This chapter presents the basic concepts behind the construction of Bayesian statistical intervals and the integration of prior information with data that Bayesian methods provide. The use of such methods has seen a rapid evolution over recent years. The development of the theory and application of Markov chain Monte Carlo (MCMC) methods and vast improvements in computational capabilities have made the use of such methods feasible.
There are, moreover, many applications for which practitioners have solid prior information on certain aspects of their applications based on knowledge of the physical-chemical mechanisms and/or relevant experience with a previously studied phenomenon. For example, engineers often have useful, but imprecise, knowledge about the effective activation energy in a temperature-accelerated life test or about the Weibull distribution shape parameter in the analysis of fatigue failure data or the amount of variability in a measurement process. In such applications, the use of Bayesian methods is compelling as it offers an appropriate compromise between assuming that such quantities are known and assuming that nothing is known.
We describe three specific methods for obtaining Bayesian intervals: