Uncertainty in Probabilistic Trust Models
Sadegh Dorri Nogoorani, Rasool Jalili
Department of Computer Engineering
Sharif University of Technology
Tehran, Iran
e-mails: [email protected], [email protected]
Abstract—Computational models of trust try to transfer the
concept of trust from the real to the virtual world. While such
models have been widely investigated in the past decade, the
uncertainty involved in trust computation has been overlooked
in the literature. In this paper, uncertainty of probabilistic trust
models is quantified using confidence intervals and its factors
are determined through simulation. The results confirm the
importance and highlight the amount of uncertainty in the Beta
and HMM (Hidden Markov Model) trust models. In addition,
an uncertainty-driven method is proposed which reduces the
risk involved in the trust-based utility maximization according
to uncertainty.
Keywords-probabilistic trust model; uncertainty; risk reduction;
I. I NTRODUCTION
Trust plays an important role in coordinating interpersonal interactions when there is a possibility of risk.
It enables a more accurate assessment of the risk, and
relaxes the strict traditional control mechanisms. Various
computational models of trust have been proposed to transfer
such a concept to the virtual world and the idea has been
successfully applied to security, routing, and collaborative
solutions.
In spite of such promising developments, many of the
proposals suffer from the fundamental problem of ignoring
uncertainty or improperly handling its factors [1], [2]. Without a proper consideration and evaluation of uncertainty,
the result of trust assessment cannot be relied on. Our
simulations in Section IV confirm that the well-known Beta
trust model [3] has a great amount of uncertainty. Even a
recent extension of this model [4], requires a relatively long
history of past interactions in order to reach an acceptable
degree of certainty.
In this paper, the uncertainty of probabilistic trust models
is analyzed and quantified in the form of confidence interc 2012 IEEE. Personal use of this material is permitted. Permission from
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Citation: S. Dorri Nogoorani and R. Jalili, “Uncertainty in Probabilistic
Trust Models,” 26th IEEE Int’l Conf. Adv. Info. Networking and Applications, Japan, Mar. 2012, pp. 511–517. DOI 10.1109/AINA.2012.73
vals. As two concrete examples, the Beta and HMM (Hidden
Markov Model) trust models are studied and their uncertainty factors are identified using simulation experiments.
In order to calculate the associated confidence intervals, the
bootstrapping method [5] is applied. The method is almost
general and can be used in other scenarios with similar trust
models.
We also propose an uncertainty-driven risk reduction
method which takes into account the risk involved in utility maximization. The applicability of the method is also
investigated on the simulation results.
The remainder of this paper is organized as follows. In
Section II, the uncertainty in probabilistic trust models is
discussed in more detail. In Section III, two trust models are
described and their uncertainty is analyzed in Section IV. In
Section V the uncertainty-driven risk reduction method is
proposed. Related work is discussed in Section VI and the
paper is concluded through Section VII.
II. U NCERTAINTY AND T RUST
Various kinds of uncertainty are associated with different
stages of trust evolution and application (observation, modeling, and prediction). Observations are inherently uncertain
as their possible error cannot be simply ignored. Trust
models are prone to uncertainty due to the existence of some
degree of abstraction or simplification in their construction.
Moreover, the trustee is not fully observable and model
inputs are barely samples of reality. The predictions of trust
models, even the most flexible ones, have some output errors
and are uncertain.
A. Basic Notation
This paper is concentrated on probabilistic uncertainty,
and other forms of uncertainty are left for further research.
Therefore, the trustee behavior is characterized by a probability distribution into which all the information about the
trustee is fed. In this case, trust is based on the probability
of the outcome of future interactions with trustee.
Assume that the outcome of an interaction can be represented by a binary variable taking one of the values of x
or x̄. Also assume that trust is based on the history of past
interactions, only. Accordingly, the probability of success in
a future interaction of a trustor tr with a trustee te at any
time instance t can be calculated from
In particular, the four arithmetic operations on two intervals
I1 = [a, b] and I2 = [c, d] are defined as
ptr,te
= Pro(Ottr,te = x | Httr,te ),
t
(1)
I1 + I2 = [a + c, b + d],
(4)
where Ottr,te represents the future outcome, and Httr,te =
{Ot1 , . . . , Otn } (t1 < t2 < . . . < tn < t) the history of
interactions until t.
