Book contents
- Frontmatter
- Contents
- List of Figures
- List of Tables
- Preface
- Part I Fundamentals of Bayesian Inference
- 1 Introduction
- 2 Basic Concepts of Probability and Inference
- 3 Posterior Distributions and Inference
- 4 Prior Distributions
- Part II Simulation
- Part III Applications
- A Probability Distributions and Matrix Theorems
- B Computer Programs for MCMC Calculations
- Bibliography
- Author Index
- Subject Index
2 - Basic Concepts of Probability and Inference
from Part I - Fundamentals of Bayesian Inference
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- List of Figures
- List of Tables
- Preface
- Part I Fundamentals of Bayesian Inference
- 1 Introduction
- 2 Basic Concepts of Probability and Inference
- 3 Posterior Distributions and Inference
- 4 Prior Distributions
- Part II Simulation
- Part III Applications
- A Probability Distributions and Matrix Theorems
- B Computer Programs for MCMC Calculations
- Bibliography
- Author Index
- Subject Index
Summary
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:
The union of A and B is the event that A or B (or both) occur; it is denoted by A ∪ B.
The intersection of A and B is the event that both A and B occur; it is denoted by AB.
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
0 ≤ P(A) ≤ 1.
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.”
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- Information
- Introduction to Bayesian Econometrics , pp. 7 - 19Publisher: Cambridge University PressPrint publication year: 2007