Bayesian inference with Subset Simulation: Strategies and improvements

Bayesian Updating with Structural reliability methods (BUS) reinterprets the Bayesian updating problem as a structural reliability problem; i.e. a rare event estimation. The BUS approach can be considered an extension of rejection sampling, where a standard uniform random variable is added to the space of random variables. Each generated sample from this extended random variable space is accepted if the realization of the uniform random variable is smaller than the likelihood function scaled by a constant c. The constant c has to be selected such that 1∕c is not smaller than the maximum of the likelihood function, which, however, is typically unknown a-priori. A c chosen too small will have negative impact on the efficiency of the BUS approach when combined with sampling-based reliability methods. For the combination of BUS with Subset Simulation, we propose an approach, termed aBUS, for adaptive BUS, that does not require c as input. The proposed algorithm requires only minimal modifications of standard BUS with Subset Simulation. We discuss why aBUS produces samples that follow the posterior distribution - even if 1∕c is selected smaller than the maximum of the likelihood function. The performance of aBUS in terms of the computed evidence required for Bayesian model class selection and in terms of the produced posterior samples is assessed numerically for different example problems. The combination of BUS with Subset Simulation (and aBUS in particular) is well suited for problems with many uncertain parameters and for Bayesian updating of models where it is computationally demanding to evaluate the likelihood function.