considered to be a Bernoulli trials
process. Assume that k scouts are sent out to estimate
ˆ Successesp k
. Then the standard
formula (Zwillinger, 2002), which we call PBern, for predicting the probability of y or more
successes in n independent trials is directly applicable:
ˆ ˆ ˆ( | )    (1 )
n
j n j
Bern
j y
nP P Y y p p pj


      


Novel Bayesian Formula
This paper assumes that the number k of scouts is between 3 and 15. Our rationale is that we
specifically want to investigate the case of scouts being a valuable and expensive commodity,
and sending more is problematic. By modeling few scouts, we are able to address the real-world
situation faced by practitioners of whether it is worthwhile sending any scouts at all if only a few
can afford to be sent. A well-known problem with using PBern is that k should be large, perhaps
more than 20 scouts, to get a reasonable estimate pˆ of p. Our reason for mentioning the number
20 in particular is that when k is less than 20, the variance of PBern tends to be high and there is a
substantial chance that either all or no scouts will reach the goal, typically leading to poor
predictions of swarm behavior. As an example, consider the case of three scouts. There is always
a risk that all three will fail, even if the environment is very easy, i.e., tiny samples can be
skewed. To overcome this limitation of PBern, we provide a Bayesian formula called PBayes.
PBayes assumes a prior distribution over the probability p of a single agent reaching the goal. A
prior distribution initializes with prior information (if such information is available), reduces the
variance of the predictions (and this lower variance has been demonstrated experimentally), and
“evens out” the random perturbations of small samples thereby enabling greater accuracy with
small k. We use a Beta distribution (α, β) over (0, 1) for the prior (Zwillinger, 2002), which is
a good choice because it can represent a variety of prior shapes when the possible values are on
the finite interval (0, 1). No other standard family of distributions has these properties. If the

Beta distribution parameters α and β are both 1, then the distribution is uniform. The probability
density function (pdf) (p) corresponding to the Beta distribution is given by:
1 1(1 )( ) ( , )
p pp B
 
 
 

where B(α, β) is the (complete) Beta function defined by
1 1 1
0 (1 )z z dz  
, which in the case
where α and β are positive integers reduces to ( 1)!( 1)!
( 1)!
 
 
 
 
. The next section and Appendix A
describe our approach to methodically varying the parameters of the (α, β) distribution and
hence the shape of its pdf (p).
Recall that our objective is to find the probability of y or more successes out of n agents. We
are allowed to observe k scouts, and their performance yields a fraction pˆ of successes that is
used to predict the true probability p of one agent reaching the goal. Our Bayesian formula is (for
the derivation, see Appendix B):
ˆ( | ) BayesP P Y y p  
1 1 1 1
0 0 · (1 ) (1 )p y n y r sM z z p p dzdp     
1where       { ( , )· ( , 1)}M B r s B y n y   
ˆ   r kp  
ˆ    (1 )s k p   
where p takes on all possible values in the outer integral, and z is a variable that ranges from 0 to
p, given the particular p chosen by the outer integral. Note that this formula is not the same
formula as one would obtain by plugging the posterior mean of p into the formula for PBern. It is
the posterior mean of the probability that y or more scouts will reach the goal, assuming a Beta
prior for p with parameters α and β. The inner integral gives the probability of y or more
successes out of n trials for a given p, weighted by the prior for p. The form of the integrand may
not be intuitive because it replaces the usual sum of Binomial probabilities with an equivalent
incomplete Beta form. The outer integral averages over all possible values p of environmental
difficulty, which ranges from 0 to 1. Note that with a uniform prior and a large sample size,
PBayes will reduce to PBern (Ferguson, 1996). The advantage of PBayes (even with a uniform prior)
over PBern is in applications with few scouts.
PBayes is implemented in our simulation using numerical methods; specifically, we use the n-
point Gaussian quadrature rule (Cheney & Kincaid, 2003; Press et al., 1994; Zwillinger, 2002).
The implementation of this rule was rigorously tested against hand-calculated values, and found
to be accurate and correct within expected bounds of precision (Rebguns, 2008).

The Prior Distribution for the Bayesian Formula
The experiments described later in the paper are designed to determine how the predictive
accuracy of the formulas changes as a function of parametric variations. One of the parameters
that can be varied in PBayes is the prior distribution. This distribution can be either more or less
confident, and more or less accurate. Confidence is measured in terms of prior strength, and

Figure 1. Graph of Beta pdf for p = 0.1, with mean μ = 0.1 and C is defined in Appendix A



Figure 2. Graph of Beta pdf for p = 0.5, with mean μ = 0.5 and C is defined in Appendix A

Figure 3. Graph of Beta pdf for p = 0.9, with mean μ = 0.9 and C is defined in Appendix A



accuracy in terms of prior correctness. See Appendix A for our methodology for varying the
prior strength and correctness. The final seven priors and their strength and correctness are (see
Appendix A for the derivation):
1. Uniform (α = β = 1). Weakest prior.
2. α = 2.1 and β = 18.9. Weak, and almost correct for p = 0.1.
3. α = 10.9 and β = 98.1. Strong, and almost correct for p = 0.1.
4. α = 2.5 and β = 2.5. Weak, and almost correct for p = 0.5.
5. α = 14.5 and β = 14.5. Strong, and almost correct for p = 0.5.
6. α = 18.9 and β = 2.1. Weak, and almost correct for p = 0.9.
7. α = 98.1 and β = 10.9. Strong, and almost correct for p = 0.9.
The uniform prior is neither correct nor incorrect; it is “non-informative.”
Figures 1, 2, and 3 show graphs of the (p) probability density functions (pdf) for three
different values of environmental difficulty p: 0.1, 0.5 and 0.9 respectively. The level of
environmental difficulty was hand-crafted -- by choosing the goal location, and the number,
sizes, shapes and locations of the obstacles, and the location of the starting square for the agents.
Finally, note that a prior that is correct for a particular value of p will be incorrect for any
other value of p. For example, a prior that is correct for p = 0.1 will be incorrect if the true p is
0.3, and will be even more incorrect if the true p is 0.5 or especially 0.9. This incorrectness is
exacerbated if the prior is strong. Also, the reason for “almost” correct is that it is usually too
hard to design environments that have a precise level of difficulty.

