
 A samplespace (e.g. S={a,c,g,t})
is the set of possible outcomes of some experiment.
 Events A, B, C, ..., H, ...
An event is a subset (possibly a singleton) of the sample space,
e.g. Purine={a,g}.
 Events have probabilities P(A), P(B), etc.
 Random variables X, Y, Z, ...
A random variable X takes values, with certain probabilities,
from the sample space.
 We may write P(X=a), P(a) or P({a}) for the probability that X=a.
Thomas Bayes (17021761)
Thomas Bayes
made an early study of probability and games of chance.
Bayes' Theorem
If B_{1}, B_{2}, ..., B_{k}
is a partition of a set B (of causes) then

P(B_{i}A) =
P(AB_{i}) P(B_{i})
/ ∑_{j=1..k} P(AB_{j}) P(B_{j})
 i = 1, 2, ..., k
One and only one of the B_{i} must occur
because they are a partition of B.
Inference
Bayes theorem is relevant to inference because
we may be entertaining a number of exclusive and exhaustive hypotheses
H_{1}, H_{2}, ..., H_{k}, and
wish to know which is the best explanation of some observed data D.
In that case P(H_{i}D) is called the posterior probability
of H_{i}, "posterior" because
it is the probability after the data has been observed.
 ∑_{j=1..k} P(DH_{j}) P(H_{j}) = P(D)

 P(H_{i}D) = P(DH_{i}) P(H_{i}) / P(D)
posterior
Note that the H_{i} can even be an infinite enumerable set.
P(H_{i}) is called the prior probability
of H_{i}, "prior" because
it is the probability before D is known.
Notes
 T. Bayes.
An essay towards solving a problem in the doctrine of chance.
Phil. Trans. of the Royal Soc. of London, 53, pp.370418, 1763.
Reprinted in Biometrika, 45, pp.296315, 1958.
Conditional Probability
The probability of B given A is written P(BA).
It is the probability of B provided that A is true;
we do not care, either way, if A is false.
Conditional probability is defined by:
 P(A&B) = P(A).P(BA) = P(B).P(AB)

 P(AB) = P(A&B) / P(B)
 P(BA) = P(A&B) / P(A)
These rules are a special case of Bayes' theorem for k=2.
There are four combinations for two Boolean variables:

 A  not A 
margin 
B 
A & B  not A & B 
(A or not A)& B = B 
not B 
A & not B  not A & not B 
(A or not A)& not B = not B 
margin 
A = A&(B or not B) 
not A = not A &(B or not B) 
LA 1999 
We can still ask what is the probability of A, say, alone
 P(A) = P(A & B) + P(A & not B)
 P(B) = P(A & B) + P(not A & B)
Independence
A and B are said to be independent
if the probability of A does not depend on B and v.v..
In that case P(AB)=P(A) and P(BA)=P(B) so
 P(A&B) = P(A).P(B)
 P(A & not B) = P(A).P(not B)
 P(not A & B) = P(not A).P(B)
 P(not A & not B) = P(not A).P(not B)
A Puzzle
I have a dice (made it myself, so it might be "tricky")
which has 1, 2, 3, 4, 5 & 6 on different faces.
Opposite faces sum to 7.
The results of rolling the dice 100 times (good vigorous rolls on carpet) were:
1 20:
3 1 1 3 3 5 1 4 4 2 3 4 3 1 2 4 6 6 6 6
21 40:
3 3 5 1 3 1 5 3 6 5 1 6 2 4 1 2 2 4 5 5
41 60:
1 1 1 1 6 6 5 5 3 5 4 3 3 3 4 3 2 2 2 3
61 80:
5 1 3 3 2 2 2 2 1 2 4 4 1 4 1 5 4 1 4 2
81100:
5 5 6 4 4 6 6 4 6 6 6 3 1 1 1 6 6 2 4 5
Can you learn anything about the dice from these results?
What would you predict might come up at the next roll?
How certain are you of your prediction?
 LA 1999

