Posterior Distribution Of Parameters Research Articles

Study on Bayesian Skew-Normal Linear Mixed Model and Its Application in Fire Insurance.

Fire insurance is a crucial component of property insurance, and its rating depends on the forecast of insurance loss claim data. Fire insurance loss claim data have complicated characteristics such as skewness and heavy tail. The traditional linear mixed model is commonly difficult to accurately describe the distribution of loss. Therefore, it is crucial to establish a scientific and reasonable distribution model of fire insurance loss claim data. In this study, the random effects and random errors in the linear mixed model are firstly assumed to obey the skew-normal distribution. Then, a skew-normal linear mixed model is established using the Bayesian MCMC method based on a set of U.S. property insurance loss claims data. Comparative analysis is conducted with the linear mixed model of logarithmic transformation. Afterward, a Bayesian skew-normal linear mixed model for Chinese fire insurance loss claims data is designed. The posterior distribution of claim data parameters and related parameter estimation are employed with the R language JAGS package to obtain the predicted and simulated loss claim values. Finally, the optimization model in this study is used to determine the insurance rate. The results demonstrate that the model established by the Bayesian MCMC method can overcome data skewness, and the fitting and correlation with the sample data are better than the log-normal linear mixed model. Hence, it can be concluded that the distribution model proposed in this paper is reasonable for describing insurance claims. This study innovates a new approach for calculating the insurance premium rate and expands the application of the Bayesian method in the fire insurance field.

Source identification in river pollution incidents using a cellular automata model and Bayesian Markov chain Monte Carlo method.

Identification of contaminant sources in rivers is crucial for river protection and emergency response. This study presents an innovative approach for identifying river pollution sources by using Bayesian inference and cellular automata (CA) modeling. A general Bayesian framework is proposed that combines the CA model with observed data to identify unknown sources of river pollution. To reduce the computational burden of the Bayesian inference, a CA contaminant transport model is developed to efficiently simulate pollutant concentration values in the river. These simulated concentration values are then used to calculate the likelihood function of available measurements. The Markov chain Monte Carlo (MCMC) method is used to produce the posterior distribution of contaminant source parameters, which is a sampling-based method that enables the estimation of complex posterior distributions. The suggested methodology is applied to a real case study of the Fen River in Yuncheng City, Shanxi Province, Northern China, and it estimates the release time, release mass, and source location with relative errors below 19%. The research indicates that the proposed methodology is an effective and flexible way to identify the location and concentrations of river contaminant sources.

Solving Geophysical Inversion Problems with Intractable Likelihoods: Linearized Gaussian Approximations Versus the Correlated Pseudo-marginal Method

A geophysical Bayesian inversion problem may target the posterior distribution of geological or hydrogeological parameters given geophysical data. To account for the scatter in the petrophysical relationship linking the target parameters to the geophysical properties, this study treats the intermediate geophysical properties as latent (unobservable) variables. To perform inversion in such a latent variable model, the intractable likelihood function of the (hydro)geological parameters given the geophysical data needs to be estimated. This can be achieved by approximation with a Gaussian probability density function based on local linearization of the geophysical forward operator, thereby, accounting for the noise in the petrophysical relationship by a corresponding addition to the data covariance matrix. The new approximate method is compared against the general correlated pseudo-marginal method, which estimates the likelihood by Monte Carlo averaging over samples of the latent variable. First, the performances of the two methods are tested on a synthetic test example, in which a multivariate Gaussian porosity field is inferred using crosshole ground-penetrating radar first-arrival travel times. For this example with rather small petrophysical uncertainty, the two methods provide near-identical estimates, while an inversion that ignores petrophysical uncertainty leads to biased estimates. The results of a sensitivity analysis are then used to suggest that the linearized Gaussian approach, while attractive due to its relative computational speed, suffers from a decreasing accuracy with increasing scatter in the petrophysical relationship. The computationally more expensive correlated pseudo-marginal method performs very well even for settings with high petrophysical uncertainty.

Inversion Framework of Reservoir Parameters Based on Deep Autoregressive Surrogate and Continual Learning Strategy

Summary History matching is a crucial process that enables the calibration of uncertain parameters of the numerical model to obtain an acceptable match between simulated and observed historical data. However, the implementation of the history-matching algorithm is usually based on iteration, which is a computationally expensive process due to the numerous runs of the simulation. To address this challenge, we propose a surrogate model for simulation based on an autoregressive model combined with a convolutional gated recurrent unit (ConvGRU). The proposed ConvGRU-based autoregressive neural network (ConvGRU-AR-Net) can accurately predict state maps (such as saturation maps) based on spatial and vector data (such as permeability and relative permeability, respectively) in an end-to-end fashion. Furthermore, history matching must be performed multiple times throughout the production cycle of the reservoir to fit the most recent production observations, making continual learning crucial. To enable the surrogate model to quickly learn recent data by transferring experience from previous tasks, an ensemble-based continual learning strategy is used. Together with the proposed neural network–based surrogate model, the randomized maximum likelihood (RML) is used to calibrate uncertain parameters. The proposed method is evaluated using 2D and 3D reservoir models. For both cases, the surrogate inversion framework successfully achieves a reasonable posterior distribution of reservoir parameters and provides a reliable assessment of the reservoir’s behaviors.

