Turbo Coding and MAP Decoding
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1
Intuitive Guide to Principles of Communications
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Turbo Coding and MAP decoding
 Part 1
Baye’s Theorem of conditional
probabilities
Let’s first state the theorem of conditional probability,
also called the Baye
’
s theorem.
)
(
)
(
)
(
A
given
B
P
A
P
B
and
A
P
Which we can write in more formal terminology as
(
, )
( )
(  )
P
P A B
A
B
P
A
A
where
P
(
B

A
) is referred to as the probability of event B given that A has already occurred.
If event A always occurs with event B, then we can write the following expression for the
absolute probability of event A.
( )
( ,
)
B
P A
P A B
B
If events A and B are independent from each other then (A) degenerates to
)
(
)
,
(
A
P
B
A
P
B
P
A
P
B
A
P
(
)
(
)
,
(
)
C
The relationship C is very important and we will use it heavily in the explanation of Turbo
decoding in this chapter.
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If there are three independent events A, B and C, then the Baye’s rule becomes
)
,

(
)

(
)

,
(
C
A
B
P
C
A
P
C
B
A
P
D
Apriori and aposteriori probabilities
Here is Bayes' theorem again.
Pr
( ,
)
(
)
( )
(
)
( )
a priori
obability of
a posteriori
probability of
both A and B
probability of
event B
event A
a priori
a posteriori
probability of
probability of
event A
event B
P A B
P AB
P B
P B A
P A
E
The probability of event A conditioned on event B, is given by the probability of A given B
times the probability of event a.
The probability of A, or P(A) is the base probability of even A and is called the
apriori
probability. The term P(A,B) the conditional probability is called the
aposteriori
probability or APP
. One is independent probability, the other depends on some event
occurring. We will be using the acronym APP a lot, so make sure you remember that is the
same aposteriori probability. In other words, the APP of an event is a function of an
another event also occurring at the same time. We can write (E) as
(
) ( )
( ,
)
(
)
()
APP
P B A P A
P A B
P AB
PB
F
This says that we can determine the APP of an event by taking the conditional probability of
that event divided by
it’s a
priori probability.
What these mean is best explained by the following two quotes.
In epistemological terms “ A priori” and “a posteriori” refer primarily to how, or on what
basis, a proposition might be known. In general terms, a proposition is knowable a priori if
it is knowable independently of experience, while a proposition knowable a posteriori is
knowable on the basis of experience. The distinction between a priori and a posteriori
knowledge thus broadly corresponds to the distinction between empirical and non
empirical knowledge.” [2]
“But how do we decide when we have gathered enough data to justify modifying our
prediction of the probabilities? That is one of the essential problems of decision theory.
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