Posterior Distribution Of Parameters Research Articles

Bayesian Model Selection of Lithium-Ion Battery Models via Bayesian Quadrature

A wide variety of battery models are available, and it is not always obvious which model ‘best’ describes a dataset. This paper presents a Bayesian model selection approach using Bayesian quadrature. The model evidence is adopted as the selection metric, choosing the simplest model that describes the data, in the spirit of Occam's razor. However, estimating this requires integral computations over parameter space, which is usually prohibitively expensive. Bayesian quadrature offers sample-efficient integration via model-based inference that minimises the number of battery model evaluations. The posterior distribution of model parameters can also be inferred as a byproduct without further computation. Here, the simplest lithium-ion battery models, equivalent circuit models, were used to analyse the sensitivity of the selection criterion to given different datasets and model configurations. We show that popular model selection criteria, such as root-mean-square error and Bayesian information criterion, can fail to select a parsimonious model in the case of a multimodal posterior. The model evidence can spot the optimal model in such cases, simultaneously providing the variance of the evidence inference itself as an indication of confidence. We also show that Bayesian quadrature can compute the evidence faster than popular Monte Carlo based solvers.

Statistical inference via conditional Bayesian posteriors in high-dimensional linear regression

We propose a new method under the Bayesian framework to perform valid inference for low dimensional parameters in high dimensional linear models under sparsity constraints. Our approach is to use surrogate Bayesian posteriors based on partial regression models to remove the effect of high dimensional nuisance variables. We name the final distribution we used to conduct inference “conditional Bayesian posterior” as it is a surrogate posterior constructed conditional on quasi posterior distributions of other parameters and does not admit a fully Bayesian interpretation. Unlike existing Bayesian regularization methods, our method can be used to quantify the estimation uncertainty for arbitrarily small signals and therefore does not require variable selection consistency to guarantee its validity. Theoretically, we show that the resulting Bayesian credible intervals achieve desired coverage probabilities in the frequentist sense. Methodologically, our proposed Bayesian framework can easily incorporate popular Bayesian regularization procedures such as those based on spike and slab priors and horseshoe priors to facilitate high accuracy estimation and inference. Numerically, our proposed method rectifies the uncertainty underestimation of Bayesian shrinkage approaches and has a comparable empirical performance with state-of-the-art frequentist methods based on extensive simulation studies and a real data analysis.

Curve Fitting Algorithm of Functional Radiation-Response Data Using Bayesian Hierarchical Gaussian Process Regression Model

We present a nonparametric Bayesian hierarchical (NBH) model and develop a variational approximation (VA) algorithm for the curve fitting of the functional radiation response data. The NBH model is based on a Bayesian hierarchical (BH) model with a Gaussian-Inverse Wishart process (G-IWP) prior, which simultaneously smooths multiple functional observations and estimates mean-covariance functions. We use the automatic differentiation variational inference (ADVI) algorithm with a Gaussian distribution as the variational distribution for approximating the posterior distribution of parameters of interest, which is applicable to a wide class of probabilistic models and can also be implemented in Stan (a probabilistic programming system). Using the NBH model and the Gaussian ADVI algorithm, we fit a dataset for the semiconductor obtained from the radiation response map (RRM) of South Korea.

IoT-Based Reinforcement Learning Using Probabilistic Model for Determining Extensive Exploration through Computational Intelligence for Next-Generation Techniques.

Computing intelligence is built on several learning and optimization techniques. Incorporating cutting-edge learning techniques to balance the interaction between exploitation and exploration is therefore an inspiring field, especially when it is combined with IoT. The reinforcement learning techniques created in recent years have largely focused on incorporating deep learning technology to improve the generalization skills of the algorithm while ignoring the issue of detecting and taking full advantage of the dilemma. To increase the effectiveness of exploration, a deep reinforcement algorithm based on computational intelligence is proposed in this study, using intelligent sensors and the Bayesian approach. In addition, the technique for computing the posterior distribution of parameters in Bayesian linear regression is expanded to nonlinear models such as artificial neural networks. The Bayesian Bootstrap Deep Q-Network (BBDQN) algorithm is created by combining the bootstrapped DQN with the recommended computing technique. Finally, tests in two scenarios demonstrate that, when faced with severe exploration problems, BBDQN outperforms DQN and bootstrapped DQN in terms of exploration efficiency.

