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Conservative claims about the probability of perfection of software-based systems

In recent years we have become interested in the problem of assessing the probability of perfection of software-based systems which are sufficiently simple that they are "possibly perfect". By "perfection" we mean that the software of interest will never fail in a specific operating environment. We can never be certain that it is perfect, so our interest lies in claims for its probability of perfection. Our approach is Bayesian: our aim is to model the changes to this probability of perfection as we see evidence of failure-free working. Much of the paper considers the difficult problem of expressing prior beliefs about the probability of failure on demand (pfd), and representing these mathematically. This requires the assessor to state his prior belief in perfection as a probability, and also to state what he believes are likely values of the pfd in the event that the system is not perfect. We take the view that it will be impractical for an assessor to express these beliefs as a complete distribution for pfd. Our approach to the problem has three threads. Firstly we assume that, although he cannot provide a full probabilistic description of his uncertainty in a single distribution, the assessor can express some precise but partial beliefs about the unknowns. Secondly, we assume that in the inevitable presence of such incompleteness, the Bayesian analysis needs to provide results that are guaranteed to be conservative (because the analyses we have in mind relate to critical systems). Finally, we seek to prune the set of prior distributions that the assessor finds acceptable in order that the conservatism of the results is no greater than it has to be, i.e. we propose, and eliminate, sets of priors that would appear generally unreasonable. We give some illustrative numerical examples of this approach, and note that the numerical values obtained for the posterior probability of perfection in this way seem potentially useful (although we make no claims for the practical realism of the numbers we use). We also note that the general approach here to the problem of expressing and using limited prior belief in a Bayesian analysis may have wider applicability than to the problem we have addressed.