Figure 4. Swarm simulation screen-shot, with no inter-agent forces. Large circles represent
obstacles, the square represents the goal and the small dots represent agents



Figure 5. Swarm simulation screen-shot, with inter-agent forces. Large circles represent
obstacles, the square represents the goal and the small dots represent agents

SWARM NAVIGATION SIMULATOR
Our 2D simulation (see Figures 4 and 5) consists of a distributed, decentralized swarm of agents
(the dots), a goal location (square), and a set of obstacles (circles/disks), whose locations can be
varied. Agents sense the goal at any distance. They sense obstacles and other agents in any
direction, but only within their sensing range, which is 1.5R, where R is 50 pixels. The choice of
this particular sensing range optimizes the formation of triangular (i.e., hexagonal if one assumes
no agent in the middle of the hexagon) agent lattices. The value of 50 pixels for R was
determined empirically; it led to good performance.
The following is a formalization of Navigation Through Obstacles to Goal problem that is
addressed here. An agent i succeeds at the problem if and only if it satisfies the following
objective after the time limit has been reached (where the arrows denote backwards implication
read as "if," and only the first matching clause executes, as in the computer language Prolog):

1. goal_reached(i) = true, where goal_reached is defined recursively as:
goal_reached(agent) d(agent, goal) < 1.5R;
goal_reached(agent)
there exists agent2 such that ([d(agent, agent2) ≤ 1.5R] & goal_reached(agent2))

Furthermore, the agent i must try to achieve the following objectives (using its force law --
see below) during as many time steps of the simulation as possible:

2. Stay in formation, i.e., d(i, j) = R for all agents j within sensing range, where j ≠ i
3. Avoid obstacles, i.e., d(i, o) ≥ R for all objects o within sensing range

The control algorithm that we use for achieving these problem objectives is physicomimetics,
also called “artificial physics” (AP). We provide a brief overview here. Potential fields (PF)
described in Khatib (1986) is the earliest approach to applying physics to agent control. AP and
PF evolved along similar lines, in parallel, although PF tends to be more control theoretic than
AP (Leonard & Fiorelli, 2001).
With physicomimetics, virtual physics forces drive a swarm to a desired configuration or
state. The desired configuration is one that minimizes overall system potential energy, and the
system acts as a dynamics F ma simulation. Each agent has a position p and a velocity v .
We use a discrete-time approximation to the continuous behavior of the agents, with time-step
Δt. At each time step, the position of each agent undergoes a perturbation p . The perturbation
depends on the agent’s velocity, i.e., p v t   . The velocity of each agent at each time step also
changes by v . The change in velocity is controlled by the force on the agent, i.e.,
/v F t m   , where m is the mass of that agent and F is the force on that agent. A frictional
force is included for self-stabilization.
Researchers have experimented with a wide variety of virtual forces for physicomimetics
(Spears & Gordon, 1999; Spears et al., 2004; Spears et al., 2005). Here we use a generalization
of the Lennard-Jones (LJ) force law (Lennard-Jones, 1931) to represent the force between i and j:
12 6
, 13 7
224 ( , ) ( , )i j
dR cRF d i j d i j
    

where the parameter є affects the strength of the force between the entities, d(i, j) is the actual
distance between i and j, R is the desired distance between i and j, and c and d control the
relative balance between attractive and repulsive forces, respectively. We have chosen the LJ
force because it is extremely effective for agents staying in geometric formations while
navigating around obstacles to get to a goal location. We use optimal parameter settings for LJ,
evolved for this task by Hettiarachchi (2007). The values of all parameters used in our simulation
can be seen on the left-hand side of Figures 4 and 5. Agents sense the range and bearing to
nearby obstacles, neighboring agents, and to the goal location.
Initially, all agents begin at uniformly randomly chosen locations within a starting square of
side length 100 pixels, approximately 1,000 pixels from the goal.i The simulation area is 800
pixels wide and 600 pixels tall, with the starting square located in the lower left corner and the
goal in the upper right corner. The simulation ends after t = 1,500 time steps, which is the time
limit. The simulation has two modes: one where the agents are completely independent of each
other (see Independent Agents section for details) and another where there are virtual LJ inter-
agent forces (see Interacting Agents section for details). Figure 4 shows the simulation with
independent agents (inter-agent forces disabled), and Figure 5 shows the simulation with
dependent agents (inter-agent forces enabled). Lines are shown in the graphics to visualize the
virtual inter-agent forces. With inter-agent forces, the agents stay in a geometric (triangular)
lattice formation while avoiding obstacles and navigating to the goal location.

EXPERIMENTAL DESIGN
The Performance Metric
Our primary experimental objective is to evaluate and compare the quality (predictive accuracy)
of the standard Bernoulli trials formula, which we call PBern and our Bayesian formula, which we
call PBayes. Evaluating the quality of a formula implies comparing its value against a form of
“ground truth.” For fairness, PBern or PBayes and ground truth all use the same parameter values
during comparisons.
The mean squared error (MSE) is our performance metric because it includes both the error
and the variance of the error. The MSE is a function of the “absolute error,” which is the
difference between the estimate and the truth. Assuming X is the true value and Xˆ is an estimate
of X, the MSE is defined as:
2ˆ ˆ( , ) [( ) ]MSE X X E X X 

where E denotes the expected value.
To measure the MSE, we need a form of “ground truth.” Our ground truth, which is called
PTruth, is the fraction out of 1,000 runs of the simulation in which y or more of the n agents reach
the goal. To compare with ground truth, we use the performance metrics MSE(PBern,PTruth) and
MSE(PBayes,PTruth).

