Mathematical Tools for Updating Probabilities

This chapter discusses two common tools used to update and to help generate failure probabilities. These are Bayesian update and Monte Carlo analysis. Both of these techniques are used widely in the probabilistic risk assessment (PRA) community and can range from very simple mathematical manipulation of data to very complex algorithms. The chapter focuses on relatively simple versions of these techniques. For reliability models, frequentist methods treat model parameters as unknown, fixed constants and employ only observed data to estimate the values of parameters. In the case of failure time data, one might assume the data is exponentially distributed. Bayesian models treat parameters as unknown random variable whose distribution, or what is called the prior, represents the current belief about the parameter. The Monte Carlo method (or Monte Carlo simulation) can be used to describe any technique that approximates solutions to quantitative problems through statistical sampling.