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mathematical lines. In that desired direction, we firstly posit and prove a fundamental
theorem necessary for such an extension to the theory of probability. Subsequently we show
some computational 'tests' to illustrate the posited framework.
3.1 A foray into higher order probabilities
It is well known that much of modern theory of probability rests upon the three
fundamental
Kolmogorov axioms
(Kolmogorov, 1956) which are conventionally stated as
follows:
1
st
axiom: The probability of any event is a non-negative real number i.e. P(E) ≥ 0
∀
E
∈
U
2
nd
axiom: The probability of any one of the elementary events in the whole event space
occurring is 1 i.e. P(U)=1
3
rd
axiom: Any countable sequence of pair-wise
non-overlapping
events E
1
, E
2
, ... E
n
satisfies
the following relation: P(E
1
E
2
... E
n
) = ∑
i
P(E
i
); i = 1, 2, ..., n.
It is basically Kolmogorov's second and third axioms as noted above that render any
extensions of the probability concept to higher orders (i.e. “probability of probability”)
superfluous as the information content of any such higher order probability can be
satisfactorily transmuted via existing set-theoretic constructs. So, extending to a higher order
would arguably yield trivial information. However the Kolmogorov axioms by themselves are
also open to 'extensions' - for instance there is previous research that has revisited the proofs
of the well-known Bell inequality based on underlying assumptions of separability and non-
contextuality and constructed a model of generalized “non-contextual contrapositive
conditional probabilities” consistent with the results of the famous Aspect experiment
showing in general such probabilities are not necessarily all positive (Atkinson, 2000). By
themselves the Kolmogorov axioms do not unequivocally rule out an extension of the
definition of the universal set U itself so as to make U possess a
time-dynamic
rather than a
time-
static
nature. So; in effect this means that if we were to consider a time-dynamic version of the
universal set; then one would suddenly find that the information content of higher order
probability no longer remains trivial i.e. an extension of the probability concept to higher
orders (i.e. “probability of probability”) is no longer superfluous - in fact it is logical! The good
thing is that no new probability calculus needs to be formulated to describe such a theory of
higher-order probabilities and this extended theory could still rest on the Kolmogorov axioms
and could still draw fundamentally from the standard set-theoretic approach (as we will be
demonstrating shortly); by merely using an extended definition of the universal set U which
would now denote not merely an event space but a broader concept, which we christen as
event-spacetime
, i.e. an event space that can
evolve
over a time dimension.
Perhaps the only academic work preceding ours to have alluded that a higher-order
probability theory is justifiable by an event space evolving over time was that by Haddawy
and others (Haddawy, 1996; Lehner, Laskey and Dubois, 1996), where they provided “a
logic that incorporates and integrates the concepts of subjective probability, objective
probability, time and causality” (Lehner, Laskey and Dubois, 1996). We take a similar
philosophical stance but go on to explicitly develop a logically tenable higher-order
probability concept in discrete time. We have no doubt that an extension in continuous time
is also attainable but it's left for later.