Experimental Parameters
Our secondary experimental objective is to vary the parametric conditions and measure how the
MSE varies. A full factorial experimental design has been performed, i.e. we vary one parameter
at a time and evaluate how varying that parameter affects performance over all combinations of
all other parameter values. The following list enumerates the parameters (the independent
variables) whose values are varied:

 Total number of agents. One variable is n, the total number of agents in the swarm. We
expect n to have a negligible effect on the errors. The values of n are 100, 200, 300, 400,
and 500. These particular swarm size values were chosen because they are variations of
the precedents that we found in the swarm agents literature (Berman et al., 2007).
 Number of scouts. Perhaps the most important independent variable is k, the number of
scouts. The values of k are 3, 5, 10, and 15. These values for the number of scouts were
selected because they are sufficiently small that we believe they would be practical for
many real-world applications. Normally, for methodical variation, we would select 5, 10,
and 15. But the value 3 was added also in order to include the smallest reasonable
number of scouts that one might wish to employ.
 Desired number of successes. Here, y is the desired number out of the n swarm agents
that are required to succeed. This varies exhaustively from 1 to n.
 Environmental difficulty. We also vary p, the “true” probability of one agent succeeding,
which is found using 1,000 runs of the simulation of n agents each. Three environments
were hand-crafted (by choosing the goal location, and the number, sizes, shapes and
locations of the obstacles, and the location of the starting square for the agents) in order
to get the following values of p: 0.1, 0.5, and 0.9. These particular numbers reflect the
need to include reasonable values for the easiest (0.9) environment, the hardest (0.1)
environment, and a mid-level of difficulty (0.5) environment. Note that throughout the
paper, the particular value of p can vary, although it is close to 0.1, 0.5, or 0.9. For
example, sometimes we use 0.51 and other times 0.55. The reason for p varying slightly
is that with hand-crafted environments we were not always able to get perfectly precise
values of environmental difficulty.
 Strength of the prior distribution. Recall that we use a Beta prior probability distribution
for PBayes. The “strength” of this prior is quantified as 1/σ
2, where σ2 denotes the variance.
In other words, a lower variance yields a “stronger” prior.
 Correctness of the prior distribution. Another variable is the prior “correctness.”
Correctness is the degree to which the prior is well-matched to the truth (p). Here,
correctness is measured as the absolute value of the difference between the mean of the
prior distribution and the true value of p. The smaller the difference, the more correct the
chosen prior distribution is considered to be.
 Independence of the agents. This parameter is Boolean. If its value is true, the agents are
assumed to be completely independent of each other.

THE EXPERIMENTAL ALGORITHMS
Next, we describe our algorithms used in the experiments. Each experiment measures the
MSE(PBern, PTruth) and MSE(PBayes, PTruth) with a single choice of values for all parameters, for
multiple runs.
The algorithm for PTruth was described above. The following is the algorithm for PBayes and
PBern:
STEP 1: Using the k scout agents obtain pˆ , which is an estimate of p.
STEP 2: Apply the implemented formula for P(Y ≥ y | pˆ ), using the pˆ obtained from Step 1.

When PBern and PBayes are calculated in the real world, the value of pˆ , which is the estimate of
p found using k scouts, is unknown until the scouts are sent out. However for the experimental
evaluation of our framework, this needs to be modified for fair comparisons. One way to obtain
pˆ in our experiments would be to deploy the k scouts in the simulation to get pˆ as the fraction
out of the k scouts that succeeded, and then plug pˆ into the formulas for PBern and PBayes. The
problem with doing this is that it does not predict performance in expectation. The correct
solution is to treat pˆ as a random variable for the experiments (though this would obviously not
be done in the real-world situation). The expectation E is calculated as an average over 1,000
runs. For each run, we randomly vary the agent locations within the starting square. The
dependent variable MSE is calculated as follows. Let PB be PBern or PBayes. Then
Bias = E(PB) − PTruth
Variance = E[(PB − E(PB))
2]
MSE = E[(PB − PTruth)
2] = Bias2 + Variance
In the next two sections we formulate and test hypotheses for independent, and then
dependent, agent scouts and swarms. There is only room here to show a few of the most
interesting and representative results. In the graphs, the horizontal axis is one of the independent
variables and, unless otherwise stated, the vertical axis is the MSE. Each curve shows the MSE
estimated by averaging over 1,000 runs.

INDEPENDENT AGENTS: EXPERIMENTAL RESULTS
The case of independent agents is practical for some search and rescue problems, as well as
chemical plume tracing tasks where the agents use anemotaxis (Zarzhitsky et al., 2004, 2005).
Before we formulate our hypotheses, consider the graphs of MSE and how it changes across
all values of y. We focus first on variations of y because the graphs have an interesting
characteristic shape that generally remains invariant as the values of all other parameters vary.
Figure 6 shows the MSE(PBayes,PTruth) as y varies. Note the bimodal curves in Figure 6, which
tend to be a common characteristic of our error curves when y is varied along the horizontal axis.
To understand this bimodal pattern, it is useful to look at the graph of PBayes versus PTruth that was
used to create one of the curves in Figure 6. Figure 7 provides the intuition for the bimodal
pattern. Compare the curve for PBayes with the curve for PTruth in Figure 7. Notice that there is an
abrupt shift in ground truth (PTruth) probability from 1 to 0, analogous to a sigmoid function (but
reversed), around y = 55. We call this a “phase transition.” In particular, if y < np, then y or more
agents are guaranteed to reach the goal – because the environment (captured by p) supports such
success. If y > np, then it is nearly impossible (probability close to 0) for y or more agents to
reach the goal. PTruth accurately reflects this phase transition. On the other hand, the curve for
PBayes very roughly approximates the transition, i.e., rather than an abrupt drop in the curve it
shows a smaller slope in going from a 1 to 0 probability of success. This explains the curves for
MSE. Near the transition point where y = np, the curves for PTruth have an abrupt phase shift but
the curves for PBayes have a more gradual shift, which causes the bimodal errors around the point
y = np. The same phenomenon occurs in all the graphs comparing PBern with PTruth as well. For
example, see Figure 8 for the MSE graph and Figure 9 for the probability graph. Although these
graphs show results for only one particular set of parameters, our factorial experiments have
confirmed that this characteristic bimodal pattern holds regardless of all the other parameter
settings, when y is the independent variable. (The odd wavy forms seen in Figures 8 and 9 for

Figure 6. Graph of MSE(PBayes, PTruth) with p = 0.55 and n = 100 (Rebguns et al., 2008)
(© 2008, IEEE. Used with permission.)