A beam on elastic foundation method for predicting deflection of braced excavations considering uncertainties

AbstractPredicting wall deflection is important to provide a critical reference to evaluate the current construction conditions and prevent potential damage risks of adjacent facilities during excavations. This paper presents a combination of a beam on elastic foundation model (BEFM) and the Bayesian framework to realize effective probabilistic predictions of wall deflection at various depths in braced excavations. First, a finite element solving algorithm to calculate wall deflection for the BEFM is developed and incorporated into the Bayesian framework. Next, the most suitable distribution pattern for soil resistance and an appropriate set of uncertain parameters in the BEFM are determined through the application of the Bayesian model selection technique. Meanwhile, the uncertain parameters are updated. A prediction is then made using the optimal model and corresponding posterior probability distributions of the updated parameters at each stage. The parameter updating and prediction process are repeated as additional field observations become available during construction. The performance of the proposed method is examined using a field case study. The results show that this method provides a satisfactory approach to predict both the magnitudes and profile of deflection when considering uncertainties. Additionally, comparisons with a Bayesian updating framework using a surrogate model (i.e., the KJHH model) indicate higher updating efficiency of the developed method.

Semiparametric Bayesian doubly robust causal estimation

Frequentist semiparametric theory has been used extensively to develop doubly robust (DR) causal estimation. DR estimation combines outcome regression (OR) and propensity score (PS) models in such a way that correct specification of just one of two models is enough to obtain consistent parameter estimation. An equivalent Bayesian solution, however, is not straightforward as there is no obvious distributional framework to the joint OR and PS model, and the DR approach involves a semiparametric estimating equation framework without a fully specified likelihood. In this paper, we develop a fully semiparametric Bayesian framework for DR causal inference by bridging a nonparametric Bayesian procedure with empirical likelihood via semiparametric linear regression. Instead of specifying a fully probabilistic model, this procedure is only realized through relevant moment conditions. Crucially, this allows the posterior distribution of the causal parameter to be simulated via Markov chain Monte Carlo methods. We show that the posterior distribution of the causal estimator satisfies consistency and the Bernstein–von Mises theorem, when either the OR or PS is correctly specified. Simulation studies suggest that our proposed method is doubly robust and can achieve the desired coverage rate. We also apply this novel Bayesian method to a real data example to assess the impact of speed cameras on car collisions in England.

Identification of a monitoring nonlinear oil damper using particle filtering approach

Oil dampers have been used in recent years for passive structural control and shock mitigation in dynamic structural systems. However, machining technical and economic problems limit its application for new and existing buildings due to the necessary precision machining of incorporating shaft bearings and pressure sealings. A kind of nonlinear oil damper that is quite different from traditional oil damper is investigated and applied to a steel building. Vibration monitoring system was instrumented on the dampers to explore their actual performance and effectivity under strong earthquakes. Based on monitoring response of the nonlinear dampers under various excitations, Bayesian model selection is employed to analyze the most probable model class which can capture main dynamic characteristics of the nonlinear oil dampers and can also be used for predicting future response as well as reliability. Then, a particle filtering approach is proposed to identify the nonlinear model of the damper and quantify the model uncertainty. The developed particle filter is capable of re-parameterizing joint posterior distribution of states and parameters of the nonlinear oil damper without augmented state estimation, which combined with Markov chain Monte Carlo algorithm so as to be able to sample high-dimensional posterior distribution. The identified models and posterior distributions of parameters show that the developed particle filter approach can be appropriately used for nonlinear parameter identification without stuck to special particles. Furthermore, the dynamic properties of the nonlinear oil damper with respect to various excitations involving different spectral characteristics are discussed.

