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A Bayesian analysis of the software reliability model of Jelinski and Moranda is given, based upon Meinhold and Singpurwalla. Important extensions are provided to the stopping rule and prior distribution of the number of defects, as well as permitting uncertainty in the failure rate. It is easy to calculate the predictive distribution of unfound errors at the end of software testing, and to see the relative effects of uncertainty in the number of errors and in the detection efficiency. The behavior of the predictive mode and mean over time are examined as possible point estimators, but are clearly inferior to calculating the full predictive distribution.