Template:Bayesian test design

Bayeisan Nonparameteric Test Design

The nonparametric analysis performed in the last section made no assumptions about the underlying distribution of the data and was performed with only the immediate data at hand. However, if prior information regarding system performance is available, it can be incorporated into a Bayesian nonparametric analysis. This subsection will demonstrate how to incorporate prior information about system reliability and also how to incorporate prior information from subsystem tests.

Assumption on System Reliablity

If we assume the system reliablity follows a Beta distribuiton, the values of system reliability R, confidence level CL, number of units tested n, and number of failures r are related by the equation.

[math]\displaystyle{ 1-CL=\text{Beta}\left(R,\alpha,\beta\right)=\text{Beta}\left(R,n-r+\alpha_{0},r+\beta_{0}\right) }[/math]

where Beta is the incomplete Beta function. If [math]\displaystyle{ \alpha\,\!_{0} }[/math] and [math]\displaystyle{ \beta\,\!_{0} }[/math] are known, then any quantity of interest can be calculated using the remaining three. The next two examples demonstrate how to calculate [math]\displaystyle{ \alpha\,\!_{0} }[/math] and [math]\displaystyle{ \beta\,\!_{0} }[/math] depending on the type of prior information available.