Two methods are presented below that yield the same posterior distribution.
We tested 18 * 30 = 540 blades, of which 1 failed. That leads to an estimate of:
P(TurbineFail) = 1-(1- P(BladeFail))30
The object of this analysis was to determine P(BladeFail) and from that determine P(TurbineFail). A Bayesian estimate of P(BladeFail) required the parameter F, which was uncertain. The distribution for F is a hyperparameter. By using simulation to arrive at P(BladeFail) we have a natural way of integrating the extra uncertainty that F introduces into the calculation.
The links to the Turbine Blade Simulation software specific models are provided here:
Here is teh result of the above model:
The graph shows that we are 90% confident that the probability a turbine fails within period T is less than 28%.
Here is the result of the above model:
The graph shows that we are 90% confident that the probability a turbine fails within period T is less than 29.4%.
Method 2: Turbine Blade Construction
We can construct a confidence distribution for P(BladeFail) = q as follows:
This leads to the following posterior distribution:
From that analysis, Crystal Ball's Custom distribution can then be used to sample values from the uncertainty distribution, as shown in the Turbine blade construction model. The analysis has not yet taken into account the uncertainty around P(detect), but we can do this with numerical integration. We can create a spreadsheet cell that simulates simulation for P(detect), as before:
P(detect) = Beta(4+1,5-4+1,1) = Beta(5,2,1)
and then link it to the BINOMDIST likelihood calculation. Crystal Ball's CB.GetForeStatFN(..,2) function can then be used to calculate Then we calculate the posterior after this hyperparameter has been taken into account, by simply taking the mean of the simulated posterior density:
If you did not need a posterior distribution for P(BladeFail), the extra step of using the CB.GetForeStatFN(..,2) function is unnecessary: it is sufficient to simply run a simulation that samples from the Beta distribution for P(detect), constructs the General distribution for P(BladeFail), samples from that General distribution and calculates a value for P(TurbineFail), your output. All options are demonstrated in the accompanying model.
The links to the Turbine Blade Construction software specific models are provided here:
In the model, Crystal Ball's Custom distribution was used to sample values from the uncertainty distribution.
In the model, @Risk's General distribution was used to sample values from the uncertainty distribution.