Figure 7. Graph of PBayes and PTruth with p = 0.55, n = 100, and k = 5 (Rebguns et al., 2008)
(© 2008, IEEE. Used with permission.)

Figure 8. Graph of MSE(PBern, PTruth) with p = 0.55 and n = 100



Figure 9. Graph of PBern and PTruth with p = 0.55, n = 100, and k = 5

Bernoulli estimators with small k are an artifact of the property that only k+1 possible estimates
exist for k scouts.)
The variable y is not the focus of any of our hypotheses below (and therefore we usually
average over it) because we understand its behavior based on the above discussion, and because
the proximity (which predicts the MSE) of y to np is not controllable by the practitioner (because
p is unknown). The remaining graphs show variations in other parameters besides y.
We next formulate and test four hypotheses regarding performance, measured with the MSE,
as parameters values are varied. Throughout this paper, we will consider a hypothesis to be
“confirmed” if it is true for the overall trends in the graphs, even if it does not necessarily hold
for every data point on every curve of every graph. In other words, our hypotheses capture
performance trends/patterns, rather than absolute performance guarantees.
To test each hypothesis, we vary the parameter of interest and hold the others (except y)
constant. When y is not on the horizontal axis, each curve is an average over all values of y
(called “Average MSE” on the vertical axis). Conclusions about both the MSE(PBern, PTruth) and
MSE(PBayes, PTruth) are presented.
The first hypothesis is about how the predictive accuracy varies when the number of scout
agents is varied. Scout and swarm agents are both independent.

Hypothesis 1 (about the number of scouts) Increasing the number of scouts, k, will improve
the predictive accuracy.

For independent agents, this hypothesis is intuitive. A bigger sample size produces a better
estimator/predictor of truth. The experimental results confirm that this hypothesis is indeed
correct for both PBern and PBayes. Let us consider Figure 6 more carefully. It shows the graph of
how the MSE of PBayes varies as we change the number of scouts k, namely, as we increase k
from 3 to 15 the maximum MSE decreases from about 0.20 to about 0.12. Next, look at Figure 8,
which shows the graph of how the MSE of PBern varies as we change the number of scouts k,
namely, as we increase k from 3 to 15 the maximum MSE decreases from about 0.40 to about
0.19. Note that PBern is much more dependent on the number of scouts for its predictions,
whereas this effect is much less pronounced for the PBayes formula. For PBayes, the prior
distribution compensates for a small sample size.
The reduction in error with more scouts can be explained with the Law of Large Numbers. In
particular, the Law of Large Numbers states that as the sample size increases, the sample means
converge to the true mean of the population. The fraction of successes found by the scouts can be
considered a sample mean, which is estimating the true mean found by ground truth. Two
phenomena occur as k increases. First, the sample mean approaches the true mean. Second, our
Bayesian formula for PBayes puts more emphasis on the samples than on the prior when k is large.
The following three hypotheses have also been tested and confirmed with independent
scouts/agents:

Hypothesis 2 (about the prior distribution for PBayes)
1. It is better to have a correct rather than incorrect prior.
2. If the prior is incorrect it is better to be weak rather than strong. Also, the non-
informative uniform (weakest) prior is better than any incorrect prior.

Figure 10. Graph of MSEs with p = 0.51, n = 300, k ranging from 3 to 15, with inter-agent
forces (Rebguns et al., 2008) (© 2008, IEEE. Used with permission.)



3. If the prior is correct it is better to be strong than weak.
As is well known in Bayesian statistics, a strong correct prior is optimal, and a uniform prior
is the best solution when prior knowledge is weak or lacking. Our results confirm this.
Based on our confirmation of Hypotheses 1 and 2, we find that there are two main ways to
reduce MSE in the predictions made by PBayes. One is to increase the number k of scouts, which
increases the sample size, and the other is to choose a better prior distribution.

Hypothesis 3 (about swarm size/scalability) If the ratio k/n is held constant, then increasing
the swarm size will not significantly increase the prediction error.

Hypothesis 4 (about environmental difficulty/robustness) The predictions will be robust
(i.e., will not change significantly in value) across a wide range of environmental difficulties.

Both Hypotheses 3 and 4 have been confirmed in all of our experiments, which demonstrated
that both formulas are good predictors, and PBayes is usually better than PBern – even with a
uniform prior. In particular, the MSE(PBayes, PTruth) is usually lower than the MSE(PBern, PTruth),
and in many cases much lower. Interestingly, based on further experimental results we have
found that with few exceptions, the variance for PBayes is much lower than for PBern.