Usually a Bayesian approach is taken and ptr,te
itself is
t
assumed to be a random variable following a probability
distribution. Therefore, the problem of trust assessment is
translated into estimating the expected probability of success. Hence, the trust of tr on te at time t is evaluated
using
I1 − I2 = [a − d, b − c],
(5)
τttr,te = E [ptr,te
].
t
(2)
I1 × I2 = [min {ac, ad, bc, bd}, max {ac, ad, bc, bd}], (6)
I1 /I2 = I1 × [1/d, 1/c]
(0 6∈ I2 ).
(7)
Note that these operators do not take dependencies between the values represented by the intervals into account.
Consequently, dependencies should be resolved before using
them.
III. C ASE S TUDIES
The super/subscripts in the above formula emphasize on
the subjectivity (the point of view of a specific trustor tr),
and time-dependency of trust. From now on, a specific
trustor tr, trustee te, and time instance t are implicitly assumed and the super/subscripts are only used for clarification
where different parties or another time instance are involved
in a formula.
In this section, the well known probabilistic Beta trust
model (proposed in the Beta reputation system) [3], and
one of its extensions, HMM (Hidden Markov Model) trust
model [4] are used to evaluate and exemplify the uncertainty
of trust assessment.
B. Quantifying Uncertainty
In the Beta trust model, behavior of the trustee is assumed
to follow a Bernoulli distribution with a different success
probability for each trustee. Hence, the probability of success
(p) follows the Beta distribution [3]:
In this paper, confidence intervals are used to represent
uncertainty of the result of trust assessment. For example, the
0.95 confidence interval of [0.4, 0.6] means that the real trust
value is possibly between 0.4 and 0.6 with the probability
of 0.95. Confidence intervals are widely used to represent
uncertainty of measurement information, and have a clear
probabilistic interpretation. Additionally, there is an almost
general method (the bootstrapping [5]) to calculate them.
This is in contrast with some ad hoc uncertainty measures
which are bound to specific factors, or do not have a clear
interpretation.
If H = {O1 , O2 , . . . , On } is the history of interactions
of tr with te, (2) is used to build a point estimator of τ .
On the other hand, an interval estimator of τ is the interval
∆τ = [τ1 , τ2 ] where τ1 and τ2 are functions of H. We refer
to [τ1 , τ2 ] as the δ confidence interval of τ if
Pro(τ1 ≤ τ ≤ τ2 ) = δ,
(3)
where the constant δ is the confidence coefficient of the
estimated trust (usually 0.95 or 0.99). The smaller the
confidence interval is, the more certain is the estimation of
trust τ . In this regard, the length of the confidence interval
is a good measure of uncertainty.
C. Propagation of Uncertainty
If the trust value is to be used in a calculation, interval
arithmetic is used to propagate its uncertainty to the result.
A. The Beta Trust Model with Forgetting Factor
Γ(α + β) α−1
p
(1 − p)β−1
Γ(α)Γ(β)
where α = r + 1, β = s + 1
f (p | α, β) =
(8)
(9)
where r and s are respectively the number of past successful
and unsuccessful interactions between the trustee and trustor.
Hence, the trust can be estimated using [3]
τ=
α
r+1
=
.
α+β
r+s+2
(10)
In order to account for changes in the trustee behavior, a
forgetting factor λ ∈ [0, 1] is introduced in the model which
controls the effect of past history on τ according to (11)
and (12) [3]:
rtn = rt(n−1) .λ + I{x} (Otn ),
(11)
stn = st(n−1) .λ + I{x̄} (Otn ),
(12)
where IA (.) is an indicator function checking if the outcome
is (un)successful, and rti (or sti ) is the value of r (or s) after
feeding the ith interaction outcome into the model. With
λ = 1, all interactions have the same effect (no forgetting)
while with λ = 0, trustor is very forgetful and no history is
taken into account.
