Immediate Consequences of the Independent Evidence Conditions
When neither independence condition holds, we at least
have:
|
P[en | hj·b·cn]
|
= |
P[en | hj·b·cn·en−1] ×
P[en−1 | hj·b·cn]
|
|
= |
… |
|
= |
n
Π
k = 1
|
P[ek | hj·b·cn·ek−1]
|
|
When condition-independence holds we have:
|
P[en | hj·b·cn]
|
= |
P[en | hj·b·cn·(cn−1·en−1)] × P[en−1 | hj·b·cn·cn−1]
|
|
= |
P[en | hj·b·cn·(cn−1·en−1)] × P[en−1 | hj·b·cn−1]
|
|
= |
… |
|
= |
n
Π
k = 1
|
P[ek | hj·b·ck·(ck−1·ek−1)]
|
|
If we add result-independence to
condition-independence, the occurrences of
‘(ck−1·ek−1)’
may be removed from the previous formula, giving:
|
P[en | hj·b·cn]
|
= |
n
Π
k = 1
|
P[ek | hj·b·ck]
|
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