8.
Assessing Product Reliability
8.1. Introduction


Several Bayesian Methods overview topics are covered in this section  This section gives an overview
of the application of Bayesian techniques in reliability investigations.
The following topics are covered:


Bayesian analysis considers population parameters
to be random, not fixed
Old information, or subjective judgment, is used to determine a prior distribution for these population parameters 
It makes a great deal of practical sense to
use all the information available, old and/or new, objective or subjective,
when making decisions under uncertainty. This is especially true when the
consequences of the decisions can have a significant impact, financial
or otherwise. Most of us make everyday personal decisions this way, using
an intuitive process based on our experience and subjective judgments.
Mainstream statistical analysis, however, seeks objectivity by generally restricting the information used in an analysis to that obtained from a current set of clearly relevant data. Prior knowledge is not used except to suggest the choice of a particular population model to "fit" to the data, and this choice is later checked against the data for reasonableness. Lifetime or repair models, as we saw earlier when we looked at repairable and non repairable reliability population models, have one or more unknown parameters. The classical statistical approach considers these parameters as fixed but unknown constants to be estimated (i.e., "guessed at") using sample data taken randomly from the population of interest. A confidence interval for an unknown parameter is really a frequency statement about the likelihood that numbers calculated from a sample capture the true parameter. Strictly speaking, one cannot make probability statements about the true parameter since it is fixed, not random. The Bayesian approach, on the other hand, treats these population model parameters as random, not fixed, quantities. Before looking at the current data, we use old information, or even subjective judgments, to construct a prior distribution model for these parameters. This model expresses our starting assessment about how likely various values of the unknown parameters are. We then make use of the current data (via Baye's formula) to revise this starting assessment, deriving what is called the posterior distribution model for the population model parameters. Parameter estimates, along with confidence intervals (known as credibility intervals), are calculated directly from the posterior distribution. Credibility intervals are legitimate probability statements about the unknown parameters, since these parameters now are considered random, not fixed. It is unlikely in most applications that data will ever exist to validate a chosen prior distribution model. Parametric Bayesian prior models are chosen because of their flexibility and mathematical convenience. In particular, conjugate priors (defined below) are a natural and popular choice of Bayesian prior distribution models. Bayes Formula, Prior and Posterior Distribution Models, and Conjugate Priors 

Bayes formula provides the mathematical tool that combines prior knowledge with current data to produce a posterior distribution 
Bayes formula is a useful equation from probability
theory that expresses the conditional probability of an event A occurring,
given that the event \(B\)
has occurred (written P\((AB)\)),
in terms of
unconditional probabilities and the probability the event \(B\)
has occurred, given that \(A\)
has occurred. In other words, Bayes formula
inverts which of the events is the conditioning event. The formula is
$$ \mbox{P}(AB) = \frac{\mbox{P}(A,B)}{\mbox{P}(B)} = \frac{\mbox{P}(A) \cdot \mbox{P}(BA)}{\mbox{P}(B)} \, , $$
and P(\(B\))
in the denominator is further expanded by using the socalled
"Law of Total Probability" to write
$$ \mbox{P}(B) = \sum_{i=1}^n \mbox{P}(BA_i) \mbox{ P}(A_i) \, , $$
with the events \(A_i\)
being mutually exclusive and exhausting
all possibilities and including the event \(A\)
as one of the \(A_i\).
The same formula, written in terms of probability density function models, takes the form: $$ g(\lambda  x) = \frac{f(x  \lambda) g(\lambda)}{\int_0^\infty f(x  \lambda) g(\lambda) d \lambda} \, , $$ where \(f(x  \lambda)\) is the probability model, or likelihood function, for the observed data \(x\) given the unknown parameter (or parameters) \(\lambda\), \(g(\lambda)\) is the prior distribution model for \(\lambda\), and \(g(\lambda  x)\) is the posterior distribution model for \(\lambda\) given that the data \(x\) have been observed. When \(g(\lambda  x)\) and \(g(\lambda)\) both belong to the same distribution family, \(g(\lambda)\) and \(f(x  \lambda)\) are called conjugate distributions and \(g(\lambda)\) is the conjugate prior for \(f(x  \lambda)\). For example, the Beta distribution model is a conjugate prior for the proportion of successes \(p\) when samples have a binomial distribution. And the Gamma model is a conjugate prior for the failure rate \(\lambda\) when sampling failure times or repair times from an exponentially distributed population. This latter conjugate pair (gamma, exponential) is used extensively in Bayesian system reliability applications. How Bayes Methodology is used in System Reliability Evaluation 

Bayesian system reliability evaluation assumes the system MTBF is a random quantity "chosen" according to a prior distribution model  Models and assumptions for using Bayes methodology
will be described in a later section.
Here we compare the classical paradigm versus the Bayesian paradigm when
system reliability follows the HPP or exponential
model (i.e., the flat portion of the Bathtub Curve).
Classical Paradigm For System Reliability Evaluation:


Pro's and con's for using Bayesian methods  While the primary motivation to use Bayesian
reliability methods is typically a desire to save on test time and materials
cost, there are other factors that should also be taken into account. The
table below summarizes some of these "good news" and "bad news" considerations.
Bayesian Paradigm: Advantages and Disadvantages
