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Characteristic values of spatially varying material properties in existing structures
Published in Airong Chen, Xin Ruan, Dan M. Frangopol, Life-Cycle Civil Engineering: Innovation, Theory and Practice, 2021
S. Geyer, I. Papaioannou, D. Straub
In many cases, the solution of Equation (4) needs to be found numerically as the proportionality constant cannot be evaluated directly. This can be avoided by the use of conjugate priors, which allow direct evaluation of the posterior distribution Raiffa_1961,Gelman_2013. Conjugate priors are characterized by the fact that the posterior distribution is of the same distribution type as the prior distribution, which can be achieved by specific choices of the prior distribution and the likelihood function. In such case, the prior distribution is termed a conjugate prior for the likelihood function and the Bayesian updating reduces to a parameter update of the chosen parametric family Gelman_2013. The presented model in this contribution makes use of the conjugacy of the chosen prior distribution and likelihood function.
Variational Bayesian Super-Resolution Reconstruction
Published in Peyman Milanfar, Super-Resolution Imaging, 2017
S. Derin Babacan, Rafael Molina, Aggelos K. Katsaggelos
The hyperparameters are crucial in determining the performance of the SR algorithm. Most super-resolution methods leave their estimation to the user, which requires a long parameter-tuning process and therefore limits the applicability of the super-resolution method. On the other hand, employing a fully Bayesian analysis allows their estimation as well. To obtain tractable Bayesian inference, generally conjugate hyperprior distributions are utilized, which lead to straightforward calculation or approximation of the posterior distribution p(x, sk, Hk, Ω|y). Conjugate priors allow one to begin with a certain functional form for the conditional and prior distributions and end up with the posterior of the same functional form, but with parameters updated from the observed samples.
Probabilistic NDT data fusion of Ferroscan test data using Bayesian inference
Published in Koen Van Balen, Els Verstrynge, Structural Analysis of Historical Constructions: Anamnesis, Diagnosis, Therapy, Controls, 2016
from previous knowledge including some uncertainties (rather than randomness). The choice of prior can be Jeffrey's prior if no knowledge about prior is available or a conjugate prior where some initial concrete cover is known from construction drawings. The flowchart shown in figure 4 can be referred to calculate posterior density function with different knowledge about initial prior.As the data is gathered concerning the parameters it is updated to a posterior. The main key for performing a Bayesian analysis is to update the prior as the new data is obtained. Choice of prior can vary from a very accurate prior with less standard deviation to even no knowledge which is non-informative Jeffrey's prior. It is reasonable to assume prior as a normal distribution (for mathematical considerations) for modelling of many mechanical parameters (LNEC Report (LNEC Report (1983)). When the posterior and the prior have the same parametric form is called conjugacy. For example, the Gaussian family is conjugate to itself (or self-conjugate) with respect to a Gaussian likelihood function. If the likelihood function is Gaussian, choosing a Gaussian prior over the mean will ensure that the posterior distribution is also Gaussian. A conjugate prior is an algebraic convenience, giving a closedform expression for the posterior, otherwise a difficult numerical integration may be necessary. All members of the exponential family have conjugate priors. For details of the Bayesian algorithm used to combine NDT test data and fusion operator (Miranda et al. 2009, Mishra 2013, Ramos et al. 2015) can be referred. 5 RESULTS
Adversarial data poisoning attacks against the PC learning algorithm
Published in International Journal of General Systems, 2020
Emad Alsuwat, Hatim Alsuwat, Marco Valtorta, Csilla Farkas
Bayesian statistics treats parameters as random variables; data is treated as fixed. For example, let θ be a parameter, and D be a dataset, then Bayes' theorem can be expressed mathematically as follows: . Since is constant (Lynch 2007), we can write Bayes' theorem in its most useful form in Bayesian update and inference as follows: It is convenient mathematically for the prior and the likelihood to be conjugate. A prior distribution is a conjugate prior for the likelihood function if the posterior distribution belongs to the same distribution as the prior (Raiffa and Schlaifer 1961). For example, the beta distribution is a conjugate prior for the binomial distribution (as a likelihood function). Equation 5 is the formula that we will use in this paper for prior to posterior update. Starting with a prior distribution , we add the count of successes,y, and the count of failures, n−y, from the dataset D (where n is total number of entries in D) to α and β, respectively. Thus, is the posterior distribution.
Performance analysis of unreliable manufacturing systems with uncertain reliability parameters estimated from production data
Published in International Journal of Computer Integrated Manufacturing, 2019
Theoretically, the choice of different priors can be considered in a Bayesian approach. In this work, the authors limit the use of conjugate priors corresponding to the geometric distribution. One of the main motivations to use conjugate priors is that they provide simplistic algebraic forms, particularly in a cyclic application of Bayesian updating from new data. By using conjugate priors, the updating of the density function with new data to compute for the posterior has a similar form to the prior. In this particular case of inference on the parameter p from a geometrically distributed TTFs then conjugate prior for the parameter p is the two parameter Beta distribution. If the observed data follows a different distributions than the assumptions adopted in this paper, i.e., geometrically distributed failure and repair times assumed in (Section 3.1), the same approach can be followed for the parameter inference. However, in those cases, the appropriate conjugate distributions need to be selected. Tables with the list of conjugate priors and associated likelihood functions can be found in statistical references, such as (Gelman et al. 2014).
Dynamic risk management of assets susceptible to pitting corrosion
Published in Corrosion Engineering, Science and Technology, 2019
Elahe Shekari, Faisal Khan, Salim Ahmed
One of the important advantages of Bayesian analysis is its ability to use inspection data to update the prior belief about the pit's behaviour. For instance, Straub [9] used new pit depth measurement data to estimate the likelihood distribution of deteriorating components and update the prior distribution of the model predictions for MPD. Using the prior and likelihood distributions, the posterior MPD distribution can be estimated using Bayes’ rule [9]:where the denominator is known as the normalising factor, L(MPD0|MPD) is the likelihood function, f′MPD is the prior MPD distribution, and f″MPD is the posterior MPD distribution. The posterior probability is the likelihood that a variable will be in a particular state, given the values of the input variables, the conditional probabilities and an associated set of rules governing how the probabilities are combined [9]. The traditional Bayesian updating approach assumes conjugate prior and likelihood distributions. Conjugate priors provide computational ease and flexibility that facilitate the development of analytical solutions for the posterior distribution. As conjugate pairs are often unable to capture the realistic behaviour of the parameters [26], use of the traditional Bayesian approach (conjugate-likelihood pair) may introduce significant uncertainty. To address this limitation, the MCMC and Metropolis–Hasting (or M–H) algorithm [28] are used to estimate posterior distribution for non-conjugate distributions.