Noisy-Or Distribution (DM prob. dist. for equations)

 Usage: NoisyOrDist (e, leak, b1, p1, ... bn, pn) Definition: P(e) = 1–[(1-leak) product i=1 to n (bi? (1-pi): 1)] Required: 0 £ leak £ 1     e, bi Boolean       0 £ pi £ 1

Use this distribution when there are several possible causes for an event, any of which can cause the event by itself, but only with a certain probability.  Also, the event can occur spontaneously (without any of the known causes being true), with probability leak (make this zero if it can’t occur spontaneously).

Each bi is a boolean variable, which may cause the event when its true.  e is also a boolean, which indicates whether the event occurs.  Each of the pi are the probability that e will occur if bi is true in isolation.

If leak is zero, and only one possible cause is true, say bk, then the probability for e is pk.  If more possible causes are true, P(e) will be greater.  And if leak is nonzero, P(e) will be greater.  Reducing a pi always results in the same or lower P(e).

pi can be considered the “strength” of the relation between e and bi, with zero indicating independence (link could be removed), and 1 indicating maximum effect.

See Pearl88, page 184 for more information (his qi = 1 – pi).

 Example: P (Effect | Cause1, Cause2) = NoisyOrDist (Effect, 0.1,              Cause1, 0.2,              Cause2, 0.4)

Another Example

See also NoisyAndDist, NoisyMaxDist (for nodes with more than just 2 states), NoisySumDist.