INTERACTING AGENTS: EXPERIMENTAL RESULTS
This section assumes dependent swarm agents and dependent scout agents (so that the scout
behavior is representative of the swarm behavior).
Here, we test the same four hypotheses that were tested for the independent agents case. The
conclusions are more interesting in the case of dependent agents. To test each hypothesis, the
parameter of interest is varied and the other parameter values (except y) are held constant. When
y is not on the horizontal axis, each curve is an average over all values of y. Conclusions about
both the MSE(PBern, PTruth) and MSE(PBayes, PTruth) are presented.
Recall Hypothesis 1, which predicts that increasing the number of scouts will increase the
predictive accuracy of the formulas. Although this hypothesis is intuitive and holds for the
independent agents case, the results are surprisingly mixed for dependent agents for both
formulas, PBern and PBayes. Figure 10 shows a typical result where Hypothesis 1 is violated. In this
figure the swarm size n is held constant and the number of scouts k is varied. By watching the
simulation one can see why Hypothesis 1 is violated for dependent scouts. It is because the
scouts stay in formation and act as a quasi-unit (just like the swarm does). The scouts are
predictive, but because they act as a quasi-unit, adding more usually does not increase their
predictive ability. However the harder the environment, the more it helps to use additional
scouts. This is because the quasi-unit gets fractured in hard environments (e.g., with many
obstacles) and it becomes more like the case of independent agents.
In general, we have found that very few scouts are needed for a very good assessment of the
swarm success rate, and adding more scouts does little to improve the predictions in the case of
dependent agents. This is fortunate because it implies that just a few scouts are enough to obtain
a good estimate of the swarm success rate.
Next, recall Hypothesis 2 about the prior distribution for PBayes. It states that a correct prior is
preferable if one is available and, if correct, then it should be strong. On the other hand, if a
correct prior is unavailable then it is better to choose a weak incorrect prior than a strong one.
Hypothesis 2 has been confirmed by our experimental results. For example, Figure 11 confirms
that if a prior is correct (or almost correct), then it should be strong; the confirmation is over
most values of y, where y is varied along the horizontal axis. Overall, we have found that a
strong, correct prior is optimal, and a uniform prior is the best choice when prior knowledge is
lacking and an incorrect prior must be used.
In Figure 11 we show PBayes with different priors. We show one curve per prior, which means
that we were able to vary y along the horizontal axis in the graph. For the sake of conciseness of
comparisons of PBayes versus PBern we hereafter resume the convention of showing curves that
average over all values of y and vary the parameter of interest along the horizontal axis.
Next, consider Hypothesis 3, which states that increasing the swarm size will not significantly
increase the prediction error, if the ratio k/n is maintained as a constant. The results from our
tests of this hypothesis display characteristic trends, and these trends depend upon the
environmental difficulty. We report three trends corresponding to easy, moderate, and hard
environments, respectively. The uniform prior is used with PBayes because we wish to show worst
case results, and in the worst case when there is no knowledge to help select a prior the uniform
can always be used to get the best results overall.

Figure 11. Graph for Hypothesis 2, part 3, with p = 0.55, n = 100, k = 5, and inter-agent forces
(Rebguns et al., 2008) (© 2008, IEEE. Used with permission.)



Figure 12. Graph of MSE with varying n and k, k/n = 0.03, p = 0.9, uniform prior for PBayes, and
inter-agent forces

Figure 13. Graph of MSE with varying n and k, k/n = 0.03, p = 0.5, uniform prior for PBayes, and
inter-agent forces



Figure 14. Graph of MSE with varying n and k, k/n = 0.03, p = 0.1, uniform prior for PBayes, and
inter-agent forces

Figure 12 shows typical results for an easy environment, where p = 0.9. As we can see in the
figure, for easy environments the MSE actually decreases when going from 100 to 300 agents,
and then it increases somewhat when going from 300 to 500 agents, but with 500 agents the MSE
is still less than with 100 agents. This assumes a constant k/n ratio of 0.03.
Figure 13 shows a typical result in a moderate environment, where p = 0.5. There is a slight
increase in MSE for PBayes as the swarm size scales up while the ratio of scouts to swarm agents
is held constant. The maximum increase in MSE in either curve is less than 0.0025, which is tiny.
Finally, consider the typical Figure 14, where the environment is most difficult (p = 0.1). In
this case as the number of agents increases while the ratio is maintained, the MSE is reduced for
PBern and reduced or maintained for PBayes, except for a second peak when there are 400 agents.
With PBayes, the MSE is highest with the fewest (100) agents. This demonstrates nice scalability.
Hypothesis 3 is true in almost all of our experiments, for both MSE(PBayes,PTruth) (with any
prior, though the results with a strong almost correct prior are, as usual, much better than with a
uniform prior) and for MSE(PBern,PTruth). Note that PBayes has a lower MSE than PBern, except
when the environment is maximally difficult with p = 0.1 (see Figures 12, 13, and 14). This latter
issue will be addressed in the next hypothesis regarding environmental difficulty.
Finally, consider Hypothesis 4, which states that the predictions will be robust, i.e., not
change significantly in value, despite wide variations in the level of environmental difficulty.
There are two trends in the graphs when varying the environmental difficulty along the
horizontal axis. Figure 15 exemplifies the first pattern. Note the contrast between the predictions
of PBayes with a uniform prior and PBern, as the environmental difficulty varies in Figure 15. PBayes
is considerably more robust than PBern over the range of environments. This figure and all those
similar to it confirm Hypothesis 4 for PBayes, e.g., in this figure MSE(PBayes, PTruth) never goes
significantly above 0.05. On the other hand, the predictive ability of PBern degrades a lot in easy
environments, e.g., MSE(PBern, PTruth) → 0.09 as p → 0.9 in Figure 15. PBayes with a uniform (but
not strong correct) prior performs worse than PBern in the most difficult environment (p = 0.1),
but both perform well at that extreme.
Figure 16 is typical of the second trend. In the set of graphs typifying this trend, both the PBern
and PBayes with uniform prior curves have a similar shape, but the MSE(PBern,PTruth) is much
higher for moderate environments than the MSE(PBayes,PTruth) with a uniform prior. Again, PBayes
with a uniform prior performs worse than PBern when the environment is most difficult, but
otherwise it performs better than PBern.
We conclude that when averaged over all environments (which is an uncontrollable
parameter), PBayes appears to be preferable. PBern shows a lack of robustness that can make it
barely acceptable in moderate to easy environments.
Finally, note that both Figure 15 and Figure 16 reinforce the value of a strong, almost correct
prior – if feasible to obtain. The MSE is always lower, and often much lower, when the prior is
well-matched to the environment.

Figure 15. Graph of MSE(PBern, PTruth) and MSE(PBayes, PTruth), with inter-agent forces, as
environmental difficulty varies. The first trend (Rebguns et al., 2008)
(© 2008, IEEE. Used with permission.)



Figure 16. Graph of MSE(PBern, PTruth) and MSE(PBayes, PTruth), with inter-agent forces, as
environmental difficulty varies. The second trend (Rebguns et al., 2008)
(© 2008, IEEE. Used with permission.)

Figure 17. Graph of MSEs with p = 0.55, n = 300, k ranging from 3 to 15, with no inter-agent
forces (Rebguns et al., 2008) (© 2008, IEEE. Used with permission.)



