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2.4.4 Probabilistic Inference

Knowledge Base > Math> Probability Theory

There is a method for probabilistic inference, using enumeration, which compute from observed evidence of posterior probabilities for query propositions. We take our Toothache versus Cavity versus Catch example, which takes up to 2 times 2 times 2 matrix table:[5]

toothache ¬ toothache
catch ¬ catch catch¬ catch
cavity 0.108 0.012 0.072 0.008
¬ cavity 0.016 0.64 0.144 0.576
The probabilities of having a toothache is P(toothache) = 0.108+0.012+0.016+0.064 = 0.2
The next example extracts a distribution over a single variable or some subset of variables, called marginal distribution. We can write the marginalization rule for any sets of variables Y and Z, that is:
∑ P (Y ) = P (Y,z),∀z ∈ Z z
(2.4.9)

In other words, a distribution over Y can be obtained by summing out all other variables, such as Z, from any joint distribution containing Y.
By using product rule (P(a,b) = P(a|b) ⋅ P(b)),

P(Y) = ∑ P(Y|z)⋅P (z),∀z ∈ Z z
(2.4.10)

This rule is called conditioning.

Thus, we can now compute the probability of a cavity, given a evidence of a toothache. That is, P(cavity|toothache) = ∧ P(cavity--toothache)- P(toothache) = -----0.108+0.012----- 0.108+0.012+0.016+0.064 = 0.6
In the contract, the probability that yields no cavity can be obtained by: P(¬cavity|toothache) = P(¬cavity∧-toothache) P(toothache) = -----0.016+0.064----- 0.108+0.012+0.016+0.064 = 0.4

Normalization

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