Uncertainty: Probability, Syntax, and Inference in Uncertain Domains, Slides of Artificial Intelligence

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Uncertainty

Chapter 13

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Outline

• Uncertainty

• Probability

• Syntax and Semantics

• Inference

• Independence and Bayes' Rule

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Methods for handling uncertainty

• Default or nonmonotonic logic:

• Assume my car does not have a flat tire
• Assume A 25 works unless contradicted by evidence
• Issues: What assumptions are reasonable? How to handle

contradiction?

• Rules with fudge factors:

• A25 | →0.3 get there on time
• Sprinkler | → (^) 0.99 WetGrass
• WetGrass | → (^) 0.7 Rain
• Issues: Problems with combination, e.g., Sprinkler causes Rain ??
• Docsity.com

Probability

Probabilistic assertions summarize effects of

• laziness: failure to enumerate exceptions, qualifications, etc.

• ignorance: lack of relevant facts, initial conditions, etc.

Subjective probability:

• Probabilities relate propositions to agent's own state of knowledge e.g., P(A 25 | no reported accidents) = 0.

These are not assertions about the world

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Syntax

• Basic element: random variable
• Similar to propositional logic: possible worlds defined by assignment of values to random variables.
• Boolean random variables
• e.g., Cavity (do I have a cavity?)
• Discrete random variables
• e.g., Weather is one of < sunny,rainy,cloudy,snow >
• Domain values must be exhaustive and mutually exclusive
• Elementary proposition constructed by assignment of a value to a
• random variable: e.g., Weather = sunny , Cavity = false
• (abbreviated as ¬ cavity )
• Complex propositions formed from elementary propositions and standardDocsity.com

Syntax

• Atomic event: A complete specification of the

state of the world about which the agent is

uncertain

E.g., if the world consists of only two Boolean variables Cavity and Toothache , then there are 4 distinct atomic events:

Cavity = false ∧ Toothache = false

Cavity = false ∧ Toothache = true

Cavity = true ∧ Toothache = false

Cavity = true ∧ Toothache = true

• Atomic events are mutually exclusive and

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Prior probability

• Prior or unconditional probabilities of propositions
• e.g., P( Cavity = true) = 0.1 and P( Weather = sunny) = 0.72 correspond to belief prior to arrival of any (new) evidence
• Probability distribution gives values for all possible assignments:
• P ( Weather ) = <0.72,0.1,0.08,0.1> (normalized, i.e., sums to 1)
• Joint probability distribution for a set of random variables gives the probability of every atomic event on those random variables
• P ( Weather,Cavity ) = a 4 × 2 matrix of values:

Weather = sunny rainy cloudy snow Cavity = true 0.144 0.02 0.016 0. Cavity = false 0.576 0.08 0.064 0.

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Conditional probability

• Conditional or posterior probabilities

e.g., P( cavity | toothache ) = 0.

i.e., given that toothache is all I know

• (Notation for conditional distributions:

P ( Cavity | Toothache ) = 2-element vector of 2-element vectors)

• If we know more, e.g., cavity is also given, then we have

P( cavity | toothache,cavity ) = 1

• New evidence may be irrelevant, allowing simplification, e.g., Docsity.com

Inference by enumeration

• Start with the joint probability distribution:
• For any proposition φ, sum the atomic events where it is

true: P(φ) = Σ (^) ω:ω╞φ P(ω)

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Inference by enumeration

• Start with the joint probability distribution:
• For any proposition φ, sum the atomic events where it is

true: P(φ) = Σ (^) ω:ω╞φ P(ω)

• P( toothache ) = 0.108 + 0.012 + 0.016 + 0.064 = 0.

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Inference by enumeration

• Start with the joint probability distribution:
• Can also compute conditional probabilities:

P(¬ cavity | toothache ) = P(¬ cavity ∧ toothache ) P( toothache ) = 0.016+0. 0.108 + 0.012 + 0.016 + 0. = 0.4 Docsity.com

Normalization

• Denominator can be viewed as a normalization constant α

P ( Cavity | toothache ) = α, P ( Cavity,toothache )

= α, [ P ( Cavity,toothache,catch ) + P ( Cavity,toothache ,¬ catch )] = α, [<0.108,0.016> + <0.012,0.064>] = α, <0.12,0.08> = <0.6,0.4>

General idea: compute distribution on query variable by fixing evidence

variables and summing over hidden variables

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Independence

• A and B are independent iff

P ( A|B ) = P ( A ) or P ( B|A ) = P ( B ) or P (A, B) = P ( A ) P ( B )

P ( Toothache, Catch, Cavity, Weather ) = P ( Toothache, Catch, Cavity ) P ( Weather )

• 32 entries reduced to 12; for n independent biased coins, O(2 n^ )

→ O(n)

• Absolute independence powerful but rare

• Dentistry is a large field with hundreds of variables, none of which

are independent. What to do? Docsity.com

Conditional independence

• P ( Toothache, Cavity, Catch ) has 2 3 – 1 = 7 independent entries

• If I have a cavity, the probability that the probe catches in it doesn't

depend on whether I have a toothache:

(1) P ( catch | toothache, cavity ) = P ( catch | cavity )

• The same independence holds if I haven't got a cavity:

(2) P ( catch | toothache, ¬ cavity ) = P ( catch | ¬ cavity )

• Catch is conditionally independent of Toothache given Cavity :

P ( Catch | Toothache,Cavity ) = P ( Catch | Cavity )

• Equivalent statements: Docsity.com