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| == Decomposition == | |
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| {{attachment:decomposition.svg}} | == Description == |
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| This is the probability of an event given that some event(s) has (have) already occurred. This is generally notated as ''P(A|B)'' or ''P(A;B)'', where ''B'' has already occurred. | |
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=== Bayes Theorem === Bayes combined the decomposition with [[Statistics/JointProbability|joint probability identities]] to arrive at this more solvable theorem. |
It is generically decomposed as ''P(A|B) = P(A∩B) / P(B)''. Importantly though, '''Bayes theorem''' provides the following decomposition based on [[Statistics/JointProbability|joint probabilities]]: |
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| ---- | === Independence === If two events are [[Statistics/JointProbability#Independence|independent]] (notated as ''A⫫B''), then probabilities of one do not change from being conditioned on the other. Put simply, if the conditioning probability is not 0, then: * ''P(A|B) = P(A)'' * ''P(B|A) = P(B)'' A conditioning probability of 0 will cause the conditional probability to be undefined. |
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| == Independence == If two events are [[Statistics/Independence|independent]], then probabilities of one do not change from being conditioned on the other. Put simply, if the conditioning probability is not 0, then: ''P(A|B) = P(A)'' ''P(B|A) = P(B)'' A conditioning probability of 0 will cause the conditional probability to be undefined. ---- == Conditional Independence == |
=== Conditional Independence === |
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| * ''P(A|B,C) = P(A|C)'' * ''P(A,B|C) = P(A|C) P(B|C)'' |
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| ''P(A|B,C) = P(A|C)'' ''P(A,B|C) = P(A|C) P(B|C)'' This interrelation is sometimes notated as ''((A⫫B)|C)''. |
This interrelation is sometimes notated as ''(A⫫B)|C''. |
Conditional Probability
A conditional probability is the likelihood of an event happening given that another event happens. The math notation is P(A|B), as in the probability of A given B.
Description
This is the probability of an event given that some event(s) has (have) already occurred. This is generally notated as P(A|B) or P(A;B), where B has already occurred.
It is generically decomposed as P(A|B) = P(A∩B) / P(B). Importantly though, Bayes theorem provides the following decomposition based on joint probabilities:
Independence
If two events are independent (notated as A⫫B), then probabilities of one do not change from being conditioned on the other.
Put simply, if the conditioning probability is not 0, then:
P(A|B) = P(A)
P(B|A) = P(B)
A conditioning probability of 0 will cause the conditional probability to be undefined.
Conditional Independence
If events A and B are conditionally independent, then:
P(A|B,C) = P(A|C)
P(A,B|C) = P(A|C) P(B|C)
This interrelation is sometimes notated as (A⫫B)|C.