Reliability Engineering and System Safety 172 (2018) 12–24 Contents lists available at ScienceDirect Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress Bayesian model updating with summarized statistical and reliability data Eric VanDerHorn a,b , Sankaran Mahadevan a,∗ a Department of Civil & Environmental Engineering, Vanderbilt University, Nashville, TN 37235, USA b American Bureau of Shipping, Houston, TX, USA a r t i c l e i n f o Keywords: Reliability Calibration Summary statistics Sufficient statistics Bayesian network a b s t r a c t The accuracy of model-based reliability analysis is affected by the uncertainty regarding the model parameters used to predict the behavior of the engineering system. The uncertainty in the model parameters can be reduced by combining prior knowledge about the parameters with observed data regarding system inputs and outputs. In some cases, the information about the observations is only available as abstracted data, where the original raw data have been reduced to a summarized representation. Common forms of abstracted data include summary statistics, such as the mean and variance for continuous variables and observed frequencies for discrete variables. In the context of reliability analysis, a common form of available information is summarized reliability data for various mechanical components (e.g., failure rates or failure probabilities) instead of detailed actual test data. This paper presents a methodology for updating the model parameters using these abstracted data forms through a Bayesian network. First, the concept of a statistics function is developed and linked to the abstracted data forms. The concept of arc reversal is then exploited to transform the Bayesian network to a form that can be used to incorporate the statistics function and thereby enable the updating of the model parameters. Several numerical examples are used to demonstrate the applicability and generality of the proposed method for several different forms of abstracted data. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction Decision making in engineering applications based on the results of reliability analysis often relies on the use of mathematical or computa- tional models to predict the behavior of complex engineering systems. Reliability analysis is affected by both aleatory uncertainty (natural vari- ability) and epistemic uncertainty lack of knowledge regarding the vari- ables or the models). The epistemic uncertainty can further be classified into statistical uncertainty and model uncertainty to represent the lack of knowledge in variables and models respectively. The model uncer- tainty is related to model approximations as well as the uncertainty in the model parameters. It is important that the model parameters be cali- brated based on the available information so that the model predictions accurately reflect the physical reality. This updating process is informed by data and requires that all available information be properly incorpo- rated into the modeling and simulation. The model calibration data may be available in many different forms, including but not limited to, experimental and operational data, in- spection reports, health monitoring data, engineering plans, rules and standards, and expert opinion. These heterogeneous sources of infor- mation can lead to significant challenges for model calibration, as the data may often be imprecise, uncertain, ambiguous, and/or incomplete. ∗ Corresponding author. E-mail address: sankaran.mahadevan@vanderbilt.edu (S. Mahadevan). Additional challenges may arise as the data may not be provided in a traditional format, such as point or interval data [1], but instead may be provided in abstracted formats such as sample statistics (e.g. mean, variance, median, max, etc.), probability or frequency data, or reliability data. The term “abstracted data” in this paper refers to the case where raw data has been reduced to a simplified representation of portions or the entirety of the raw dataset. There are several sources of abstracted data in practical applications [2,3]. For example, instead of receiving the full data of all the outcomes of an experiment, sometimes the only informa- tion provided from testing may be in the form of summary statistics of the observed sample distribution (e.g., mean, variance etc.) or the ob- served frequencies for categorical data. In some cases, the performance of a population of components or system may be given as reliability data [4] or summarized results from acceptance testing [5], both of which can be considered as forms of abstracted data. Sometimes, experts may provide their point or interval estimates of moments, frequencies, or probability ranges. This calibration process can be further complicated if data is provided simultaneously in several of these heterogeneous ab- stracted forms. The incorporation of abstracted data in inference is not a new con- cern. Early work focused on the use of abstracted data for distribution parameter estimation of random variables, particularly in cases where https://doi.org/10.1016/j.ress.2017.11.023 Received 3 April 2017; Received in revised form 20 October 2017; Accepted 29 November 2017 Available online 2 December 2017 0951-8320/© 2017 Elsevier Ltd. All rights reserved.