Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

= ( 0.6x 0.25x 0.2x 0.3) + ( 0.6x 0.05x 0.2x 0.6) + ( 0.1x 0.25x 0.7x 0.3) +
(0.1x 0.05x 0.7x 0.6)
= 0.01995
Suppose analogously, we find Bel (BT≥100° F) = β (say).
Then α = 1 / ( 0.01995 + β ).
So, normalized Bel (BT ≤ 98° F) = α x 0.01995
and normalized Bel (BT ≥ 100° F) = α x β
The λ and Π messages can also be calculated by the formulas supplied.
According to Pearl [2], the belief computation in the polytree is done in an
asynchronous manner, and at some point of time, the beliefs at all nodes do
not change. We call it an equilibrium condition. The belief of the nodes in
the polytree at this condition is consistent with the theory of probability.
9.2.4 Dempster-Shafer Theory for
Uncertainty Management
The Bayesian formalism assigns a positive belief to a proposition, but it does
not take into account of the disbelief of the propositions. Dempster-Shafer
(DS) theory, on the other hand, allows information integration by considering
both their belief and disbelief. To illustrate this point, let us consider an
example. Suppose that one of the three terrorist groups: A, B and C planted a
bomb in an office building in a country. Further, suppose, we have adequate
evidence to believe that group C is the guilty one with a measure of belief
P(C) = 0.8, say. On the other hand, without any additional knowledge / fact,
we do not like to say that P(B)+ P(A) = 0.2. Unfortunately, we are forced to
say so using conventional probability theory as it presumes P(¬C) = 1- P(C)
=P(B) + P(A). This prompted Dempster and his follower Shafer to develop a
new theory, well known as the DS theory in the AI community.
In the DS theory, we often use a term, frame of discernment (FOD)
θ.To illustrate this, let us consider an example of rolling a die. In rolling a
die, the set of outcomes could be described by a statement of the form: “thenumber-showing-is-i”
for 1 ≤ i ≤ 6 . The frame of discernment in the die
example is given by
FOD θ = { 1, 2, 3, 4, 5, 6 }.