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B. HMM Trust
In the HMM trust model, a 2-state Hidden Markov Model
Ω = (Q, π, A, R, B) is used to keep pace with dynamics
of the trustee behavior. In this model, the probability of a
successful outcome is again assumed to follow a Bernoulli
distribution but in this case dependent on the current internal
state of the trustee. The parameters of the model are:
• Q = {s0 , s1 }: the set of (hidden) states (e.g. being
malicious or benevolent),
• π = {π0 , π1 }: the initial probability of being in each
state,
• A = {Ai,j | i, j ∈ {0, 1}}: the probability of a
transition between two states (from i to j),
• R = {x, x̄}: the possible outcomes,
• B = {Bi | i ∈ {0, 1}}: the probability distribution of
the possible outcomes in each state (Bernoulli).
In the HMM model, the trustor uses the history of
past interactions with the trustee to calculate (estimate) the
parameters π, A, and B of the HMM by the Baum-Welch
algorithm (see [6] for a tutorial on Hidden Markov Models
and related algorithms). Having the estimated model of
trustee, the probability of success in the future interaction is
calculated using [4]
p = Pro (O = x | H, Ω) =
Pro (O = x, H | Ω)
,
Pro (H | Ω)
(13)
where Ω is the estimated trustee HMM. Accordingly, the
expected probability of success (trust) can be calculated
using the Forward-Backward algorithm [6].
C. Theoretical Analysis
In order to analyze the uncertainty of the Beta and HMM
models, the confidence intervals associated with their trust
estimates must be calculated. Bearing in mind that τ in the
Beta model follows the Beta distribution (8), the bounds of
the δ confidence interval ∆τ = [τ1 , τ2 ] of the Beta model
can be calculated from
τ1 = F−1
α,β (c),
(14)
F−1
α,β (c
(15)
τ2 =
+ δ),
max{F(τ ) − δ, 0} ≤ c ≤ min{F(τ ), 1 − δ},
(16)
where Fα,β and F−1
α,β are the Beta cumulative distribution
function (CDF) and its inverse respectively, and τ is the
estimated trust value. c is an arbitrary constant real number
(e.g. c = 1−δ
2 ) which controls the relative position of τ
in the interval. The constant is constrained to (16) in order
to ensure that the interval is a probability interval (subset of
[0, 1]) around τ . The parameters α and β are calculated from
H, and there are efficient computational methods which
can be used to calculate Fα,β and F−1
α,β . The equations
can be straightforwardly derived from (3) using the general
properties of CDFs.
In order to take advantage of (14) and (15), the real
behavior of the trustee must be assumed to follow a fixed
known probability distribution (the Beta distribution in this
case). Additionally, this method do not produce a closedform formula which can be mathematically analyzed. Another disadvantage of these equations is that they are not
practical for the HMM model because the CDF of p (and
its inverse) in (13) cannot be determined efficiently.
There are approximations to the confidence interval under
normality (of the estimator) and independence (of observations) assumptions which cannot be used too because
normality is already violated by (8), and a fundamental
assumption of the HMM model is the dependence of observation outcomes. Hence, we use the bootstrapping method
to determine the confidence intervals.
D. The Bootstrapping Method
Bootstrapping is a resampling-based method to measure
accuracy of almost any statistic using a simple general
procedure [5]. In order to compute the confidence interval of
τ based on the history of interaction outcomes H of length
n, the following steps must be followed in the bootstrapping
method:
1) The trust value τ is estimated according to H and the
trust model in use.
2) A new bootstrap sample H ∗ of length n is randomly
chosen with replacement from the empirical distribution of outcomes in H.
3) A new trust value τ ∗ is calculated according to H ∗
and the trust model.
4) The steps 2 and 3 are repeated B times (B is a large
number at least 1000) and the respective bootstrap
estimates τ ∗ (1), . . . , τ ∗ (B) are calculated.