Efficient sampling of the irregular probability distributions of geotechnical parameters for reliability analysis

Values of site-specific geotechnical parameters are generally estimated based on the results of in-situ and/or laboratory tests. In most cases, the field test data in geotechnical engineering practice are so limited and sparse that their histograms may only be described using an irregular probability distribution. In addition, the posterior distributions of geotechnical parameters obtained from the Bayesian updating may also be an irregular probability distribution. The irregular probability distributions that exhibit a multi-modal nature cannot be well fitted using theoretical probability distributions such as normal, lognormal or beta distributions. Therefore, challenges may arise in sampling of such irregular probability distributions, which could hinder the subsequent geotechnical reliability analyses. To facilitate the geotechnical reliability analyses with limited data, an efficient sampling method that is based on a mixture of uniform distributions is proposed in this study. This sampling method is capable of drawing random samples from the irregular probability distribution of a single geotechnical parameter as well as the joint probability distribution of cross-correlated spatially varied geotechnical parameters. The proposed sampling method is first illustrated and validated using three examples that employ both simulated and real data. The results confirm that the proposed sampling method is accurate and highly efficient. Last, the proposed method is implemented to sample from the posterior distributions of two cross-correlated spatially varied geotechnical parameters obtained from the Bayesian updating of a soil slope. The results indicate that the proposed method can provide an effective and versatile tool for the random sampling of the joint probability distribution and posterior distributions of geotechnical parameters.

A mathematical-mechanical hybrid driven approach for determining the deformation monitoring indexes of concrete dam

This paper aims at quantifying the uncertainty of mechanical parameters of the concrete dam based on prototype measured data, and puts forward a novel method for determining deformation monitoring indexes based on the quantification results of the uncertainty of deformation causes. Firstly, the deformation components are separated by the HST model for deformation of concrete dam. Secondly, on the basis of the separated hydraulic component and numerical simulation, the posterior distribution of main mechanical parameters involved in the finite element model (FEM) is updated by utilizing Markov chain Monte Carlo sampling method under Bayesian framework. Meanwhile, in order to improve the computational efficiency of the parameter calibration procedure, this study adopts a surrogate model based on the multi-layer perceptron algorithm to replace the finite element calculation. The training set of the surrogate model is generated by Latin hypercube sampling in the sample space and the optimum sample size is discussed. Thirdly, based on the probability inversion analysis results of mechanical parameters, the uncertainty quantification of hydraulic component is realized by forward analysis based on FEM. The uncertainty of temperature and aging component is characterized by confidence interval method. Then, the deformation monitoring indexes are established by integrating the uncertainty quantization results of three components. Finally, the feasibility of the proposed method is verified by using the long-term deformation monitoring data of Jinping-I arch dam. The results show that the monitoring indexes determined by this method is more sensitive to the abnormal measurement due to the uncertainty quantification of hydraulic component.

Bayesian Calibration and Uncertainty Assessment of HYDRUS-1D Model Using GLUE Algorithm for Simulating Corn Root Zone Salinity under Linear Move Sprinkle Irrigation System

Soil salinization is one of the significant concerns regarding irrigation with saline waters as an alternative resource for limited freshwater resources in arid and semi-arid regions. Thus, the investigation of proper management methods to control soil salinity for irrigation with saline waters is inevitable. The HYDRUS-1D model is a well-known numerical model that can facilitate the exploration of management scenarios to mitigate the consequences of irrigation with saline waters, especially soil salinization. However, before using the model as a decision support system, it is crucial to calibrate the model and analyze the model’s parameters and outputs’ uncertainty. Therefore, the generalized likelihood uncertainty estimation (GLUE) algorithm was implemented for the HYDRUS-1D model in the R environment to calibrate the model and assess the uncertainty aspects for simulating soil salinity of corn root zone under saline irrigation with linear move sprinkle irrigation system. The results of the study have detected a lower level of uncertainty in the α, n, and θs (saturated soil water content) parameters of water flow simulations, dispersivity (λ), and adsorption isotherm coefficient (Kd) parameters of solute transport simulations comparing to the other parameters. A higher level of uncertainty was found for the diffusion coefficient as its corresponding posterior distribution was not considerably changed from its prior distribution. The reason for this phenomenon could be the minor contribution of diffusion to the solute transport process in the soil compared with advection and hydrodynamic dispersion under saline water irrigation conditions. Predictive uncertainty results revealed a lower level of uncertainty in the model outputs for the initial growth stages of corn. The analysis of the predictive uncertainty band also declared that the uncertainty in the model parameters was the predominant source of uncertainty in the model outputs. In addition, the excellent performance of the calibrated model based on 50% quantiles of the posterior distributions of the model parameters was observed in terms of simulating soil water content (SWC) and electrical conductivity of soil water (ECsw) at the corn root zone. The ranges of NRMSE for SWC and ECsw simulations at different soil depths were 0.003 to 0.01 and 0.09 to 0.11, respectively. The results of this study have demonstrated the authenticity of the GLUE algorithm to seek uncertainty aspects and calibration of the HYDRUS-1D model to simulate the soil salinity at the corn root zone at field scale under a linear move irrigation system.