Figure 18. Graph of MSEs with p = 0.51, n = 300, k ranging from 3 to 15, with inter-agent
forces. Identical to Figure 10 but shown here again for ease of comparison with Figure 17

INDEPENDENT VERSUS DEPENDENT AGENTS: EXPERIMENTAL RESULTS
The most interesting result obtained from our experiments is that although our two formulas
make independence assumptions, when these assumptions are violated (and the LJ force law is
used) the predictions of both formulas usually improve! Figures 17 and 18 illustrate this result.
All parameter values are identical (and the value of p is nearly identical) in these two figures,
other than inter-agent (in)dependence (which is the same for both the scouts and the swarm).
This result is both surprising and exciting. It is surprising because any mathematical formula
will typically be more accurate when its assumptions hold, and are not violated. Note that we are
not stating that the swarm performance is better when the agents are dependent – that would be
intuitively obvious. Rather, we are stating that the formulas’ predictions are closer to “ground
truth,” which is not at all obvious. In hindsight, we have a better intuitive understanding for why
dependent scouts are so adept at predicting the performance of dependent swarm agents.
In the case of independent agents, the only forces affecting agents are the attractive goal force,
which is felt from any distance, and the repulsive forces of obstacles, which are felt when an
agent gets close enough to an obstacle. In this situation the initial (starting) position of every
agent in the swarm is very important. Figure 19 illustrates the independent agents case. (Here, as
in simulation screen-shots in Figures 4 and 5, a square denotes the goal, big circles denote
obstacles, and small circles denote agents; arrows show the direction of movement that an agent
is likely to take.) It is very likely that agents 1, 2 and 3 will get stuck behind the cul-de-sac that
lies between them and the goal, and the rest of swarm will successfully reach the goal. If we
send out three scouts that start out from the same positions as agents 1, 2 and 3, and they also get
stuck, then this is certainly not representative of the swarm. The variance is high – as you change
the scout starting positions, the results differ drastically. With inter-agent forces enabled, on the
other hand, the starting position does not matter as much; variations in the starting positions do
not affect how the swarm as a whole moves. (See Figure 20; small lighter dots are other possible
starting positions.) Another way to think about it is as if we were reducing the size of the starting
square, and hence reducing the variance. Reduced variance improves the results.
Another explanation of this phenomenon is related to the Lennard-Jones force law used for
dependent agents. LJ produces swarm behavior similar to the flow of a liquid. Liquid agent
movement is advantageous because:
1. A small drop of liquid behaves similarly to a large body of liquid. Analogously, scouts
behave very similarly to the swarm when using LJ.
2. Liquid particles interact, whereas gas particles do not. Interacting particles (or agents)
are more predictable than independent ones.
Interacting particles can behave like a liquid or a solid. To test whether LJ liquid-like forces
improve the predictions with dependent agents, we applied the Newtonian virtual force law of
Spears and Gordon (1999), which generates solid-like swarm behavior. Our results (which agree
with the earlier results of Hettiarachchi (2007)) demonstrate that small groups of agents can
squeeze through tight paths between obstacles, whereas larger solid-like groups get stuck. This
implies that scouts and swarms will often behave differently when using a Newtonian force. In
fact, when using the Newtonian law our formulas do not produce better predictions with
dependent than with independent agents. Therefore, our choice of the Lennard-Jones force law,
in particular, is responsible for the outstanding predictive accuracy of our swarms with
dependent agents.

Figure 19. Independent agents; initial position is very important






Figure 20. Dependent agents; initial position not as important

EXPERIMENTAL CONCLUSIONS
The following results/conclusions hold for both independent and dependent agents:
1. Both formulas are usually good predictors. Their accuracy can be seen in all of our
graphs.
2. Observations about prior distribution selection for our Bayes formula are:
 It is better to have a correct rather than an incorrect prior.
 If the prior is incorrect it is better to be weak rather than strong. Uniform is the
best incorrect prior.
 If the prior is correct it is better to be strong than weak.
3. PBayes with a uniform prior is usually better (i.e., has a lower MSE) than PBern, and is often
much better. This conclusion has significant practical implications. It means that even in
the case when prior knowledge is absent, PBayes typically outperforms PBern. All graphs in
this paper that compare PBayes with PBern provide evidence to support this conclusion. Of
course, theoretically we know that as the number of scouts gets large, the difference
between PBayes and PBern will vanish to 0. Therefore, the advantage of PBayes is for
situations of few scouts.
4. Increasing the swarm size n with a proportional increase in the number of scouts (i.e.,
holding the ratio k/n constant) has a negligible effect on predictability.
5. PBayes is “robust” across wide variations in environmental difficulty. By “robust” we
mean that the MSE does not substantially increase as the environmental difficulty is
varied. PBern is less robust than PBayes. In particular, when averaged over all environments
(which is an uncontrollable parameter), PBayes appears to be preferable. PBern shows a lack
of robustness that can make it barely acceptable in moderate to easy environments.

6. With the Lennard-Jones force law, PBern and PBayes predict better when the agents are
dependent, i.e., when their independence assumptions are violated.

PORTING TO REAL ROBOTS
Our theoretical framework is a general approach to risk assessment for swarm applications. It
models agents at an abstract level. Agents are “abstract” because the types and levels of sensor
and actuator noise, vehicle limitations and dynamics, and the environmental details are not
modeled. However, note that in terms of the probability of success, which is the meta-parameter
of variation here, noise that causes performance degradation is equivalent (at an abstract level) to
a “hard” environment, which is in fact modeled in our simulator. Furthermore, if AP is used then
we have confidence (based on applying AP with real robots) that the success rate will not be
significantly adversely affected by sensor noise, even if the noise level is quite high (e.g., see
Maxim et al. (2009) for robust simulated multi-robot chain formations despite 50% sensor noise
and 50% motor noise, and robust real multi-robot chain formations despite unknown but very
high noise levels; furthermore, W. Spears and D. Green - personal communication - found AP to
be exceptionally robust in simulated swarm optimization problems where throughout the
environment the sensor noise exceeds the signal).