5) The confidence interval of τ is determined according
to the (empirical) distribution of τ ∗ (.) (e.g. by calculating its percentiles). Before that, the distribution
must be shifted so that its mean becomes equal to τ
(in order to ensure the interval forms around τ ).
Note that the bootstrapping method is supposed to analyze
the (possible) deviation of the estimator around a specific
estimation. Hence, the relative distribution of τ ∗ (.) around
its mean is of concern in the aforementioned procedure, not
the absolute distribution.
This method approximates the real unknown or complex
distribution of τ with the resampling technique. Our coverage analysis shows that the resulting confidence intervals
are nearly tight. Using a larger B reduces the resampling
error; however, it does not eliminate all the errors and does
not produce an exact confidence interval. The bootstrapping
method is based on few assumptions and is very general,
therefore can be applied to other similar trust models.
The bootstrapping method in contrast with the analytic
method of (14) and (15) is applicable to estimators with a
complex or unknown probability distribution (such as the
513
B0 (x) = 1.0
B0 (x̄) = 0.0
b,w
1−s
3
b,f
B1 (x) = 0.7
1−s
3
π0 = 0.25
1−s
3
B2 (x) = 0.3
B2 (x̄) = 0.7
m,f
π2 = 0.25
π1 = 0.25
1−s
3
1−s
3
1−s
3
m,w
λ = 0.1
λ = 0.3
λ = 0.5
λ = 0.7
λ = 0.9
HMM
0.1
B1 (x̄) = 0.3
8 · 10−2
6 · 10−2
B3 (x) = 0.0
B3 (x̄) = 1.0
π3 = 0.25
4 · 10−2
2 · 10−2
0.2
Figure 1. The HMM which simulates the trustee behavior (Ω0 ). State
labels: b/m: benevolent/malicious, w/f: working, faulty.
result of the forward-backward algorithm in the HMM trust
model). However, it does not produce a closed-form formula
again.
We have also used a variant of the basic bootstrapping
(described earlier) which is called parametric bootstrapping.
In this variant, the bootstrap samples (step 2) are chosen
from the parametric distribution of the estimated model (the
trust model instantiated with τ ).
Although the bootstrapping method imposes an overhead
proportional to B on the computation of trust, the assessment
of uncertainty is very valuable to decision processes. We
believe that the overhead is tolerable with regard to the
nowadays computationally powerful computers and devices.
Besides that, uncertainty information can be used to reduce
costs and overhead in other aspects of trust computation.
IV. S IMULATION R ESULTS AND D ISCUSSION
The Beta trust model with various forgetting factors has
already been compared to the HMM model with respect
to prediction error [4]. However, we studied the effect of
different settings and various history sizes on the uncertainty
of the models, as well as their error.
0.4
0.6
0.8
Stability (s)
Figure 2. Average relative entropy error of the Beta and HMM models
with 300 observations.
·10−2
λ = 0.1
λ = 0.3
λ = 0.5
λ = 0.7
λ = 0.9
HMM
3
2
1
0
0
200
400
600
Observation Count (n)
Figure 3. Average relative entropy error of the Beta and HMM models
with s = 0.4.
in 100 independent rounds and average of the results is
reported. Without loss of generality, the 0.95 confidence
interval is of interest in all simulations.
A. Simulation Setting
The behavior of the trustee is simulated using a 4-state
HMM Ω0 (similar to the one used in [4]) depicted in
Figure 1. The transition probabilities of Ω0 are dependent
on the stability of trustee (s ∈ [0, 1]). The more stable the
trustee, the less probable is a transition to another state.
In each round of a simulation, a sample outcome sequence
(history) of specified length is generated from Ω0 and
performances of the trust models are compared with respect
to the sample. Their accuracies are compared using the
relative entropy [4] (with reference to the real distribution
calculated from Ω0 and the distribution suggested by each
trust model). To compare uncertainties of the models, the
length of their confidence intervals are considered.
