# Probabilistic Reasoning

**Probabilistic Reasoning**: Using logic to represent and reason we can represent knowledge about the world with facts and rules, like the following ones:

bird(tweety).

fly(X) :- bird(X).

We can also use a theorem-prover to reason about the world and deduct new facts about the world, for e.g.,

?- fly(tweety).

Yes

However, this often does not work outside of toy domains - non-tautologous certain

rules are hard to find.

A way to handle knowledge representation in real problems is to extend logic by using certainty factors.

In other words, replace

**IF condition THEN fact
with
IF condition with certainty x THEN fact with certainty f(x)**

Unfortunately cannot really adapt logical inference to probabilistic inference, since the latter is not context-free.

Replacing rules with conditional probabilities makes inferencing simpler.

**Replace
smoking -> lung cancer
or
lotsofconditions, smoking -> lung cancer
with
P(lung cancer | smoking) = 0.6
**

Uncertainty is represented explicitly and quantitatively within probability theory, a formalism that has been developed over centuries.

A probabilistic model describes the world in terms of a set S of possible states - the sample space. We don’t know the true state of the world, so we (somehow) come up with a probability distribution over S which gives the probability of any state being the true one. The world usually described by a set of variables or attributes.

Consider the probabilistic model of a fictitious medical expert system. The ‘world’ is described by 8 binary valued variables:

**Visit to Asia? A
Tuberculosis? T
Either tub. or lung cancer? E
Lung cancer? L
Smoking? S
Bronchitis? B
Dyspnoea? D
Positive X-ray? X
**

We have 28 = 256 possible states or configurations and so 256 probabilities to find.