Approximate Bayesian Computation by Subset Simulation

A new Approximate Bayesian Computation (ABC) algorithm for Bayesian updating of model parameters is proposed in this paper, which combines the ABC principles with the technique of Subset Simulation for efficient rare-event simulation, first developed in S.K. Au and J.L. Beck [1]. It has been named ABC- SubSim. The idea is to choose the nested decreasing sequence of regions in Subset Simulation as the regions that correspond to increasingly closer approximations of the actual data vector in observation space. The efficiency of the algorithm is demonstrated in two examples that illustrate some of the challenges faced in real-world applications of ABC. We show that the proposed algorithm outperforms other recent sequential ABC algorithms in terms of computational efficiency while achieving the same, or better, measure of ac- curacy in the posterior distribution. We also show that ABC-SubSim readily provides an estimate of the evidence (marginal likelihood) for posterior model class assessment, as a by-product

Approximate Bayesian Computation by Subset Simulation for model selection in dynamical systems

Approximate Bayesian Computation (ABC) methods are originally conceived to expand the horizon of Bayesian inference methods to the range of models for which only forward simulation is available. However, there are well-known limitations of the ABC approach to the Bayesian model selection problem, mainly due to lack of a sufficient summary statistics that work across models. In this paper, we show that formulating the standard ABC posterior distribution as the exact posterior PDF for a hierarchical state-space model class allows us to independently estimate the evidence for each alternative candidate model. We also show that the model evidence is a simple by-product of the ABC-SubSim algorithm. The validity of the proposed approach to ABC model selection is illustrated using simulated data from a three-story shear building with Masing hysteresis

Using Approximate Bayesian Computation by Subset Simulation for Efficient Posterior Assessment of Dynamic State-Space Model Classes

Approximate Bayesian Computation (ABC) methods have gained in popularity over the last decade because they expand the horizon of Bayesian parameter inference methods to the range of models for which an analytical formula for the likelihood function might be difficult, or even impossible, to establish. The majority of the ABC methods rely on the choice of a set of summary statistics to reduce the dimension of the data. However, as has been noted in the ABC literature, the lack of convergence guarantees induced by the absence of a vector of sufficient summary statistics that assures intermodel sufficiency over the set of competing models hinders the use of the usual ABC methods when applied to Bayesian model selection or assessment. In this paper, we present a novel ABC model selection procedure for dynamical systems based on a recently introduced multilevel Markov chain Monte Carlo method, self-regulating ABC-SubSim, and a hierarchical state-space formulation of dynamic models. We show that this formulation makes it possible to independently approximate the model evidence required for assessing the posterior probability of each of the competing models. We also show that ABC-SubSim not only provides an estimate of the model evidence as a simple by-product but also gives the posterior probability of each model as a function of the tolerance level, which allows the ABC model choices made in previous studies to be understood. We illustrate the performance of the proposed framework for ABC model updating and model class selection by applying it to two problems in Bayesian system identification: a single-degree-of-freedom bilinear hysteretic oscillator and a three-story shear building with Masing hysteresis, both of which are subject to a seismic excitation

Guided wave-based characterisation of cracks in pipes utilising approximate Bayesian computation

Available online 29 August 2023This paper proposed a probabilistic framework of damage characterisation to detect and identify early-state cracks in pipe-like structures using ultrasonic guided waves. The crack location, crack sizes (e.g., depth and width of the crack), and Young’s modulus are considered as unknown parameters in the model updating using a Bayesian approach, by which their values and the associated uncertainties are quantified. The proposed framework is developed based on approximate Bayesian computation (ABC) by subset simulation, which is a likelihood-free Bayesian approach. This algorithm estimates the posterior distributions of unknown parameters by directly accessing the similarity between the measured signals from experiments and the simulated guided wave (GW) signals from the numerical model. In this case, the evaluation of likelihood functions can be smartly circumvented during Bayesian inference. A time-domain spectral finite element (SFE) method with a cracked finite element model is employed to model the pipes to enhance the computational efficiency of the simulation and model updating. Numerical and experimental case studies are carried out to evaluate the performance of the proposed likelihood-free approach. Numerical results show the accuracy and robustness of the proposed approach in identifying unknown parameters under different scenarios. The associated uncertainties of each parameter are also quantified by analysing the statistical properties of the sample set, such as mean and coefficient of variation (COV) values. Experimental results show that the proposed method can accurately identify the unknown parameters, which further verifies the accuracy and practicability of the probabilistic damage characterisation framework.Zijie Zeng, Min Gao, Ching Tai Ng, Abdul Hamid Sheik

Bayesian System Identification based on Hierarchical Sparse Bayesian Learning and Gibbs Sampling with Application to Structural Damage Assessment

The focus in this paper is Bayesian system identification based on noisy incomplete modal data where we can impose spatially-sparse stiffness changes when updating a structural model. To this end, based on a similar hierarchical sparse Bayesian learning model from our previous work, we propose two Gibbs sampling algorithms. The algorithms differ in their strategies to deal with the posterior uncertainty of the equation-error precision parameter, but both sample from the conditional posterior probability density functions (PDFs) for the structural stiffness parameters and system modal parameters. The effective dimension for the Gibbs sampling is low because iterative sampling is done from only three conditional posterior PDFs that correspond to three parameter groups, along with sampling of the equation-error precision parameter from another conditional posterior PDF in one of the algorithms where it is not integrated out as a "nuisance" parameter. A nice feature from a computational perspective is that it is not necessary to solve a nonlinear eigenvalue problem of a structural model. The effectiveness and robustness of the proposed algorithms are illustrated by applying them to the IASE-ASCE Phase II simulated and experimental benchmark studies. The goal is to use incomplete modal data identified before and after possible damage to detect and assess spatially-sparse stiffness reductions induced by any damage. Our past and current focus on meeting challenges arising from Bayesian inference of structural stiffness serve to strengthen the capability of vibration-based structural system identification but our methods also have much broader applicability for inverse problems in science and technology where system matrices are to be inferred from noisy partial information about their eigenquantities.Comment: 12 figure

Model selection and parameter estimation of dynamical systems using a novel variant of approximate Bayesian computation

Model selection is a challenging problem that is of importance in many branches of the sciences and engineering, particularly in structural dynamics. By definition, it is intended to select the most likely model among a set of competing models that best matches the dynamic behaviour of a real structure and better predicts the measured data. The Bayesian approach which is based essentially on the evaluation of a likelihood function is one of the most popular approach to deal with model selection and parameter estimation issues. However, in some circumstances, the likelihood function is either intractable or not available even in a closed form. To overcome this issue, the likelihood-free or approximate Bayesian computation (ABC) algorithm has been introduced in the literature, which relaxes the need for an explicit likelihood function to measure the level of agreement between model predictions and measurements. However, ABC algorithms suffer from a low acceptance rate of samples which is actually a common problem with the traditional Bayesian methods. To overcome this shortcoming and alleviate the computational burden, a new variant of the ABC algorithm based on an ellipsoidal Nested Sampling (NS) technique is introduced in this paper; it has been called ABC-NS. Through this paper, it will be shown how the new algorithm is a promising alternative to deal with parameter estimation and model selection issues. It promises drastic speedups and provides a good approximation of the posterior distributions. To demonstrate its robust computational efficiency, four illustrative examples are given. Firstly, the efficiency of the algorithm is demonstrated to deal with parameter estimation. Secondly, two examples based on simulated and real data are given to demonstrate the efficiency of the algorithm to deal with model selection in structural dynamics