8.2.5. What models and assumptions are typically made when Bayesian methods are usedfor reliability evaluation?

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 non repairable component populations, we will restrict ourselves to the system application in this Handbook.

2. The MTBF for the system can be thought of as chosen from a prior distribution model which 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 = 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 . There are many possible ways to convert "knowledge" to gamma parameters, depending on the form of the "knowledge" - we will describe three approaches.

i) If you have actual data from previous testing done on the system (or a system believed to have the same reliability as the one under investigation) this is the most credible prior knowledge, and the easiest to use. Simply set the gamma parameter a equal to the total number of failuress from all the previous data, and set the parameter b equal to the total of all the previous test hours.

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 sub components well from a reliability viewpoint.



• Have the group reach agreement on a reasonable MTBF they expect the system to have. They could each pick a number they would be willing to bet even money that the system would either meet or miss, and the average or median of these numbers would be their 50% best guess for the MTBF. Or they could just discuss even money MTBF candidates until a consensus is reached
  
  
• Repeat the process again, this time reaching agreement on a low MTBF they expect the system to exceed. A "5%" value that they are "95% confident" the system will exceed (i.e., they would give 19 to 1 odds) is a good choice. Or a "10%" value might be chosen (i.e., they would give 9 to 1 odds the actual MTBF exceeds the low MTBF). Use whichever percentile choice the group prefers.
  
  
• Call the reasonable MTBF MTBF₅₀ and the low MTBF you are 95% confident the system will exceed MTBF₀₅. These two numbers uniquely determine gamma parameters a and b that have percentile values at the right locations We call this method of specifying gamma prior parameters the 50/95 method (or the 50/90 method, if we use MTBF₁₀ , etc.). A simple way to calculate a and b for this method, using EXCEL, is described below.

iii) A third way of choosing prior parameters starts the same way as the second method. Consensus is reached on an reasonable MTBF, MTBF₅₀. Next, however, the group decides they want a somewhat weak prior that will change rapidly, based on new test information. If the prior parameter "a" is set to 1, the gamma has a standard deviation equal to its mean, which makes it spread out, or "weak". To insure the 50th percentile is set at ₅₀ = 1/ MTBF₅₀ , we have to choose b = ln 2 × MTBF₅₀, which is approximately .6931 × MTBF₅₀.

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

Many variations are possible, based on the above three methods. For example, you might have prior data from sources that you  don't completely trust. Or you might question whether the data really applies to the system under investigation. You might decide to "weight" the prior data by .5, to "weaken" it. This can be implemented by setting a = .5 x the number of fails in the prior data and b = .5 times the number of test hours. That spreads out the prior distribution more, and lets it react quicker to new test data.

Consequences

Under assumptions 1 and 2, when a new test is run with T system operating hours and r failures, the posterior distribution for is still a gamma, with new parameters:

In other words, add to a the number of new failures and add to b the number of new test hours to get 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. 

Using EXCEL To Obtain Gamma Parameters

1. Calculate the ratio RT = MTBF₅₀/MTBF₀₅.
   
2. Open an EXCEL spreadsheet and put any starting value guess for a in A1 - say 2.
   
   

3. Move to B1 and type the following expression:

   = GAMMAINV(.95,A1,1)/GAMMAINV(.5,A1,1)
   
   

4. Press enter and a number will appear in B1. We are going to use the "Goal Seek" tool EXCEL has to vary A1 until the number in B1 equals RT.
   
   

5. Click on "Tools" (on the top menu bar) and then on "Goal Seek". A box will open. Click on "Set cell" and highlight cell B1. $B$1 will appear in the "Set Cell" window. Click on "To value" and type in the numerical value for RT. Click on "By changing cell" and highlight A1 ($A$1 will appear in "By changing cell"). Now click "OK" and watch the value of the "a" parameter appear in A1.
   
   
6. Go to C1 and type
   = .5*MTBF₅₀*GAMMAINV(.5, A1, 2)

   and the value of b will appear in C1 when you hit enter.

Example

After clicking "OK", the value in A1 changes from 2 to 2.862978. This new value is the prior a parameter. (Note: if the group felt 250 was a MTBF₁₀ value, instead of a MTBF₀₅ 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.)

The figure below shows what to enter in C1 to obtain the prior "b" parameter value of 1522.46.

The gamma prior with parameters a = 2.863 and b = 1522.46 will have (approximately) a probability of 50% of l being below 1/600 = .001667 and a probability of 95% of being below 1/250 = .004. This can be checked by typing

as described when gamma EXCEl functions were introduced in Section 1. 

This example will be continued in Section 3, where the Bayesian test time needed to confirm a 500 hour MTBF at 80% confidence, will be derived.