Probability

AI, Decision Making, and Probability

• About decision-making

• 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](http://github.com/pdx-cs-ai/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)
    

Bayes's Rule

• Given

  • evidence E with prior probability pr(E)

  • hypothesis H with pp pr(H)

  • probability pr(E|H) of the evidence given that the hypothesis holds

• We want

  • probability pr(H|E) that the hypothesis holds given the evidence

• By Bayes's Rule

  • pr(H|E) = pr(E|H) pr(H) / pr(E)

The Medical Example

• H = "You have Glaubner's Disease"
  E = "Reaper's Test is positive"

• Rare disease: pr(H) = 1e-6

• Low false positive rate: pr(E|¬H) = 1e-4

• Perfect false negative rate: pr(E|H) = 1

  pr(H|E) = pr(H) pr(E|H) / pr(E)
          = pr(H) / pr(E∧H ∨ E∧¬H)
          = pr(H) / (pr(E|¬H) pr(¬H) + pr(E|H) pr(H))
          = 1e-6 / (1e-4 × (1-1e-6) + 1e-6)
          ~= 1e-6 / 1e-4 = 1e-2 = 0.01
  

• IRL the false negative rate will be nonzero too, so you will not learn a ton from the test either way

• Been there

Bayesian Belief Networks

• Bayes Net (BBN, influence diagram) : indicate which priors and conditionals have significant influence in practice

                  Cloudy
                 /      \
            Sprinkler   Rain
                 \      /
                Wet Grass
  

• Usually reason one of two ways:

  • causal/top-down:

     p(W|C) = p(W|S∨R)
            = 1 - p(W|¬S) p(¬S|C) p(C) ×
                  p(W|¬R) p(¬R|C) p(C)
    

  • diagnostic/bottom-up: p(C|W)

• Polytrees: special case for easy computation

• Problem: everything depends on everything else

  • Need to know impossible number of prior and conditional probabilities to conclude anything

  • Maybe learn probabilities?

Is Your Probabilistic Model Meaningful?

• Difference between 0.5 and "don't know" and "don't care"

• MYCIN and probabilities vs "likelihoods"

• Consequence of Cox's Theorem: under reasonable assumptions, any labeling of logical sentences with real numbers will be consistent with probability

• Measurement issues; numeric issues