Bayesian statistics: A statistical theory of evidence

event A     P(A) - the probability of A     A real number between 0 and 1 inclusive (Rule 1 in O'Reilly text Ch. 12)

event B     P(B) - the probability of B     A real number between 0 and 1 inclusive

If S represents the entire sample space for the event, then P(S) = 1 (Rule 2 in text)

Given the probability of event A occuring is P(A), then the probability of A not occurring is 1-P(A) (Rule 3 in text)

However, if two events are mutually exclusive (only one can occur at a time) then P(A U B)= P(A) + P(B) (Rule 4 of text)

If two events are not mutually exclusive then P(A U B)= P(A) + P(B) - P(A and B) (Rule 5 of text)

If A and B are independent (occurance of one does not depend on the occurance of the other)
P(A and B) = P(A) * P(B)     (Rule 6 in text)

Rating moves in a game (using probabilities)

Score = P(i) * rating(i)

i= 1 to # of possible outcomes

"weighting" the contribution of each possible outcome

Bayes Theorem:

Conditional probabilities - events are not independent

P(H_i | E) = probability that hypothesis H_i is true given evidence E

P(E | H_i) = probability that one will observe E given H_i

P(H_i) = a priori probability

( in the absence of evidence - "priors")

if k = number of possibilities

P(H_i | E) = P(E | H_i) * P(H_i)
                P(E | H_n) * P(H_n))

(the above should be P(E | H_i) * P(H_i) divided by P(E | H_n) * P(H_n)) if it came out right

(if exact 1/1 = 1 )

(notice complete set of hypothesis needed)

(it is often difficult to collect all the a priori conditional and joint probabilities required.)

In a complex world, n may be very large and as new evidence (E) is given, the prior body of evidence (e) changes.

P(H|E, e) = P(H|E) * [P(e|E,H) / P(e|E)]

Examples: