Probability

SINCE STATISTICAL INFERENCE is based on probability theory, the major difference between Bayesian and frequentist approaches to inference can be traced to the different views that each have about the interpretation and scope of probability theory. We therefore begin by stating the basic axioms of probability and explaining the two views.

A probability is a number assigned to statements or events. We use the terms “statements” and “events” interchangeably. Examples of such statements are

• A1 = “A coin tossed three times will come up heads either two or three times.”

• A2 = “A six-sided die rolled once shows an even number of spots.”

• A3 = “There will be measurable precipitation on January 1, 2008, at your local airport.”

Before presenting the probability axioms, we review some standard notation:

1. The union of A and B is the event that A or B (or both) occur; it is denoted by A ∪ B.

2. The intersection of A and B is the event that both A and B occur; it is denoted by AB.

3. The complement of A is the event that A does not occur; it is denoted by Ac.

The probability of event A is denoted by P(A). Probabilities are assumed to satisfy the following axioms:

Probability Axioms

1. 0 ≤ P(A) ≤ 1.

2. P(A) = 1 if A represents a logical truth, that is, a statement that must be true; for example, “A coin comes up either heads or tails.”

3. […]