Russell and Norvig, Chapter 14: Uncertainty

14.1 Acting Under Uncertainty
- in the real world, an agent can never be 100% sure
- but, agent can make a Rational Decision
  - action benefits vs certainty of success
- probability
  - degree of belief in either being true or false
  - summarizes uncertainty
  - usually statistically determined
  - not degree of truth (fuzzy logic)
  - prior or unconditional: P(Sentence)
  - posterior or conditional: P(Sentence|Percepts)
- utility theory
  - states have utility values or preferences
  - agents prefer states with higher utility
- decision theory = probability theory + utility theory
  - Maximum Expected Utility (MEU) principle
    - agent is rational if chooses actions yielding highest expected
      utility, averaged over all possible action outcomes
  - weight utility of an outcome by its probability of occurring
- decision-theoretic agent [Fig14.1, p419]

14.2 Basic Probability Notation
- random variables
- domain: possible values of a random variable
  - usually discrete
- prior probability
  - {\bf P}(X) = < P(X=x1), P(X=x2), ... >
  - {\bf P}(X,Y) = < P(X=xi,Y=yi) >
  - P(X and (not Y)), logical sentences allowed
- conditional probability
  - P(X|Y): probability of X when we know Y and nothing else
  - {\bf P}(X|Y) = < P(X=xi|Y=yi) >
  - P(X and Y) = P(X|Y) * P(Y) = P(Y|X) * P(X)
  - {\bf P}(X,Y) = {\bf P}(X|Y) * {\bf P}(Y)
                 = < P(X=xi,Y=yi) = P(X=xi|Y=yi) * P(Y=yi) >
  - P(X|Y) = p does not imply "if Y is true, P(X) = p"
    - P(X) = P(X| )
    - P(X|Y) applicable only when Y is only evidence

14.3 Axioms of Probability
- 0 <= P(A) <= 1
- P(True) = 1, P(False) = 0
- P(A or B) = P(A) + P(B) - P(A and B)
- joint probability distribution
  - {\bf P}(X1,...,Xn) is an n-dimensional table of probabilities
  - complete joint table allows computation of any probability
  - complete table typically infeasible

14.4 Bayes Rule
- P(B|A) = P(A|B) * P(B) / P(A)
- {\bf P}(Y|X) = {\bf P}(X|Y) * {\bf P}(Y) / {\bf P}(X)
- {\bf P}(Y|X,E) = {\bf P}(X|Y,E) * {\bf P}(Y|E) / {\bf P}(X|E)
  - the conditional probability of Y given X and background evidence E
- P(rain|wet) = P(wet|rain) * P(rain) / P(wet)
  - more likely to know probability of rain causing wet P(wet|rain)
- finding ratio P(rain|wet)/P(sprinkler|wet) does not need P(wet)
- can also avoid P(wet) through Normalization
  - need P(wet|~rain)
  - P(rain|wet) =              P(wet|rain) * P(rain)
                  -------------------------------------------------
                   P(wet|rain) * P(rain) + P(wet|~rain) * P(~rain)
  - in general, {\bf P}(Y|X) = alpha * {\bf P}(X|Y) * {\bf P}(Y)
    - where alpha chosen such that sum({\bf P}(Y|X)) = 1
- combining evidence
  - how to compute P(rain|wet ^ thunder) = P(w^t|r) * P(r) / P(w^t)
    - know P(w^t|r) possibly, but tedious as evidence increases
  - conditional independence of "symptoms"
    - thunder does not cause wet, and vice versa
    - P(r|w^t) = alpha * P(r) * P(w|r) * P(t|r)  ( with normalization)
    - true in the wumpus world ?

14.5 Where Do Probabilities Come From?
- statistical sampling
- universal principles
- individual beliefs