Therefore, it is our hypothesis that our scouts approach will carry over to realistically
simulated robots, as well as to real robots, provided a sampling assumption holds. This sampling
assumption states that the scouts are “typical” of the swarm, i.e., the scout vehicle imperfections
closely match those of the swarm vehicles. If the scouts are randomly selected from the pool of
swarm vehicles, then this assumption will be likely to hold. Our hypothesis of portability to real
robots (or even realistically-simulated robots) has not yet been tested, but will be done as future
work.

CONCLUSION
Our theoretical framework is very useful for predicting the success rate of a swarm of agents
before it is deployed, especially in difficult or dangerous situations -- when sending the entire
swarm at once is inadvisable and may result in a loss of a substantial part of the swarm. In other
words, our scouts framework has significant practical implications for agent swarm risk
assessment.
As part of our framework, we introduced a novel Bayesian formula for success probability
calculation, along with its mathematical derivation (in Appendix B). The derivation shows how
this formula is a Bayesian variant of the standard Bernoulli trials formula for predicting success
rate, and how it employs a prior distribution to reduce the variance and increase the robustness of
the formula’s predictions. We have designed and implemented a swarm agent simulator that uses
both Bernoulli and Bayesian formulas to predict the swarm success rate. The simulator has
parameterization capabilities for running rigorous experiments. We have implemented and
executed a full factorial experimental design for evaluating both the Bernoulli and Bayesian
formulas, in the context of the navigation task. Both formulas were compared against “ground
truth,” using the mean-squared error as an evaluation metric. The results were analyzed and
presented along with several important conclusions, including the accuracy and robustness of our
Bayesian formula as compared to the standard Bernoulli trials formula, and the effectiveness of
both formulas for swarms of interacting agents using a Lennard-Jones force law. The latter
conclusion is surprising because both formulas assume independence; we have provided an
explanation for this surprising result.
Future work includes transitioning our framework to a swarm of real robots. We also plan to
design strategies for increasing the success rate when a low rate is predicted. We could, for
example, record successful scout paths for the swarm members to follow. In this case, scouts
would not only predict the success rate for the swarm, but they would also provide advice to the
swarm regarding which paths are preferable and which should be avoided. We could also add
further complexity to the interactions between agents, e.g., by merging the DAEDALUS multi-
agent online evolution approach described in Hettiarachchi (2007) with scouts and path planning.
Another important direction for future research would be to apply our predictions to behavior-
based, rule-based and biomimetic swarms, and compare on other (e.g., benchmark optimization)
problems. So far we have seen that predictions improve when coupled with a physicomimetic
control strategy for inter-agent cooperation, but it would be useful to see if the same holds for
other control strategies.
In conclusion, our theoretical framework that uses scouts for predicting swarm success rate is
an important contribution that could, in the future, be complemented with a variety of control
strategies, as well as with the use of scouts for transmitting additional learning and planning
information to the swarm. Future work will focus on these directions, as well as application to
other tasks.

ACKNOWLEDGEMENT

This research was funded in part by the Office of the Secretary of Defense Joint Ground
Robotics Enterprise (JGRE) program.

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Antons Rebguns is working on his PhD in Computer Science at the University of Arizona in
Tucson, AZ. He received his BSc in Electrical Engineering from Riga Technical University, and
his MS degree in Computer Science from the University of Wyoming. This article is partially
taken from his unpublished MS thesis, entitled "Using scouts to predict swarm success rate." His
primary research interests are in multi-agent architectures, swarm robotics, planning, learning,
cognitive science, cognitive architectures and artificial intelligence. Currently his research
focuses on instructable computing with planning.
Diana Spears received her MS and PhD degrees in Computer Science from the University of
Maryland. She is currently an owner and senior research scientist at Swarmotics, LLC, and an
Adjunct Associate Professor in the Department or Mathematics at the University of Wyoming
(UW). Previously, she worked at NASA Goddard Space Flight Center, the National Institute of
Standards and Technology, and as an Associate Professor of Computer Science at UW. She is
known internationally for her expertise on swarm robotics and multi-robot chemical plume
tracing. Her other research interests include behaviorally assured adaptive and machine
learning systems, and mathematical/graphical modeling and model reconstruction.
Richard Anderson-Sprecher is a professor in the Department of Statistics at the University of
Wyoming, Laramie, Wyoming, USA. He received his BA in Mathematics from Carleton College,
Northfield, MN, his MS in Mathematics from the University of Minnesota, Minneapolis, MN, and
his PhD in Statistics from the University of Iowa, Iowa City, IA. His primary research interests
are statistical applications and time series analysis.
Aleksey Kletsov is a Teaching Assistant Professor in the Department of Physics, East Carolina
University, USA. He received his BSc in Physics from Saratov State University, Russia, PhD in
Physics and Mathematical Sciences from Saratov State University, Russia, and PhD in Electrical
Engineering from University of Wyoming, USA. His primary research interests are
computational nanoscience and nonlinear dynamics of global processes.

APPENDIX A: DERIVATION OF PRIORS
This appendix is from Rebguns (2008). The following methodology is used to vary prior
strength and correctness. We quantify the “strength” of a prior as 1/σ2, where σ2 denotes the
variance, and the correctness of a prior as the degree to which the prior is well-matched to
“ground truth” (p). Recall from above that we use a Beta distribution for our priors, which has
parameters α and β. When we vary these two parameters the shape of the probability density
function (p) changes. The mean of this Beta distribution is    
and the variance is
2 
2( ) ( 1)

     
2(1 )
( )
 
 
 
2(1 )
( 1 )
 
 
  
. It is expected that in actual usage, the prior
belief will be unimodal. The weakest unimodal prior with mean µ has one parameter equal to 1
and the other ≥ 1:
µ < 0.5: 11,    
 
, and 2
2
2
(1 )
( 1) ( 2) (1 )
     
   

µ ≥ 0.5:
,  11
  
, and 2
2
2
(1 )
( 1) ( 2) (2 )
     
   

To strengthening the prior, we shrink the variance by a factor C:
µ < 0.5: To reduce the variance from 2(1 )
(1 )
 



to 2(1 )
(1 )C
 



we solve the equation 2
*
(1 )
( )
 
 


=
2(1 )
(1 )C
 



for α* and then solve for
* *1  

. The results are
* ( (1 ) )C     and
* 1 ( (1 ) )C  
  
.
µ ≥ 0.5: To reduce the variance from 2(1 )
(2 )
 



to 2(1 )
(2 )C
 



we solve the equation 2
*
(1 )
( 1 )
 
 

 

= 2(1 )
(2 )C
 



for β* and then solve for
* *
1
  
. The results are
* ( (2 ) (1 ))1 C
     
and
* ( (2 (1 ))C     .
Using the solutions to these equations, we have selected α*, β*, and C to find six priors (plus
uniform makes seven) that represent a good variability along the strength and correctness
dimensions.