The basic bootstrap with B = 1000 is used to compute
the confidence interval of the Beta model and a parametric
bootstrap [7] for the HMM model. Each simulation is run
B. Relative Error
The effects of changing stability on relative entropy error
is depicted in Figure 2. Not surprisingly, the HMM model
has relatively less error than the Beta model. However, as
stated in [4], the performance of the HMM model diminishes
by increasing stability, and in some settings the Beta model
performs better. Nevertheless, the trustor cannot use this
information to tune his/her model because stability and
relative entropy error are hidden from the trustor.
In Figure 3, the effect of the observation count (n) is
studied on the error of the models (not studied in [4]). The
Beta model does not have a consistent behavior in response
to changes in this respect. The HMM model has greater error
for small ns. However, in contrast with the Beta model, the
error rapidly decreases as n increases. With n < 100, the
two models have relatively the same amount of error whereas
514
λ = 0.1
λ = 0.3
λ = 0.5
λ = 0.7
λ = 0.9
HMM
0.5
0.4
0.3
0.2
factor (in most settings, the length is greater than 0.35).
In contrast, the HMM model gives more certain results
with more observations. However, the performance of the
HMM model is very unsatisfiable with a small number of
observations. For example, with n ≤ 30, the interval length
exceeds 0.45. More specifically, with n < 75, the Beta
model is more certain while the HMM model is preferred
with more observations.
D. Discussions
0.2
0.4
0.6
0.8
Stability (s)
Figure 4. Average confidence interval length of the Beta and HMM models
with 300 observations.
0.8
λ = 0.1
λ = 0.3
λ = 0.5
λ = 0.7
λ = 0.9
HMM
0.6
0.4
0.2
0
200
400
600
Observation Count (n)
Figure 5. Average confidence interval length of the Beta and HMM models
with s = 0.4.
with more observations, the HMM model is superior to the
Beta model.
C. Uncertainty of the Models
Figure 4 shows the effect of stability as well as the model
parameters on confidence interval length (which has inverse
relationship with certainty). According to this figure, the
HMM model calculates trust with higher certainty because
the interval length of the HMM model is very smaller than
that of the Beta model. The Beta model has very disappointing certainty specially with λ = 0.5, 0.7, and the best results
belong to λ = 0.1. Even in this case (λ = 0.1), interval
length is still greater than 0.35, compared to 0.11 ∼ 0.13
for the HMM model (less than half). Moreover, the interval
length in all models shows a very small decrease with
increasing stability, and is almost independent of stability
(in the Beta model) or weakly dependent on it (in the HMM
model).
Figure 5 shows the effect of observation count on the
uncertainty (confidence interval length). It confirms that
the certainty of the Beta model is not dependent on this
According to the performance results of the Beta and
HMM models, uncertainty of the Beta model is independent
of observation count and the stability of the trustee, yet
nonmonotonically dependent on the forgetting factor. By
comparing these results with relative entropy error of the
model, we conclude that a trustor with a Beta trust model,
has no trivial option to decrease uncertainty of its trust
estimations.
On the other hand, uncertainty of the HMM model weakly
depends on the stability of the trustee, and is directly related
to observation count. Therefore, a trustor with an HMM
trust model can decrease its uncertainty by testing trustee
in unimportant situations, or consulting other agents who
may have more experience with the trustee. In this way, the
trustor will reach a more certain as well as a more accurate
estimation of trust.
These results are very valuable to the trustor because
confidence intervals are based solely on the information
available to the trustor and can be computed in real time
with a moderate overhead.
In the following section an uncertainty-driven risk reduction method is proposed which can be used to take advantage
of uncertainty information.
V. U NCERTAINTY-D RIVEN R ISK R EDUCTION
In a multi-agent environment, a rational agent acts as
though it is maximizing a utility function [8]. The utility
in a trust-based agent, or trustor, is dependent on trust
relationships between agents as well as the application
specific factors. Particularly, trust supports the agent in a
better estimation of the other agents’ behavior, while the
application determines the utility function of the trustor.
