Decision Theory and Bayes's Rule

Decision theory

  • The utility equation

  • Measuring information

  • Cost of gathering information

  • A Bayesian method

Expected Value

  • The expected value of a uniformly distributed set of numbers is the average of its values

    E(S) = sum S / |S|
  • More generally, the expected value is the centroid

    E(S) = sum [e in S] e pr(e in S)

    This reduces to the average when elements are equally probable

The Utility Equation

  • Sometimes you need to pick a best action a from a set of choices A

  • When you know what effect e(a) each action will have, and you know the value v(e) of each effect, this is easy: just pick

    argmax [a in A] v(e(a))
  • But the world is uncertain. Sometimes all you know is the probability of various action effects pr(e|a)

  • You might evaluate the utility U of an action in terms of expected effect value

    U(a) = E(v(a)) = sum [e in e(a)] v(e) pr(e|a)
  • You can now just maximize again

    argmax [a in A] U(a)

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

                   /      \
              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

Last modified: Monday, 26 October 2020, 11:40 PM