Practical Applications of Bayesian Reliability

2Basic Concepts of Bayesian Statistics and Models

This chapter introduces basic concepts of Bayesian statistics. We start with the basic idea of Bayes' theorem, followed by an introduction of Bayesian inference. Selection of prior distributions and commonly used conjugate prior distributions are briefly discussed. We also summarize the difference between Bayesian inference and frequentist inference, and how Bayesian inference works with Monte Carlo simulation methods. Basic concepts of point estimation, interval estimation, Bayes' factor, and prediction are provided. In addition, we will show how to use Bayes' factor as a model selection tool.

2.1 Basic Idea of Bayesian Reasoning

Bayesian reasoning is an approach of learning from evidence as it accumulates. To illustrate this basic idea, let's think about the following two scenarios.

• Scenario 1. Tom just moved to a new city without investigating the crime rates in the area. He randomly selected a neighborhood. Over the next few months, Tom continually heard or observed crime incidents in his neighborhood. How do these observed events change Tom's impression of the community? Tom may come to the idea that he lives in a neighborhood with a relatively high crime rate.
• Scenario 2. Tom decided to move to another block with a lower crime rate. This time, he learned about the crime data from the local police department. According to the crime statistics, he moved to one of the safest districts in the city. Tom lived happily in his ...