Bayesian updating and model class selection with Subset Simulation

Abstract

Identifying the parameters of a model and rating competitive models based on measured data has been among the most important and challenging topics in modern science and engineering, with great potential of application in structural system identification, updating and development of high fidelity models. These problems in principle can be tackled using a Bayesian probabilistic approach, where the parameters to be identified are treated as uncertain and their inference information are given in terms of their posterior probability distribution. For complex models encountered in applications, efficient computational tools robust to the number of uncertain parameters in the problem are required for computing the posterior statistics, which can generally be formulated as a multi-dimensional integral over the space of the uncertain parameters. Subset Simulation has been developed for solving reliability problems involving complex systems and it is found to be robust to the number of uncertain parameters. An analogy has been recently established between a Bayesian updating problem and a reliability problem, which opens up the possibility of efficient solution by Subset Simulation. The formulation, called BUS (Bayesian Updating with Structural reliability methods), is based the standard rejection principle. Its theoretical correctness and efficiency requires the prudent choice of a multiplier, which has remained an open question. This paper presents a fundamental study of the multiplier and investigates its bias effect when it is not properly chosen. A revised formulation of BUS is proposed, which fundamentally resolves the problem such that Subset Simulation can be implemented without knowing the multiplier a priori. An automatic stopping condition is also provided. Examples are presented to illustrate the theory and applications.

Bayesian Updating for Probabilistic Classification Using Reliability Methods

“…The multiplier only affects the target threshold level b beyond which the samples can be collected as posterior samples.Under the new framework the distribution of the samples conditional on {Y > b} will remain unchanged for a sufficiently large b. The minimum value of b beyond which the distribution of the samples will settle at the posterior PDF can be shown to be D [7] where P D = R q(x)L(x)dx. It can be expected that as b increases from the left tail and to a value greater than bmin, the CCDF of Y typically changes from a decreasing function to an exponentially decaying function.…”

“…A modification of the original BUS formulation has been proposed [7] which isolates the effect of the multiplier on the simulation. The modification involves a reexpression of the failure event such that…”

“…However, correctly choosing the value of this multiplier has in general remained an open question. A revised BUS formulation [7] allows SuS to be implemented to sample from the posterior distribution whilst learning the multiplier automatically. The objective of this paper is to apply the modified BUS framework to sample from a posterior distribution in a binary classification setting.…”

Self‐calibrating Bayesian real‐time system identification

“…Field and Grigoriu () studied the model selection problem of stochastic processes with bounded range. DiazDelaO, Garbuno‐Inigo, Au, and Yoshida () incorporated the subset simulation with Bayesian updating and model class selection for engineering reliability analysis. In recent years, Bayesian model class selection has been extensively utilized in various engineering applications, such as structural health monitoring (Muto & Beck, ; Ritto & Nunes, ; Worden & Hensman, ; Yuen & Kuok, ), geotechnical properties modeling (Chiu, Yan, & Yuen, ), seismic attenuation relationship (Kuok, Ng, & Yuen, ; Mu & Yuen, ; Yuen, Ortiz, & Huang, ), modal identifiability (Kuok & Yuen, ), etc.…”

Modular Bayesian damage detection for complex civil infrastructure

“…The Bayesian approach gathered considerable interest [10,15,45]. Its conceptual simplicity and consistent treatment of uncertainties contributed to its wide dissemination.…”