An straightforward strategy is to select the trustee with
the maximum expected utility. In this strategy, the expected
utility of an interaction is calculated according to
U = E [u] = E [O = x].u(x) + E [O = x̄].u(x̄) (17)
= τ.u(x) + (1 − τ ).u(x̄)
(18)
= τ.(u(x) − u(x̄)) + u(x̄),
(19)
where u(.) is the application-specific utility function that
maps the possible outcomes to their corresponding supposed
utility.
Practically, utility on its own is insufficient to make a
rational decision, and the more risky a decision is, the
515
Table I
M AX . ACCEPTABLE U NCERTAINTY FOR C RITICALITY C LASSES .
Criticality
Low
Medium
High
Max. Confidence Interval Length
15
10
5
more cautious the trustor must be to make it. Risk has
direct relationship with criticality and inverse relationship
with certainty. Criticality of a decision is dependent on the
situation and cannot be reduced. In contrast, uncertainty
may be decreased by considering the uncertainty factors
(highlighted in Section IV). However, the cost incurred
by decreasing uncertainty (time, bandwidth, . . . ) can be
adjusted by considering the required minimum certainty
during evaluation. This way, more investment is made on
highly critical decisions (to lower risk), whereas a modest
investment will suffice for noncritical ones. The trustee can
use (20) to propagate uncertainty of the trust to the expected
utility using interval arithmetic:
∆U = ∆τ .(u(x) − u(x̄)) + u(x̄),
(20)
where a scalar value a is assumed to be a zero-length interval
[a, a] in interval arithmetic. Equation (20) is based on (19)
in order to resolve the dependency between different parts
of (18).
Having ∆U , confidence interval length (uncertainty) is
compared to the prespecified maximum acceptable interval
length threshold corresponding to the criticality of the decision. If the requirement is not met, trustor may increase
the certainty by using the uncertainty factors, or leave out
the trustee from the possible interaction alternatives. For
example it may consult other agents, or examine the trustee
in experimental scenarios. This method is exemplified with
the results of the Beta and HMM trust models in the
following sections.
A. Uncertain Utility Estimation
In this section, the estimation of the trustor’s expected
utility and its uncertainty is examined in a randomly selected
run of the simulations to illustrate its application. The
simulation settings were nearly optimal for both models:
n = 300, s = 0.9, and λ = 0.4.
Suppose the utility function of the trustor is specified by
+50 units O = x
u(O) =
.
(21)
−20 units O = x̄
Moreover, the maximum acceptable uncertainty for three
criticality classes of low, medium, and high are specified
according to Table I.
The estimated trust values and their corresponding 0.95
confidence intervals in the selected run are stated in
τ b = 0.478, ∆b = [0.285, 0.673],
(22)
τ h = 0.519,
∆h = [0.431, 0.601],
(23)
where the superscripts of b and h distinguish between the
Beta and HMM models respectively.
In this case, the resulting expected utilities and their
confidence intervals are as in (24) and (25), according to
the interval arithmetic.
U b = 13.450, ∆U,b = [−0.037, 27.087]
(24)
U h = 16.353,
∆U,h = [10.151, 22.058]
(25)
According to the results, the Beta model has a great
amount of uncertainty in its trust estimation, and the length
of the expected utility confidence interval is about two times
greater than the expected utility.
B. Risk Reduction
Both models suggest a positive expected utility. However,
according to the confidence intervals, the true value may be
far away from the expectations. With the Beta model, this
value can vary in an interval of length 27.124 units while
with the HMM model, the interval length is 11.907 units.
According to the uncertainty-driven risk reduction method
applied to the Beta model results, the trustee should not be
selected even in low-critical situations. Moreover, according
to the analyses of uncertainty factors in the past sections,
there is no way to improve the certainty. However, the result
of the HMM model can be used for low or medium-critical
situations. Moreover, in case of high-critical decisions, collecting further observations will result in a more certain
estimation with the HMM model.
