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A fundamental theorem of higher order probabilities (in discrete time)
If we set P
0
(E)
P(E), then
P
t
(E) = P(E)
.
P
t-1
(E)
.
[P{(U
t-1
= U
t
)|E}/P(U
t-1
= U
t
)] for t = 1, 2,3, ..., n
Proof
P
1
(E) = P{P
0
(E)} = P(E)
.
[P{(U
0
= U
1
)|E}/P(U
0
= U
1
)]
from lemma 2
P
2
(E) = P{P
1
(E)} = P(E)
.
P
1
(E)
.
[P{(U
1
= U
2
)|E}/P(U
1
= U
2
)]
.. from lemma 3
P
3
(E) = P[{E | (U
0
= U
1
)} ∩ {E|(U
1
= U
2
)} ∩ {E|(U
2
= U
3
)}]
= P[[{E | (U
0
= U
1
)} ∩ {E|(U
1
= U
2
)}] ∩
{E|(U
2
= U
3
)}]
= P
2
(E)
.
P{E|(U
2
= U
3
)}
= P(E)
.
P
2
(E)
.
[P{(U
2
= U
3
)|E}/P(U
2
= U
3
)
P
4
(E) = P[{E | (U
0
= U
1
)} ∩ {E|(U
1
= U
2
)} ∩ {E|(U
2
= U
3
)} ∩ {E|(U
3
= U
4
)}]
= P[[{E | (U
0
= U
1
)} ∩ {E|(U
1
= U
2
)} ∩ {E|(U
2
= U
3
)}] ∩
{E|(U
3
= U
4
)}]
= P
3
(E)
.
P{E|(U
3
= U
4
)}
= P(E)
.
P
3
(E)
.
[P{(U
3
= U
4
)|E}/P(U
3
= U
4
)]
Extending to the (t-1)-th term, we can therefore write:
P
t-1
(E) = P(E)
.
P
t-2
(E)
.
[P{(U
t-2
= U
t-1
)|E}/P(U
t-2
= U
t-1
)
(1)
The expression for the t-th term is derived from (1) as follows:
P
t
(E) = P(E)
.
P
(t-2)+1
(E)
.
[P{(U
(t-2)+1
=U
t
)|E}/P(U
(t-2)+1
= U
t
)
= P(E)
.
P
t-1
(E)
.
[P{(U
t-1
=U
t
)|E}/P(U
t-1
=U
t
)
(2)
However we may also write:
P
t
(E)=P[{E | (U
0
= U
1
)}∩{E|(U
1
= U
2
)}∩{E|(U
2
= U
3
)}∩{E|(U
3
= U
4
)}∩...∩{E|(U
t-1
= U
t
)}]
=P[[{E|(U
0
=U
1
)}∩{E|(U
1
=U
2
)}∩{E|(U
2
=U
3
)}∩{E|(U
3
=U
4
)}∩...∩{E|(U
t-2
=U
t-1
)]∩{E|(U
t-1
= U
t
)}]
= P
t-1
(E)
.
P{E|(U
t-1
= U
t
)}
= P
t-1
(E)
.
P(E)
.
[P{(U
t-1
=U
t
)|E}/P(U
t-1
=U
t
)]
(3)
As (2) is identical to (3); by
principle of mathematical induction
the general case is proved for t= n.
QED
Obviously then, if
P(U
t-1
=U
t
) =
P{
(U
t-1
=U
t
)|E}
, for all t = 1, 2, 3, ..., n; we will end up with
P
n
(E) =
[
P(E)
]
n
which makes this approach to
H-O
probability fully consistent with classical
probability theory and in fact a very natural extension thereof if one sees the fundamentally
time-dynamic characteristic of U.