Bayesian Inference of Triple Seasonal Autoregressive Models

In this paper we extend autoregressive models to fit time series that have three layers of seasonality, i.e. triple seasonal autoregressive (TSAR) models, and we introduce the Bayesian inference for these TSAR models. Assuming the TSAR model errors are normally distributed and employing three priors, i.e. Jeffreys', g, and normal-gamma priors, on the model parameters, we derive the marginal posterior distributions of the TSAR model parameters. In particular, we show that the marginal posterior distributions to be multivariate t and gamma distributions for the model coefficients and precision, respectively. We evaluate the efficiency of the proposed Bayesian inference using simulation study, and we then apply it to real-world hourly electricity load time series datasets in six European countries.

Bayesian fixed-domain asymptotics for covariance parameters in a Gaussian process model

Gaussian process models typically contain finite-dimensional parameters in the covariance function that need to be estimated from the data. We study the Bayesian fixed-domain asymptotics for the covariance parameters in a universal kriging model with an isotropic Matérn covariance function, which has many applications in spatial statistics. We show that when the dimension of domain is less than or equal to three, the joint posterior distribution of the microergodic parameter and the range parameter can be factored independently into the product of their marginal posteriors under fixed-domain asymptotics. The posterior of the microergodic parameter is asymptotically close in total variation distance to a normal distribution with shrinking variance, while the posterior distribution of the range parameter does not converge to any point mass distribution in general. Our theory allows an unbounded prior support for the range parameter and flexible designs of sampling points. We further study the asymptotic efficiency and convergence rates in posterior prediction for the Bayesian kriging predictor with covariance parameters randomly drawn from their posterior distribution. In the special case of one-dimensional Ornstein–Uhlenbeck process, we derive explicitly the limiting posterior of the range parameter and the posterior convergence rate for asymptotic efficiency in posterior prediction. We verify these asymptotic results in numerical experiments.

Fault Rate Estimation of Electric Energy Meter Based on WDPOD-NHBM

To solve the problems of the small amount of sample information in the original failure rate data of smart energy meters and the difficulty of mining under typical environmental stress, a smart energy meter failure rate prediction and reliability evaluation model based on double precision combined with nonlinear hierarchical Bayes is proposed. First, the KNN algorithm is used to test the outliers in the original data, and a double-precision outlier detection algorithm is further established to assign the weight to outliers. Then, a nonlinear hierarchical Bayesian model is built based on Weibull distribution, and couples the failure rate data of smart energy meters with various typical environmental stresses. Through multi-dimensional analysis of the data distribution rule, the most prior distribution is selected. Finally, the Markov Chain Monte Carlo method is used to sample and solve the posterior condition distribution of the model parameters. The posterior parameters and confidence interval estimation are obtained, and the reliability of the smart energy meters is calculated as 95%. The experimental results show that the method can accurately predict the trend of the failure rate of smart energy meters over time in a typical environment.

Using wavelets to capture deviations from smoothness in galaxy-scale strong lenses

Modeling the mass distribution of galaxy-scale strong gravitational lenses is a task of increasing difficulty. The high-resolution and depth of imaging data now available render simple analytical forms ineffective at capturing lens structures spanning a large range in spatial scale, mass scale, and morphology. In this work, we address the problem with a novel multiscale method based on wavelets. We tested our method on simulated Hubble Space Telescope (HST) imaging data of strong lenses containing the following different types of mass substructures making them deviate from smooth models: (1) a localized small dark matter subhalo, (2) a Gaussian random field (GRF) that mimics a nonlocalized population of subhalos along the line of sight, and (3) galaxy-scale multipoles that break elliptical symmetry. We show that wavelets are able to recover all of these structures accurately. This is made technically possible by using gradient-informed optimization based on automatic differentiation over thousands of parameters, which also allow us to sample the posterior distributions of all model parameters simultaneously. By construction, our method merges the two main modeling paradigms – analytical and pixelated – with machine-learning optimization techniques into a single modular framework. It is also well-suited for the fast modeling of large samples of lenses.