APPENDIX B: BAYESIAN FORMULA AND DERIVATION
This appendix, which is from Rebguns et al. (2008), provides our derivation of the PBayes
formula. Here, we use f to denote probability functions, and we employ the common statistics
convention of letting the argument determine which probability function is meant by f, i.e. f
varies depending on the context. Also, we use π rather than f for the prior distribution in
particular, as is traditional in the statistics literature.
The intuition behind the derivation of the Bayesian formula is as follows. We start by deriving
the probability of exactly x out of n successes, given our estimate pˆ from the fraction of k scouts
that successfully reached the goal, using the standard Bernoulli trials formula. We apply the Beta
distribution prior assumption, and perform mathematical simplification operations on it to get a
simplified version of the formula for exactly x out of n successes. To find the probability P(Y ≥
y) of y or more successes (where y is a lower bound on the desired number of successes), we next
take the sum of this expression, from x equals y to x equals n. Applying the relation between the
cumulative Binomial probability and the incomplete Beta function, along with other
mathematical operations, we simplify the expression to get our final formula. The full detailed
formula derivation follows.
Let Y be a binomial random variable, where | ~ ( , )Y p n p and p is the probability of one
agent reaching the goal and n is the total number of agents. Let pˆ be an estimate of p based on k
trials. Then ˆ | ~ ( , )kp p k p, and these variables are independent. From the Bernoulli trials
formula, we have:
( | ) (1 )x n xnf x p p px
    

ˆ ˆ(1 )ˆ( | ) (1 )ˆ
kp k pkf p p p pkp
    

Assume that ~ ( )p p with parameters α and β, so that the prior distribution
1 11( ) (1 )( , )p p pB
   
  
. We can now calculate ˆ( | )P Y y p , abbreviated P(y). Let us
begin with the probability of exactly x successes, given pˆ :
1 1
0 0
1 1
0 0
ˆ ˆ( , , ) ( , | ) ( )ˆ( , )ˆ( | )
ˆ( ) ˆ ˆ( , ) ( | ) ( )
f x p p dp f x p p p dpf x pf x p
f p f p p dp f p p p dp


   
 

by the definition of conditional probability and the product rule. Now, expand the numerator:
1 1
0 0ˆ ˆ( , | ) ( ) ( | ) ( | ) ( )f x p p p dp f x p f p p p dp  

by the conditionalized product rule. Substitution yields:
1 ˆ ˆ(1 ) 1 1
0
1(1 ) (1 ) (1 )ˆ ( , )
x n x kp k pn kp p p p p p dpx kp B
 
 
               

Then, expand the denominator:

1
0 ˆ( | ) ( )f p p p dp 

1 ˆ ˆ(1 ) 1 1
0
1(1 ) (1 )ˆ ( , )
kp k pk p p p p dpkp B
 
 
        

1 ˆ ˆ1 (1 ) 1
0
1 (1 )ˆ ( , )
kp k pk p p dpkp B
 
 
          

ˆ ˆ( , (1 ) )
ˆ ( , )
k B kp k p
kp B
 
 
      

Because the denominator is free of x, we will temporarily refer to it as D. Now, for exactly x
success we have:
1
0
ˆ( , | ) ( )
ˆ( | )
f x p p p dp
f x p D

 

1
0
1 ˆ,   ( ) ( , | ) ( )
n
x y
Thus P y f x p p p dpD  

1 ˆ ˆ(1 ) 1 1
0
1 1(1 ) (1 ) (1 )ˆ ( , )
n
x n x kp k p
x y
n kp p p p p p dpx kpD B
 
 
   

            

1 ˆ ˆ(1 ) 1 1
0
1 1 (1 ) (1 ) (1 )ˆ ( , )
n
x n x kp k p
x y
k n p p p p p p dpkp xD B
 
 
   

            

Now apply the relation between the cumulative Binomial probability and the incomplete Beta
function:
(1 ) ( , 1)
n
x n x
p
x y
n p p I y n yx


        

1
0
1 (1 )( , 1)
p y n yz z dzB y n y
   

and therefore
1 1( ) ˆ ( , ) ( , 1)
kP y kpD B B y n y 
      

1 ˆ ˆ1 1 (1 ) 1
0 0 (1 ) (1 )p y n y kp k pz z p p dzdp         

Returning to D,
1 1
ˆ ( , ) ( , 1)
k
kpD B B y n y 
      

1ˆ ( , ) ( , 1)
ˆ ˆ( , (1 ) )
ˆ ( , )
k
kp B B y n y
k B kp k p
kp B
 
 
 
 
 
   
    
 
 

1
( , 1)
ˆ ˆ( , (1 ) )
B y n y
B kp k p 
  
  

1
ˆ ˆ( , 1) ( , (1 ) )B y n y B kp k p     

Therefore,
1 ˆ ˆ1 1 (1 ) 1
0 0
(1 ) (1 )
( ) ˆ ˆ( , 1) ( , (1 ) )
p y n y kp k pz z p p dzdp
P y B y n y B kp k p
 
 
       
     
 
When simplified, this becomes our Bayesian formula from Section 3 above:
1 1 1 1
0 0( ) · (1 ) (1 )p y n y r sP y M z z p p dzdp      
1where       { ( , )· ( , 1)}M B r s B y n y   
ˆ   r kp  
ˆ    (1 )s k p   


i These numbers of pixels were chosen because they led to a reasonable running time and helped the agents and
their path show up nicely on the graphical interface.