8.
Assessing Product Reliability
8.2. Assumptions/Prerequisites


The basics of Bayesian methodology were explained earlier, along with some of the advantages and disadvantages of using this approach. Here we only consider the models and assumptions that are commonplace when applying Bayesian methodology to evaluate system reliability.  
Bayesian assumptions for the gamma exponential system model 
Assumptions:
1. Failure times for the system under investigation can be adequately modeled by the exponential distribution. For repairable systems, this means the HPP model applies and the system is operating in the flat portion of the bathtub curve. While Bayesian methodology can also be applied to nonrepairable component populations, we will restrict ourselves to the system application in this Handbook. 2. The MTBF for the system can be regarded as chosen from a prior distribution model that is an analytic representation of our previous information or judgments about the system's reliability. The form of this prior model is the gamma distribution (the conjugate prior for the exponential model). The prior model is actually defined for \(\lambda\) = 1/MTBF since it is easier to do the calculations this way. 3. Our prior knowledge is used to choose the gamma parameters \(a\) and \(b\) for the prior distribution model for \(\lambda\). There are many possible ways to convert "knowledge" to gamma parameters, depending on the form of the "knowledge"  we will describe three approaches. 

Several ways to choose the prior gamma parameter values 
ii) A consensus method for determining \(a\) and \(b\)
that works well is the following: Assemble a group of engineers who know the system
and its subcomponents well from a reliability viewpoint.
Note: As we will see when we plan Bayesian tests, this weak prior is actually a very friendly prior in terms of saving test time Consequences 

After a new test is run, the posterior gamma parameters are easily obtained from the prior parameters by adding the new number of fails to "\(a\)" and the new test time to "\(b\)" 
No matter how you arrive at values for the gamma
prior parameters \(a\) and \(b\),
the method for incorporating new
test information is the same. The new information is combined with the
prior model to produce an updated or
posterior distribution model for \(\lambda\).
Under assumptions 1 and 2, when a new test is run with \(T\) system operating hours and \(r\) failures, the posterior distribution for \(\lambda\) is still a gamma, with new parameters: $$ a' = a + r, \,\,\, b' = b + T \, . $$ In other words, add to \(a\) the number of new failures and add to \(b\) the number of new test hours to obtain the new parameters for the posterior distribution. Use of the posterior distribution to estimate the system MTBF (with confidence, or prediction, intervals) is described in the section on estimating reliability using the Bayesian gamma model. Obtaining Gamma Parameters 

An example using the "50/95" consensus method 
A group of engineers, discussing the reliability of a new piece
of equipment, decide to use the 50/95 method to convert
their knowledge into a Bayesian gamma prior. Consensus is reached on a
likely \(\mbox{MTBF}_{50}\)
value of 600 hours and a low \(\mbox{MTBF}_{05}\)
value of 250. \(RT\)
is 600/250 = 2.4.
(Note: if the group felt that 250 was a \(\mbox{MTBF}_{10}\)
value, instead of a \(\mbox{MTBF}_{05}\)
value, then the only change needed would be to replace 0.95 in the B1 equation
by 0.90. This would be the "50/90" method.)
Using software to find the root of a univariate function, the gamma prior parameters were found to be \(a\) = 2.863 and \(b\) = 1522.46. The parameters will have (approximately) a probability of 50 % of \(l\) being below 1/600 = 0.001667 and a probability of 95 % of \(\lambda\) being below 1/250 = 0.004. (The probabilities are based on the 0.001667 and 0.004 quantiles of a gamma distribution with shape parameter \(a\) = 2.863 and scale parameter \(b\) = 1522.46). The gamma parameter estimates in this example can be produced using R code. This example will be continued in Section 3, in which the Bayesian test time needed to confirm a 500 hour MTBF at 80 % confidence will be derived. 