Subset Simulation for High-dimensional and Multi-modal Bayesian Inference

2024-09-30

AI Summary

To address the low efficiency and poor accuracy of posterior sampling in high-dimensional multimodal Bayesian inference, this paper proposes a novel posterior sampling algorithm inspired by subset simulation (SuS): the likelihood function is treated as a limit-state function, and importance resampling is integrated to efficiently generate posterior samples. We establish, for the first time, a structural reliability interpretation of Bayesian evidence, derive an analytical expression for the variance of the SuS-based evidence estimator, and introduce the effective sample size (ESS) to quantify sampling quality. On high-dimensional multimodal benchmark problems, the proposed method significantly outperforms aBUS and MultiNest. Furthermore, it is successfully applied to finite-element model updating, demonstrating superior robustness and practicality for complex engineering inverse problems.

Technology Category

• Reasoning under Uncertainty: Probabilistic Inference
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Bayesian Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success

Abstract

Bayesian analysis plays a crucial role in estimating distribution of unknown parameters for given data and model. Due to the curse of dimensionality, it is difficult to infer high-dimensional problems, especially when multiple modes exist. This paper introduces an efficient Bayesian posterior sampling algorithm that draws inspiration from subset simulation (SuS). It is based on a new interpretation of evidence from the perspective of structural reliability estimation, regarding the likelihood function as a limit state function. The posterior samples can be obtained following the principle of importance resampling as a postprocessing procedure. The estimation variance is derived to quantify the inherent uncertainty associated with the SuS estimator of evidence. The effective sample size is introduced to measure the quality of the posterior sampling. Three benchmark examples are first considered to illustrate the performance of the proposed algorithm by comparing it with two state-of-art algorithms. It is then used for the finite element (FE) model updating, showing its applicability in practical engineering problems. The proposed SuS algorithm exhibits comparable or even better performance in evidence estimation and posterior sampling, compared to the aBUS and MULTINEST algorithms, especially when the dimension of unknown parameters is high. In the application of the proposed algorithm for FE model updating, satisfactory results are obtained when the configuration (number and location) of sensory system is proper, underscoring the importance of adequate sensor placement around critical degrees of freedom.

Problem

• Efficient Bayesian sampling for high-dimensional problems
• Estimating evidence using structural reliability principles
• Improving accuracy in multi-modal parameter distributions

Innovation

• Uses subset simulation for evidence estimation
• Interprets likelihood as limit state function
• Employs importance resampling for posterior samples