December 2003
The Bayesian Group
Definition: Bayesian Addition
-
The Bayesian Group is the set
A*B
A B = ----------------
A*B + (1-A)(1-B)
½ A = A
A
A (B
C) = (A
B)
C
Definition: Bayesian Subtraction
-
The definition of
A*(1-B)
A B = ----------------
A*(1-B) + (1-A)B
Theorems: (proof left to reader)
½ = A A
A B = A
(1 - B)
(A B)
C = A
(B
C)
A B = C <=> A = C
B
Definition: Bayesian Multiplication
-
We define a multiplication operator, where n
Pn
n P = ------------
Pn + (1 - P)n
This does not make the Bayesian group into a ring because n
is not in .
Theorem:
(n P)
(m
P) = (n + m)
P
Proof: (be sure your browser is displaying wide enough)
(n P)(m
P)
(n P)
(m
P) = ----------------------------------------
(n P)(m
P) + (1-(n
P))(1-(m
P))
/ Pn \ / Pm \
| ----------- | | ----------- |
\ Pn + (1-P)n / \ Pm + (1-P)m /
= ------------------------------------------------------------------
/ Pn \ / Pm \ / (1-P)n \ / (1-P)m \
| ----------- | | ----------- | + | ------------ | | ----------- |
\ Pn + (1-P)n / \ Pm + (1-P)m / \ Pn + (1-P)n / \ Pm + (1-P)m /
Pn Pm
= ---------------------
Pn Pm + (1-P)n (1-P)m
Pn+m
= ---------------
Pn+m + (1-P)n+m
= (n + m) P
Intuitive Multiplication Theorem:
The definition for was originally derived from the desire to have n
P
mean "add P
to itself n
times". This theorem proves that this intuitive meaning still holds.
n P = P
P
...
P
n
Naturals
(n times)
Proof:
-
Base Case:
P1
1 P = ------------ = P
P1 + (1-P)1
n P = (n - 1 + 1)
P = (1
P)
((n-1)
P)
= P (P
P
...
P)
(n-1 times)
= P P
...
P
(n times)
More Theorems:
0 P = ½
n (P
Q) = (n
P)
(n
Q)
n (m
P) = (n * m)
P
(n + m) P = (n
P)
(m
P)
(n - m) P = (n
P)
(m
P)
December 2003, first draft