Decision Theory and Bayes's Rule
The utility equation
Cost of gathering information
A Bayesian method
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)
- 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
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:
p(W|C) = p(W|S∨R) = 1 - p(W|¬S) p(¬S|C) p(C) × p(W|¬R) p(¬R|C) p(C)
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