Bayes' Theory
Representing Uncertainty

Q = Set of possible answers (frame of discernment)

Bel(T) = Belief that the right answer is on T
T contained in Q

Ex.

Q = {hepatitis, cirrhosis, gallstones, pancreatic cancer}

Bel({Hepatitis, Cirrhosis}) = strength of belief that either hepatitis or cirrhosis is the diagnosis

Rules:

1. Bel(Q) = 1     Belief that there is an answer

2. Bel ({}) = 0     Belief that there is no answer

3. Bel (A OR B) = Bel(A) + Bel(B) when A OR B = {} (disjoint)
   Bayes' Rule of Additivity

4. Bel (B | A) = Bel(B AND A) / Bel(A) = Bel(B) * Bel(A | B) / Bel(A)
   remember P(Hi|E) = probability that hypothesis Hi is true given evidence E
   
   P(E|Hi) = probability that one will observe E given Hi

Sherlock Holmes and the Sweet Shop Burglary

Q= {LI, LO, RI, RO} (4 suspects)

1. initially ignorant but need priors
   so set Bel(A) = 1/4 for all A in Q

2. Evidence , E1, that thief is a Lefty = 3/4

3. Compute posteriors
   Bel (LI|E1 )= Bel(LI) * Bel(E1|LI) / Bel(E1) = ?
   Bel (LO|E1)
   Bel (RI|E1)
   Bel (RI|E1)

   normalize

4. Evidence , E2, that thief is an Insider = 2/3

5. Compute posteriors
   
   Bel (LI|E2)
   Bel (LO|E2)
   Bel (RI|E2)
   Bel (RI|E2)

   normalize

Note that undistributed belief can be given to non-zero propositions as easily as to the zeros. An argument for the latter is that it keeps propositions alive

Some Objections to Bayes

1. Additivity requires that the Bel(not A) = 1- Bel(A)

2. No consistent way to represent ignorance
   usually distribute eqully

3. Not easy to represent disconfirming evidence
   evidence against A can be given by Bel(not A|E) but how to distribute this among the propositions making up not A?

4. Modifications due to new evidence is strange
   the new evidence is represented as a proposition not in Q (i.e., in Q x {E}, which gives a new Q). The new evidence must be treated as certain. But the priors contain old evidence

5. Forces detailed prior opinions over Q