IEEE Systems, Man, and Cybernetics Magazine - January 2018 - 36

```Effective
P (e )

Deployed
P (d )

Not
Deployed
1 - P (d )

Not
Effective
1 - P (e )

P (e, d ) =
P(e | d )P (d )
= P (e)

P (~e, d ) =
P (d ) - P (e)

P (e, ~d )

P (~e, ~d )

=0

= 1 - P (d )

Figure 2. The decomposition and math model of

missile negation.

object of interest. Upon activity completion, a unique
event is activated. This event performs a random draw,
imports a probability distribution as a function, and computes Ps for that activity. If there is only one activity, the
process ends. If additional activities occur, more events
are activated, and more probabilities are computed and
accumulated until all of the activities and events are complete. This process iterates many times to derive confidence intervals for the computed probabilities through
Monte Carlo analysis.
Supporting this architecture is the stochastic math model
(SMM) [16], which performs the derivation of Ps and its associated confidence interval. The AMADS uses SMM results to
enable an automatic analysis of the actual versus specified
performance for a range of data sources, as modeled in the
scenarios considered here. We illustrate the application of
the SMM through the missile-defense scenario.
For missile defense, Ps corresponds to the probability
of negating a missile, i.e., that a missile threat target is
negated based on a threat vulnerability (V ) and a threat
attack technique (T ) that together create a vulnerability/
r ^VT h be a discrete random
technique (VT ) pair. Let N
variable called missile negation that can take on the valr is a funcues " negated ^ n h, not negated ^~n h,, where N
tion of the combination of several VT pairs. Likewise,
Pn ^VT h is the probability of negating the missile or, in
r ^VT h taking on the value
other words, the probability of N
" negated ^ n h, . The probability of not negating the missile,
r = not negated ^~n h,, is 1 - Pn ^VT h .
given by " N
r ^VT h, we will do so for a single VT
As we decompose N
pair composed of vulnerability Vi and attack technique T j,
so the probability of negating the missile is Pn ^Vi T j h = Ps .
For a missile to be negated from a VT pair, Vi T j, the attack
technique T j must first be deployed against vulnerability Vi, and then it must be effective, so the probability of
missile negation is the probability that Vi T j is both
r ^Vi T j h be a discrete, random
deployed and effective. Let D
36

IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE Janu ar y 20 18

variable called technique deployment that can take on valr is a
ues " Deployed ^d h, Not Deployed ^~d h,, where D
function of the VT pair Vi T j . The probability of a technique T j being deployed against a vulnerability Vi is
given by Pd ^Vi T j h, and the probability of a technique
T j not being deployed against a vulnerability Vi is
r ^Vi T j h be a discrete ranP~d ^Vi T j h = 1 - Pd ^Vi T j h . Let E
dom variable called technique effectiveness that can take
on the values " Effective ^ e h, Not Effective ^~e h,, where
r is a function of the VT pair Vi T j . The corresponding
E
probabilities of being effective and not effective are
Pe ^Vi T j h and P~e ^Vi T j h = 1 - Pe ^Vi T j h, respectively. Figure 2 features the 2 × 2 matrix showing all of the possible
relationships among technique effectiveness and technique
deployment and their corresponding joint probabilities,
P ^ e, dh, P ^~e, dh, P ^e, ~dh, and P ^~e, ~d h . The independent variable Vi T j has been dropped for easier reading.
From basic probability theory, we know that
P ^e, dh + P ^~e, d h + P ^e, ~dh + P ^~e, ~dh = 1,

(1)

P ^e, d h + P ^~e, d h = P ^d h,

(2)

P ^e, d h + P ^e, ~d h = P ^ e h .

(3)

Applying Bayes' theorem a nd recalling that Ps =
Pn ^Vi T j h = P ^e, d h, we see that
Ps = P ^e, d h = P ^e | d h P ^d h = P ^d | e h P ^ e h .

(4)

Because the probability of a technique being effective and
not deployed is zero- P ^e, ~d h = 0 -combining (3) and
(4) gives
Ps = P ^e, d h = P ^e | d h P ^d h = P ^ e h .

(5)

Thus, the probability of negating a missile is the probability that the technique used is effective. Equating (4) and
(5) gives P ^d | e h = 1, i.e., the probability of a technique
having been deployed, given that it has been effective, is
one. If a technique has been effective, then it must have
been deployed.
For the remote-object-location scenario, the SMM computes Ps with respect to the probability of success that a
group of object-detection and location algorithms will
have in correctly identifying the object of interest from
among many images. These algorithms are structured as
a chain to achieve this goal. This algorithm chain structure is modeled using a discrete event simulator to calculate Ps, including the confidence intervals, for all of the
algorithms for each step. A Ps is assigned to each step
based on appropriate metrics, including size, color, number of pixels, land/water, longitude/latitude, image shape,
and image identification. Confidence intervals are then
derived by applying Monte Carlo analysis. With respect to
the number of steps, SMM scalability is limited only by the
processing requirements.

```

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