## AI, Decision Making, and Probability

• Uncertainty, likelihood, and probability

• Decision-theoretic methods

## Probability

• Idea: chance and likelihood are important concepts for real reasoning

• Method: assign probabilities to events and combinations of events

• Reason from model using calculation

• This is a general plan:

    Evidence -----------------> Conclusions
|                            ^
|                            |
v                            |
Model   -----------------> Extrapolation


## Probability of Events

• Domain: events

• pr(E) is probability that event E happens

• pr(E) = #E / (#E + #¬E)

• For a coin pr(H) = #H / (#H + #T) = 1 / 2

• For a pair of dice:

• pr(R7) = #R7 / #R

• #R7 = #{(1, 6), (2, 5), (3, 4), …} = 6

• #R = #{D×D} = 36

• pr(R7) = 6 / 36 = 1/6

• Check with computer program prob7

## Probability of Logical Situations

• Domain: propositional formula

• pr(p) is probability of logical combination of events

• p is a prop formula

• Priors and conditionals

• pr(p|q) is prob of p given q (easy to get backward)

## Axioms Of Probability

• equivalence: if p ≡ q then pr(p) = pr(q)

• range: 0 ≤ pr(p) ≤ 1

• negation: pr(¬p) = 1 - pr(p)

• conjunction: pr(p ∧ q) = pr(p) pr(q|p) = pr(q) pr(p|q)

## Derived Probability Rules

• disjunction: pr(p ∨ q) = pr(¬(¬p ∧ ¬q)) = 1 - pr(¬p ∧ ¬q)

• Bayes's Rule:

  pr(q) pr(p|q) = pr(p ∧ q) = pr(p) pr(q|p)
pr(q) pr(p|q) = pr(q|p) pr(p)
pr(p|q) = pr(q|p) pr(p) / pr(q)


## Independence

• When p and q are conditionally independent pr(p|q) = pr(p)

• By Bayes's Rule

pr(q|p) = pr(p|q) pr(q) / pr(p)
= pr(p) pr(q) / pr(p)
= pr(q)

• In this case, conjunction gets easier

• pr(p ∧ q) = pr(p|q) pr(q) = pr(p) pr(q)
• When p and q are strictly independent pr(p ∧ q) = 0

• In this case, disjunction gets easier

pr(p ∨ q)
= 1 - pr(¬p ∧ ¬q)
= 1 - pr(¬p) pr(¬q)
= 1 - (1 - pr(p))(1 - pr(q))
= 1 - (1 - pr(p) - pr(q) - pr(p) pr(q))
= pr(p) + pr(q) + pr(p) pr(q)
= pr(p) + pr(q)