VI. R ELATED W ORK
Traditionally, security researchers were concerned about
trust in their systems (e.g. in Public Key Infrastructure, and
Trusted Computing). Yet, initial formation and evolution of
trust were implicit or external to the proposed systems [9],
[10]. The current trend in the literature, however, is to
explicitly consider these aspects and many trust models are
proposed to compute the trust between parties based on
past history, social networks, and other social factors (for
a survey on computational trust models see [9], [11], [12]).
Uncertainty is disregarded in many of the proposals,
or combined into trust [1]. The probabilistic and fuzzy
trust models are the most susceptible ones to uncertainty
analysis. A fuzzy trust model for peer-to-peer systems has
been proposed in [13] with special focus on uncertainty.
Particularly, a threshold on observation count has been put
in order to decide whether to include history information in
the trust evaluation or not.
Referential trust as an uncertainty factor is put forward
in SUNNY which is a trust derivation algorithm in social
networks [1], [14]. Referential trust in SUNNY is represented by a binary value and is used to calculate confidence.
Confidence in this model is the probability of positive trustor
belief in the correctness of trust information received from
another agent. Unlike the confidence intervals in this paper,
516
the confidence in SUNNY is an internal uncertainty measure
that cannot be used in trust-based decision making.
TRAVOS [15] has followed an approach which is more
comparable to ours. In TRAVOS, evaluation of trust follows
the same base formula of the Beta model, and the confidence
coefficient of a specific interval around the trust value is used
as the measure of uncertainty. The coefficient is calculated
using the Beta distribution and does not take the effect
of forgetting factor into account. In the case where past
interaction history does not estimate a confident trust value,
TRAVOS uses other agents’ recommendations to improve
the result. Nevertheless, we have shown in Section IV
that confidence of the Beta model is (at least sometimes)
independent of observation count. Moreover, according to
our analyses which are not reported in this paper, the
Beta distribution produces relatively wide confidence intervals when the trustee behavior does not exactly follow a
Bernoulli distribution.
Confidence intervals are superior to other proposals because as our simulations indicate, uncertainty factors are not
the same in all models, and these intervals are not bound to
a specific uncertainty factor. Additionally, the bootstrapping
method helps in studying the effect of various parameters of
a trust model on uncertainty, and all the effects are unified
in the confidence interval.
The risk involved in the decision of a trustor with respect to uncertainty is not considered in the existing trust
models even in the most general ones such as [12]. To our
knowledge, our uncertainty-driven risk reduction is the first
proposal which propagates the uncertainty to the decisionmaking process and helps to maintain a balance between the
risk induced by uncertainty, and cost.
relying on a bigger history that may not be affordable in
many applications.
The disappointing amount of uncertainty in the models
cannot be avoided because it is inherent to the models. We
believe that new models of trust must be proposed with
special attention to uncertainty, and all forms of uncertainty
should be considered in the models. In order to achieve this
goal, a mathematical framework other than the probability
theory should be used that is capable to handle various forms
of uncertainty.
VII. C ONCLUSIONS AND F UTURE W ORK
[6] L. R. Rabiner, “A tutorial on hidden markov models and
selected applications in speech recognition,” Proceedings of
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Computational models of trust are useful tools in the assessment of the risk involved in future interactions and help
agents in a multi-agent environment maximize their utility.
Confidence intervals calculated via the bootstrapping method
were proposed in this paper to quantify the uncertainty of
probabilistic trust models.
The uncertainty of two trust models was studied and
the related factors were determined. Both models suffered
from a great amount of uncertainty when a small number
of past interaction outcomes were used to calculate trust.
However, the performance of the HMM model was improved
by increasing the size of history. In contrast, the Beta model
was indifferent to the size of history, and nonmonotonically
dependent on its forgetting factor (fixed over time).
We also proposed a risk reduction method which takes
advantage of trust uncertainty to include risk in decision
making.
Studying the effect of referential trust and other factors on
uncertainty of the models are planned for further research.
These factors can help in reducing uncertainty without solely
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