Probability

Elementary Probability

A sample-space (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 (1702-1761)

Thomas Bayes made an early study of probability and games of chance.

Bayes's Theorem

If B₁, B₂, ..., B_k is a partition of a set B (of causes) then

P(B_i|A) = P(A|B_i) P(B_i) / ∑_j=1..k P(A|B_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's theorem is relevant to inference because we may be entertaining a number of exclusive and exhaustive hypotheses H₁, H₂, ..., 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(D|H_j) P(H_j) = P(D)

P(H_i|D) = P(D|H_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.370-418, 1763.
Reprinted in Biometrika, 45, pp.296-315, 1958.

Conditional Probability

The probability of B given A is written P(B|A). 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(B|A) = P(B).P(A|B)

P(A|B) = P(A&B) / P(B)

P(B|A) = P(A&B) / P(A)

These rules are a special case of Bayes's 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(A|B)=P(A) and P(B|A)=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

81-100: 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