Russell and Norvig, Chapter 15: Probabilistic Reasoning Systems

15.1 Representing Knowledge in an Uncertain Domain
- represention of uncertain knowledge from Chapter 14
  - complete joint probability distribution
  - conditional probabilities and Bayes rule
    - assuming conditional independence
- belief networks
  - nodes represent random variables
  - directed link between X and Y implies that X "directly influences" Y
  - each node has a conditional probability table (CPT) quantifying the
    effects that the parents (incoming links) have on the node
  - network is a DAG (no directed cycles)
- example belief network [Fig15.2, p439]
  - only represent what we know
  - which summarizes what we don't know
    - reasons alarm doesn't go off
    - reasons John and Mary don't call

15.2 Semantics of Belief Networks
- network represents the joint probability distribution
  - every entry of the JPD can be calculated form the network
    - P(X1=x1 ^ ... ^ Xn=xn) = P(x1,...,xn)
                             = Pi(i=1,n) P(xi|Parents(Xi))
    - how to construct network ?
- network encodes conditional independence statements
  - P(x1,...,xn) = Pi(i=1,n) P(xi|x(i-1),...,x1)
  - {\bf P}(Xi|X(i-1),...,X1) = P(Xi|Parents(Xi))                       (1)
    - provided Parents(Xi) subset of {X(i-1),...,X1}
      - i.e., order the nodes breadth first
    - those X(i-1),...,X1 not in Parents(Xi) implies conditional
      independence between the two
  - conditional independence implies no direct influence
    - e.g., MaryCalls and EarthQuake are conditionally independent
            MaryCalls and Alarm are not
  - network construction
    - algorithm
        choose variables (nodes) Xi
        order variables
        while variables left
          pick variables Xi and add node for it
          select Parents(Xi) such that conditional independence satisfied (1)
          define CPT for Xi
    - designer cannot (i.e., unable to) violate axioms of probability
    - order variables from causes to effects (causal model)
    - may not need to specify each entry of CPT
      - e.g., node represents disjunction of parents (add parent's probs.)
- conditional independence relations
  - are X and Y conditionally independent given evidence E
    - depends on whether E "blocks" the path from X to Y
    - graphical depiction [Fig15.4, p445], example [Fig15.5, p446]
    - needed for inference

15.3 Inference in Belief Networks
- given network, compute P(Query|Evidence)
  - evidence obtained from percepts
- possible inferences
  - diagnostic: P(Burglary|JohnCalls)
  - causal: P(JohnCalls|Burglary)
  - intercausal: P(Burglary|Alarm ^ Earthquake)
  - mixed
- algorithm for polytrees [Fig 15.7, p449; Fig15.8, p452]
  - at most one directed path between two nodes
  - example P(Burglary|JohnCalls) = 0.016

15.4 Inference in Multiply-Connected Belief Networks
- clustering [Fig15.9-15.10, p454]
  - convert to polytree by combining alternative nodes into a meganode
  - use polytree algorithm
  - queries on clustered variables averaged over values of others
  - CPTs grow exponentially in size
- cutset conditioning [Fig15.11, p455]
  - form multiple polytrees
    - choose cutset
    - form a polytree for each instantiation of variables in cutset
    - average results
  - exponential number of polytrees
  - bounded cutset conditioning
    - only use a subset of polytrees
- stochastic simulation
  - logic sampling
    - randomly chose values of root-node variables according to their priors
      - continue with next layer, using CPTs
    - iterate for a number of trials
    - P(Q|E) = [freq(Q and E) / freq(Q)] in trials
    - difficult for rare values of E, so ...
  - likelihood weighting
    - use given evidence values, instead of random
    - randomly choose non-evidence values
    - use CPTs to compute likelihood of this value
    - use accumulated CPTs to weight query value result
    - iterate (choosing different random values)
    - compute P(Q|E) using frequencies weighted by likelihoods
    - example in text [p456]

15.5 Knowledge Engineering for Uncertain Reasoning
- decide what to talk about
- decide on vocabulary of random variables
  - discretize continuous variables
- encode knowledge about variable dependencies
  - links and CPTs
- encode case description
- pose queries
- case study: PathFinder IV
  - lymph node diseases
  - almost 90% accurate
  - better than the experts involved in design

15.6 Other Approaches to Uncertain Reasoning
- default reasoning
  - statements assumed true in the absence of contradictory evidence
- rule-based methods
  - locality
    - if A=>B and A, then B without considering other evidence
  - detachment
    - once B concluded, then used despite dependence on A
  - these allow more efficiency
  - but not appropriate for uncertain reasoning
  - add certainty factors (MYCIN)
    - but can lead to incorrect inferences
- representing ignorance
  - Dempster-Shafer theory
  - computes belief function
    - probability evidence supports proposition
    - not probability of proposition itself
- representing vagueness
  - fuzzy sets and fuzzy logic
  - propositions are no longer true or false, but a number between 0 and 1
  - evaluation of sentences similar to certainty factors
    - fuzzy logic
  - vagueness on meaning is not uncertainty