Information Technology Reference
In-Depth Information
3.2 Simple computational 'tests' to better illustrate the above-posited concept of
H-O
probability
To provide a simple illustration of how the
H-O
probabilities would pan out in discrete
event-spacetime we have done a series of computations the results of which are graphically
represented below. The graphs show the temporal evolution of the event-spacetime in
discrete “time steps” and the resulting P
t
(
E
) values for t = 1, 2, ..., 5. We assume three
temporal evolution forms - “
expanding event-spacetime
”, “
contracting event-spacetime
” and
“
oscillating event-spacetime
” and plot the P
t
(
E
) values for each of these three forms starting
with a pervading assumption that P(U
t-1
= U
t
) = 1. This assumption simplifies a lot of the
computations as P
t
(
E
) then depends totally on P{(U
t-1
= U
t
)/
E
}. When P{(U
t-1
= U
t
)/
E
} = 1,
we see that P
t
(
E
) converges to P(
E
)
t
for all values of t. On the other hand, when P{(U
t-1
=
U
t
)/
E
} = 0, P
t
(
E
) converges to zero for all values of t. So, holding P(
E
) = 0.10, in an
“expanding event-spacetime”, P
1
(
E
) = P(
E
) = 0.10, p
2
(
E
) = 0.10
2
= 0.01 and so on for P{(U
t-
1
=U
t
)/
E
} = 1. For P{(U
t-1
=U
t
)/
E
} = 0, P
1
(
E
) = p
2
(
E
) = P
3
(
E
) = p
4
(
E
) = P
5
(
E
) = 0, while P
t
(
E
)
values are seen to oscillate for P{(U
t-1
=U
t
)/
E
} values randomly oscillating about 0.50 - the
degree of oscillation decreasing with increasing order of probability i.e. P
1
(
E
) oscillates more
than P
2
(
E
), P
2
(
E
) more than P
3
(
E
) and so on.
Fig. 2. Expanding Event-Spacetime, [P(
E
) = 0.10]
Plot of P
t
(
E
); t = 1, 2, ..., 5 for P{(U
t -1
= U
t
)/
E
} increasing from 0 to 1 in steps of 0.05
The expanding event-spacetime represents the situation where, with passage of time and
evolution of the current stock of domain knowledge, there is a steadily increasing
“probability of probability” of the occurrence of the elementary event of interest. The
contracting event-spacetime represents the situation where, with passage of time and
evolution of the current stock of domain knowledge, there is a steadily decreasing
“probability of probability” of the occurrence of the elementary event of interest. The
oscillating event-spacetime represents the situation where, with passage of time and