Bayesian approach for validation of runaway electron simulations

Plasma-terminating disruptions in future fusion reactors may result in conversion of the initial current to a relativistic runaway electron beam. Validated predictive tools are required to optimise the scenarios and mitigation actuators to avoid the excessive damage that can be caused by such events. Many of the simulation tools applied in fusion energy research require the user to specify input parameters that are not constrained by the available experimental information. The conventional approach, where an expert modeller calibrates these input parameters based on domain knowledge, is prone to lead to an intractable validation challenge without systematic uncertainty quantification. Bayesian inference algorithms offer a promising alternative approach that naturally includes uncertainty quantification and is less subject to user bias in choosing the input parameters. The main challenge in using these methods is the computational cost of simulating enough samples to construct the posterior distributions for the uncertain input parameters. This challenge can be overcome by combining probabilistic surrogate modelling, such as Gaussian process regression, with Bayesian optimisation, which can reduce the number of required simulations by several orders of magnitude. Here, we implement this type of Bayesian optimisation framework for a model for analysis of disruption runaway electrons, and explore for simulations of current quench in a JET plasma discharge with an argon induced disruption. We use this proof-of-principle framework to explore the optimum input parameters with uncertainties in optimisation tasks ranging from one to seven dimensions. The relevant Python codes that are used in the analysis are available via https://github.com/aejarvin/BO_FOR_RE_SIMULATIONS/.

Some statistical inferences of parameter in MCMC approach and the application in uncertainty analysis of hydrological simulation

Markov Chain Monte Carlo (MCMC) method has been increasingly popular in uncertainty analysis of hydrological simulation. In MCMC approach, deviations between model outputs and observations are commonly assumed to follow Gaussian distribution with zero medium and constant standard deviation σ2. However, the estimation of σ2 is a difficulty in terms of that it was assigned subjectively in previous studies, hindering the improvement of performance for uncertainty assessment. This work systemically investigates the statistical meaning of parameter σ2. σ could be expressed as the product of data length and two standard deviations, one of which is for observations (i.e. σObs) and the other for Nash-Sutcliffe Coefficient of Efficiency (NSCE) (i.e.σs). A new label called Confidence Level of Model (CLM) is developed to interpretσs. The natural logarithm of the posterior probability distribution for NSCE is a first-order linear equation associated with CLM. The CLM could be employed to guide the construction of σs and then the estimation of σ2. Uncertainty analysis of a flow duration curve (FDC) model is conducted using the MCMC method based on CLM, and the generalized likelihood uncertainty estimation (GLUE) method is employed for comparison. Results show that the CLM affects the MCMC results by three kinds of trade-offs, and the MCMC method based on CLM performs well in generating regular posterior distributions of model parameters and discharges. The MCMC method also yields narrow and symmetrical confidence intervals. Findings of this paper could interpret typical uncertainty behaviors commonly existing in hydrological modeling, and provide beneficial insights for the uncertainty analysis of other environmental modeling.

Variational inference of fractional Brownian motion with linear computational complexity.

We introduce a simulation-based, amortized Bayesian inference scheme to infer the parameters of random walks. Our approach learns the posterior distribution of the walks' parameters with a likelihood-free method. In the first step a graph neural network is trained on simulated data to learn optimized low-dimensional summary statistics of the random walk. In the second step an invertible neural network generates the posterior distribution of the parameters from the learned summary statistics using variational inference. We apply our method to infer the parameters of the fractional Brownian motion model from single trajectories. The computational complexity of the amortized inference procedure scales linearly with trajectory length, and its precision scales similarly to the Cramér-Rao bound over a wide range of lengths. The approach is robust to positional noise, and generalizes to trajectories longer than those seen during training. Finally, we adapt this scheme to show that a finite decorrelation time in the environment can furthermore be inferred from individual trajectories.

