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

                  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