Combining pre- and post-model information in the uncertainty quantification of non-deterministic models using an extended Bayesian melding approach

Abstract

Due to the increasing complexity of manufacturing process and the diversity of information sources, it is not rare in practical engineering that multiple priors are simultaneously available on the same quantity. To address this issue, which occurs due to inconsistent information from different sources, we propose a probability framework to quantify the uncertainty of a general propagation model. An extended Bayesian melding approach is developed to eliminate the limitations inherent in traditional Bayesian methods. It is found that the aggregation error, which is caused by inconsistent information from multi-sources, can be alleviated by combining the pre- and post- model information. Novel features of our approach involve a modified sampling importance resampling algorithm in which a distribution mixture technique is adopted to reduce the computational cost. To meet practical engineering requirements, this approach is extended to a non-deterministic scenario that has not been covered by existing studies. We use several case studies to validate our proposal as well as its benefits in practical applications.

Introduction

The rapidly growing pace and continuous evolution of modern society have made products and systems increasingly complex. Such products and equipment are installed to meet advanced functional and performance requirements for mission-critical fields, e.g., infrastructure, manufacturing, aviation and high-level national security. The unexpected failures during lifespan may lead to catastrophic consequences. Among all factors affecting the accuracy of system response prediction, the uncertainty arising due to either the inherent stochasticity (aleatory uncertainty) or a lack of knowledge (epistemic uncertainty) should not be overlooked [15]. Without a deep insight into the nature of uncertainty, stakeholders could hardly assess the results in a reasonable confidence level and make advisable decision [36].

The importance of multi-source information aggregation and its corresponding representation in uncertainty propagation has been recognized and emphasized in a series of publications. For example, Zhao et al. [44] proposed a new method that describes information in different granular spaces by using hierarchical coordinates. Gullo at al. [11] developed a prototype-based agglomerative hierarchical clustering method. Chen et al. [2] proposed a decision making approach on the basis of the information axiom in the presence of both fuzziness and randomness. Petry et al. [26] developed an approach of probability aggregation using rational consensus with equi-weighting. Saadi et al. [31] proposed a hierarchical algorithm in urban and transportation research which allows for a combination of unlimited datasets. Che et al. [1] addressed the numerical characterization of uncertain data as well as the information fusion through a combination of several uncertain theories. Li. [19] predicted the reliability of turbine blades by aggregating multi-source information.

There are many approaches in information fusion based on different uncertainty theories, such as Dempster-Shafer evidence theory [33], possibility theory [5], fuzzy set [6] and rough set [37]. Among all in-used information fusion methodologies, Bayesian-based approaches have drawn extensive attentions due to the ability to combine both subjective experts judgements and objective test data in a principled way to perform inference [21], [39]. To integrate all available information in reliability analysis, Johnson et al. [17] developed a full-Bayesian approach for a multi-level system. Hamada et al. [14] generalized this approach to simultaneously use both basic event data and higher-level event data in the uncertainty quantification in a fault tree. Stein et al. [35] considered several possible scenarios to fuzzify the Bayesian approach and developed several computation techniques to implement the fuzzy Bayesian formula. Li et al. [20] established an innovative semi-parametric modeling framework for hierarchical systems, in which the lower-level information is extracted and aggregated to compensate the inadequacy of higher-level information. Recently, Wilson et al. [38] provided several examples about the application of Bayesian methods. Pan et al. [24] made a comprehensive review of the modern advances for Bayesian approaches in the reliability field.

The motivation of our research stems from the desire to reduce the so-called aggregation error [14] in uncertainty quantification of multiple inconsistent information. The study to aggregation error has a long history in econometrics, typical examples are [3], [16], [34]. However, the notion of aggregation error has not been sufficiently addressed in other research fields. In effect, the case that multiple priors simultaneously available on the same variable appears frequently in practical engineering, which could lead to a significant aggregation error due to the inherent inconsistency of available information [41]. Considering a general motion model, one side, we have prior knowledge about basic variables (e.g. velocity, acceleration, initial position, etc.) of the motion, which induces priors on the model output through the governing equation. Alternatively, we monitor the motion process and collect sensor data. The observed data could directly lead to a prior on the model output. Since the two priors are elicited from different information sources, they usually do not match each other and may even be controversial in some scenarios. As a consequence, parameter estimation results contain notable aggregation error because the evaluation process could follow different information propagation paths, i.e. an aggregated result or a disaggregated result. A comprehensive approach for multi-source information fusion is then in demand.

To fill this research gap, The Bayesian Melding (BM) was proposed by Poole and Raftery [27], [28] as a general statistical method of merging two inconsistent prior distributions in a coherent model. The BM method is pervasively employed in a wide range of research fields and has gain dozens of successful application examples, such as uncertainty assessment in urban simulation [32], prediction of spatial tracks [22], reliability evaluation [12], [13], [40], estimation of bowhead whale [30] and calibration of forecast ensembles [29]. However, the original form of this method exhibits some drawbacks which make the estimation results problematic in certain conditions, e.g. it is initially developed for deterministic models and is not applicable to non-deterministic transfer models [27], [28]; the weight α in traditional BM is arbitrary selected [7] that lacks robustness in an information imbalance scenario [43]. Although some studies [32] are devoted in this research area, the aggregation of inconsistent information in a general probabilistic framework has not been well developed. In this paper, we follow this research line and propose a comprehensive probability methodology to incorporate pre- and post-model information from multi-sources. Addressing some flaws in the existing studies, we develop an Adaptive Bayesian Melding (ABM) approach which is capacity in fusing the inconsistent prior information in a more flexible way. We also extend this approach to non-deterministic propagation models—a situation that is not covered by traditional melding methods [27], [28]. It will show that our approach has some superiorities compared with existing works in a more general situation.

The rest of this paper is organized as follows. We start with a simple motivating example to demonstrate the problem at hand in Section 2. The rationality of BM is illustrated in Section 3. We describe the mathematical foundation of our approach by revising the motivating example. Section 4 extends our approach to non-deterministic models. A Modified Sampling Importance Resampling (MSIR) algorithm is developed for routine implementation, and in addition, a distribution mixture technique is introduced to save the computation cost. We then demonstrate the benefits of our approach via a numerical case study in Section 5. Finally, we present a comprehensive application example in Section 6, where the experimental test data is involved for validation.