Posterior estimation using deep learning: a simulation study of compartmental modeling in dynamic positron emission tomography.

In medical imaging, images are usually treated as deterministic, while their uncertainties are largely underexplored. This work aims at using deep learning to efficiently estimate posterior distributions of imaging parameters, which in turn can be used to derive the most probable parameters as well as theiruncertainties. Our deep learning-based approaches are based on a variational Bayesian inference framework, which is implemented using two different deep neural networks based on conditional variational auto-encoder (CVAE), CVAE-dual-encoder, and CVAE-dual-decoder. The conventional CVAE framework, that is, CVAE-vanilla, can be regarded as a simplified case of these two neural networks. We applied these approaches to a simulation study of dynamic brain PET imaging using a reference region-based kineticmodel. In the simulation study, we estimated posterior distributions of PET kinetic parameters given a measurement of the time-activity curve. Our proposed CVAE-dual-encoder and CVAE-dual-decoder yield results that are in good agreement with the asymptotically unbiased posterior distributions sampled by Markov Chain Monte Carlo (MCMC). The CVAE-vanilla can also be used for estimating posterior distributions, although it has an inferior performance to both CVAE-dual-encoder andCVAE-dual-decoder. We have evaluated the performance of our deep learning approaches for estimating posterior distributions in dynamic brain PET. Our deep learning approaches yield posterior distributions, which are in good agreement with unbiased distributions estimated by MCMC. All these neural networks have different characteristics and can be chosen by the user for specific applications. The proposed methods are general and can be adapted to otherproblems.

A Bayesian approach for torque modelling of BeXRB pulsars with application to super-Eddington accretors

ABSTRACT In this study, we present a method to estimate posterior distributions for standard accretion torque model parameters and binary orbital parameters for X-ray binaries using a nested sampling algorithm for Bayesian parameter estimation. We study the spin evolution of two Be X-ray binary systems in the Magellanic Clouds, RX J0520.5−6932 and RX J0209−7427, during major outbursts, in which they surpassed the Eddington limit. Moreover, we apply our method to the recently discovered Swift J0243.6+6124, the only known Galactic pulsating ultra-luminous X-ray source. This is an excellent candidate for studying the disc evolution at super-Eddington accretion rates, because its luminosity spans several orders of magnitude during its outburst, with a maximum LX that exceeded the Eddington limit by a factor of ∼10. Our method, when applied to RX J0520.5−6932 and RX J0209−7427, is able to identify the more favourable torque model for each system, while yielding meaningful ranges for the NS and orbital parameters. Our analysis for Swift J0243.6+6124 illustrates that, contrary to the standard torque model predictions, the magnetospheric radius (Rm) and the Alfvén radius (RA) are not proportional to each other when surpassing the Eddington limit. Reported distance estimates of this source range between 5 and 7 kpc. Smaller distances require non-typical neutron star properties (i.e. mass and radius) and possibly lower radiative efficiency of the accretion column.

Corrosion fatigue crack growth prediction of bridge suspender wires using Bayesian gaussian process

This paper proposes a Bayesian gaussian process-based corrosion fatigue damage assessment framework for bridge suspender wires. Prior statistical parameters in the fatigue life prediction model are obtained by corrosion fatigue crack growth test under different stress ratios. A numerical simulation method of corrosion fatigue crack growth is developed combined with ABAQUS and FRANC3D. Following this, a Bayesian network-based framework is established by integrating fatigue crack growth model, stress concentration factor and observed fatigue crack length information to reduce the uncertainties. The uncertainties exist in stress concentration factor are quantified by gaussian process regression-based surrogate model, and the pit geometry of length, width and depth are selected as three-dimensional input. Markov Chain Monte Carlo simulation is used to obtain the posterior distribution of parameters in the Bayesian network. The results show that the fatigue crack growth predictions agree well with experimental observations. The proposed method can simplify the complicated modeling process of stress concentration factor under the influence of corrosion pit, and can effectively update the model parameters to accurately predict the corrosion fatigue crack of bridge suspender wires.