A Bayesian framework for uncertainty quantification of perturbed gamma process based on simulated likelihood

The perturbed gamma process (PGP) has recently been widely used in modeling the noisy degradation data collected from engineering structures and components since it can simultaneously consider the temporal variability of degradation and measurement uncertainty. As a result of the sampling and inspection uncertainty in engineering practice, it is necessary to account for the resulting parameter uncertainty. Meanwhile, the flexibility of the form of measurement error motivates a potential demand for quantifying the model uncertainty and selecting the most fitting error model for the given inspection data. The Bayesian approach is well-suited to quantity the parameter uncertainty induced by imperfect inspection and limited inspection data, but its practical implementation is extremely challenging due to the intractable likelihood function of PGP. In the paper, a novel Bayesian framework for quantifying parameter and model uncertainty of PGP is presented, where the simulated likelihood that is an unbiased estimator generated by Sequential Monte Carlo (SMC) is introduced to overcome the intractable likelihood of PGP. More specifically, an Adaptive Particle Markov chain Monte Carlo (APMCMC) is proposed to perform the Bayesian sampling from the posterior distributions of parameters, achieving the requirement for the quantification of parameter uncertainty. By utilizing the posterior samples from APMCMC, a model selection method based on the Bayes factor is employed to determine the most fitting one from some alternative error models. Finally, two simulation examples are presented to illustrate the efficiency and accuracy of the proposed framework and its applicability is confirmed by a practical case involving the corrosion modeling of a group of pipelines.