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Uncertainty quantification of axle weight estimated by Bayesian bridge weigh-in-motion
Published in Joan-Ramon Casas, Dan M. Frangopol, Jose Turmo, Bridge Safety, Maintenance, Management, Life-Cycle, Resilience and Sustainability, 2022
K. Maruyama, I. Yoshida, H. Sekiya, S. Mustafa
The uncertainties in the estimated weight can be evaluated by BBWIM. The formulation of Bayesian BWIM is summarized from Yoshida et al. (2021). In Bayesian framework, the prior information is updated by observed information. p(x|z)=p(z|x)p(x)p(z)
Probabilistic design of foundations and earth structures
Published in K.S. Li, S-C.R. Lo, Probabilistic Methods in Geotechnical Engineering, 2020
If the critical value of a geotechnical installation such as the ultimate load capacity is known to vary over a site, a preliminary estimate of the value including representation of its uncertainty can be rationally improved by the application of Bayes’ theorem in conjunction with a measurement of the failure load from a prototype test conducted on the site. The prerequisite distributions are (a) a probabilistic representation for the estimated critical value based on available prior information and computational methods such as those from fundamental soil mechanics and (b) the state of nature distribution. Both were discussed in the foregoing. For (a) both the mean and the standard deviation are required and for (b) only the coefficient of variation is required. For both distributions, rational arguments can be made and used together with factual evidence to justify the use of lognormal distributions as reasonable approximations in the above. It is highly significant, however, that they are convenient mathematically and are readily converted to normal distributions of the parameter logarithms. These forms are directly amenable to application of Bayes’ theorem. For the case of normal distributions, the estimation of the variation in the mean of the state of nature (which can be considered to represent a distribution of the reducible component of uncertainty) can readily be obtained. It is similarly normally distributed and has a mean equal to the estimated distribution mean and a variance equal to the difference in the two variances. In Bayesian terminology it is known as the prior distribution.
Bayesian Statistical Methods
Published in Simon Washington, Matthew Karlaftis, Fred Mannering, Panagiotis Anastasopoulos, Statistical and Econometric Methods for Transportation Data Analysis, 2020
Simon Washington, Matthew Karlaftis, Fred Mannering, Panagiotis Anastasopoulos
Bayesian models are becoming popular within the transportation profession, not primarily due to the benefits of Bayes’ theorem, but because of the accessibility of Bayesian models through the use of Markov Chain Monte Carlo (MCMC) methods. MCMC is a sampling-based approach to estimation that is well suited for Bayesian models. In addition, MCMC enables the estimation of complex functional forms—forms that are often difficult to estimate using maximum likelihood methods. As a result, many Bayes model applications in transportation involve noninformative (also known as diffuse, vague, or ignorance) prior information and complex model functional forms. Because of the accessibility and flexibility offered by MCMC estimation methods for estimating Bayesian statistical models, the transportation profession is seeing an increasing number of applications. As a result, general guidance on these methods is needed.
A Bayesian approach for site-specific extreme load prediction of large scale bridges
Published in Structure and Infrastructure Engineering, 2023
Xiang Xu, Michael C. Forde, Yuan Ren, Qiao Huang
Under Bayesian inferences, the prior distribution indicates prior knowledge regarding the parameters. In general, two types of prior distributions are available, including informative and non-informative priors. The informative is determined on the foundation of the prior knowledge usually obtained from existing literature or experts. While, the non-informative is employed when litter or no prior knowledge is known. Owing to the lack of prior information regarding parameters, a flat distribution is always used for the non-informative prior distribution to cover a wide range of possible values. In this paper, the non-informative prior distribution is employed since rare knowledge regarding parameters of the GPD for SEL estimation is known. The selected flat distribution is N(0, 100), which will cover from −400 to 400.
Fragility Estimates for High-Rise Buildings with Outrigger Systems Under Seismic and Wind Loads
Published in Journal of Earthquake Engineering, 2023
Lili Xing, Paolo Gardoni, Ying Zhou
where = posterior distribution representing our updated state of knowledge about ; = likelihood function representing the objective information on contained in a set of observations; = prior distribution reflecting our state information of knowledge about prior to obtaining the observations; and = normalized factor. The Bayesian inference relies on the feasibility of computing the posterior statistics. However, computing can be challenging in the case of multidimensional problems. In this paper, we use an adaptive delayed rejection MCMC simulation method, also known as the DRAM method (Laine 2008), to obtain posterior statistics of the unknown model parameters.
Joint Diagnosis of High-Dimensional Process Mean and Covariance Matrix based on Bayesian Model Selection
Published in Technometrics, 2023
Feng Xu, Lianjie Shu, Yanting Li
In the Bayesian framework, model comparison is based on posterior probabilities. Denote a model by () if and only if and all other elements of are 0. The cardinality of model k is denoted by or more simply by k when there is no risk of confusion. Assume that model has a prior probability , where refers to the model space. According to the Bayesian theory, the posterior probability of model can be written where is the parameter vector of model k, is the prior density function of the parameter , and is the density function of data. The implementation of the Bayesian model selection involves three steps: (i) to set the density function of data, ; (ii) to select the prior density function of the parameter ; and (iii) to select the prior probability of model .