"Computing with Odds and Certainty Factors"
Motivation: General Means are Needed for Propagating Certainty Information While Reasoning
Much reasoning is done with "chaining";
It should be possible to chain rule applications and maintain appropriate
certainty values at the same time.
.
We often need, however, a means
to compute P(H | E) where E is given with a certain probability that is
less than 1 but more than 0.
.
Probability Updating Functions
for Relationships of Sufficiency, Necessity, etc.
A probabilistic version of a rule
P -> Q can be thought of as a function. The function maps a probability
of P into a probability of Q. Let us use instead of P and Q the symbols
E and H for evidence and hypothesis. Then the modus ponens expression
E and E -> H implies H becomes in the probabilistic realm,
P(E | E') and f_{EH}:[0, 1]
-> [0, 1] implies P(H | E')
Here P(E | E') represents the current
probability of the evidence.
P(H | E') represents the current
probability of the hypothesis, taking into account the uncertain evidence.
f_{EH} is a function relating E
and H in terms of their a posteriori probabilities.
Given a value of P(E | E'),
f_{EH} returns a corresponding value of P(H | E').
If f_{EH} is the identity function,
then we can say that E is necessary and sufficient for H.
However, many other shapes are
possible for f_{EH}.
(see the text for examples of sufficiency-only
and necessity-only updating functions).
Fuzzy Logic
Bayes' rule and its generalization with the PROSPECTOR mechanisms give us a good way to handle logical combinations analogous to modus ponens.
However, we also need ways to propagate probability or certainty values through other logical expressions such as (A ^ B), (A V B), etc.
One approach is used when items
being combined are not assumed to be independent:
P(A ^ B) = min [ P(A), P(B)
]
P(A V B) = max[ P(A),
P(B) ]
P(~A) = 1 - P(A)
Another approach is used when items being combined can be considered independent.
P(A ^ B) = P(A) P(B)
P(A V B) = 1 - P(~A ^ ~B)
= 1 - [ (1 - P(A))
(1 - P(B)) ]
= P(A) + P(B) - P(A)
P(B)
Last modified: November 18, 1998
Steve Tanimoto