The physics-of-failure (POF) modeling approach is a proven and powerful method to predict the reliability of mechanical components and systems. Most of POF models have been originally developed based upon empirical data from a wide range of applications (e.g. fracture mechanics approach to the fatigue life). Available curve fitting methods such as least square for example, calculate the best estimate of parameters by minimizing the distance function. Such point estimate approaches, basically overlook the other possibilities for the parameters and fail to incorporate the real uncertainty of empirical data into the process. The other important issue with traditional methods is when new data points become available. In such conditions, the best estimate methods need to be recalculated using the new and old data sets all together. But the original data sets, used to develop POF models may be no longer available to be combined with new data in a point estimate framework. In this research, for efficient uncertainty management in POF models, a powerful Bayesian framework is proposed. Bayesian approach provides many practical features such as a fair coverage of uncertainty and the updating concept that provide a powerful means for knowledge management, meaning that the Bayesian models allow the available information to be stored in a probability density format over the model parameters. These distributions may be considered as prior to be updated in the light of new data when they become available. At the first part of this article a brief review of classical and probabilistic approach to regression is presented. In this part the accuracy of traditional normal distribution assumption for error is examined and a new flexible likelihood function is proposed. The Bayesian approach to regression and its bonds with classical and probabilistic methods are explained next. In Bayesian section we shall discuss how the likelihood functions introduced in probabilistic approach, can be combined with prior information using the conditional probability concept. In order to highlight the advantages, the Bayesian approach is further clarified with case studies in which the result of calculation is compared with other traditional methods such as least square and maximum likelihood estimation (MLE) method. In this research, the mathematical complexity of Bayesian inference equations was overcome utilizing Markov Chain Monte Carlo simulation technique.
The promise of Bayesian analysis for prominence seismology