TD-CARMA: Painless, Accurate, and Scalable Estimates of Gravitational Lens Time Delays with Flexible CARMA Processes

Cosmological parameters encoding our understanding of the expansion history of the universe can be constrained by the accurate estimation of time delays arising in gravitationally lensed systems. We propose TD-CARMA, a Bayesian method to estimate cosmological time delays by modeling observed and irregularly sampled light curves as realizations of a continuous auto-regressive moving average (CARMA) process. Our model accounts for heteroskedastic measurement errors and microlensing, an additional source of independent extrinsic long-term variability in the source brightness. The semiseparable structure of the CARMA covariance matrix allows for fast and scalable likelihood computation using Gaussian process modeling. We obtain a sample from the joint posterior distribution of the model parameters using a nested sampling approach. This allows for “painless” Bayesian computation, dealing with the expected multimodality of the posterior distribution in a straightforward manner and not requiring the specification of starting values or an initial guess for the time delay, unlike existing methods. In addition, the proposed sampling procedure automatically evaluates the Bayesian evidence, allowing us to perform principled Bayesian model selection. TD-CARMA is parsimonious, and typically includes no more than a dozen unknown parameters. We apply TD-CARMA to six doubly lensed quasars HS2209+1914, SDSS J1001+5027, SDSS J1206+4332, SDSS J1515+1511, SDSS J1455+1447, and SDSS J1349+1227, estimating their time delays as −21.96 ± 1.448, 120.93 ± 1.015, 111.51 ± 1.452, 210.80 ± 2.18, 45.36 ± 1.93, and 432.05 ± 1.950, respectively. These estimates are consistent with those derived in the relevant literature, but are typically two to four times more precise.

Spectral classification of young stars using conditional invertible neural networks

Aims. We introduce a new deep-learning tool that estimates stellar parameters (e.g. effective temperature, surface gravity, and extinction) of young low-mass stars by coupling the Phoenix stellar atmosphere model with a conditional invertible neural network (cINN). Our networks allow us to infer the posterior distribution of each stellar parameter from the optical spectrum. Methods. We discuss cINNs trained on three different Phoenix grids: Settl, NextGen, and Dusty. We evaluate the performance of these cINNs on unlearned Phoenix synthetic spectra and on the spectra of 36 class III template stars with well-characterised stellar parameters. Results. We confirm that the cINNs estimate the considered stellar parameters almost perfectly when tested on unlearned Phoenix synthetic spectra. Applying our networks to class III stars, we find good agreement with deviations of 5–10% at most. The cINNs perform slightly better for earlier-type stars than for later-type stars such as late M-type stars, but we conclude that estimates of effective temperature and surface gravity are reliable for all spectral types within the training range of the network. Conclusions. Our networks are time-efficient tools that are applicable to large numbers of observations. Among the three networks, we recommend using the cINN trained on the Settl library (Settl-Net) because it provides the best performance across the widest range of temperature and gravity.

Bayesian optimal sensor placement for parameter estimation under modeling and input uncertainties

A Bayesian optimal sensor placement (OSP) framework for parameter estimation in nonlinear structural dynamics models is proposed, based on maximizing a utility function built from appropriate measures of information contained in the input–output response time history data. The information gain is quantified using Kullback–Leibler divergence (KL-div) between the prior and posterior distribution of the model parameters. The design variables may include the type and location of sensors. Asymptotic approximations, valid for large number of data, provide valuable insight into the measure of information. Robustness to uncertainties in nuisance (non-updatable) parameters associated with modeling and excitation uncertainties is considered by maximizing the expected information gain over all possible values of the nuisance parameters. In particular, the framework handles the case where the excitation time history is measured by installed sensors but remains unknown at the experimental design phase. Introducing stochastic excitation models, the expected information gain is taken over the large number of uncertain parameters used to model the random variability in the input time histories. Monte Carlo or sparse grid methods estimate the multidimensional probability integrals arising in the formulation. Heuristic algorithms are used to solve the optimization problem. The effectiveness of the method is demonstrated for a multi-degree of freedom (DOF) spring–mass chain system with restoring elements that exhibit hysteretic nonlinearities.

A Stochastic Analysis of the Effect of Trading Parameters on the Stability of the Financial Markets Using a Bayesian Approach

The purpose of this study was to identify and measure the impact of the different effects of entropy states over the high-frequency trade of the cryptocurrency market, especially in Bitcoin, using and selecting optimal parameters of the Bayesian approach, specifically through approximate Bayesian computation (ABC). ABC corresponds to a class of computational methods rooted in Bayesian statistics that could be used to estimate the posterior distributions of model parameters. For this research, ABC was applied to estimate the daily prices of the Bitcoin cryptocurrency from May 2013 to December 2021. The findings suggest that the behaviour of the parameters for our tested trading algorithms, in which sudden jumps are observed, can be interpreted as changes in states of the generated time series. Additionally, it is possible to identify and model the effects of the COVID-19 pandemic on the series analysed in the research. Finally, the main contribution of this research is that we have characterised the relationship between entropy and the evolution of parameters defining the optimal selection of trading algorithms in the financial industry.

Using Bayesian updating for monopile offshore wind turbines monitoring

Digital twins (DTs) are thought to be promising tools for offshore wind turbines (OWTs) in real-time updating, optimized design, intelligent operation, and maintenance. When creating DT analysis tools, it is critical to update model parameters to create high-fidelity, high-accuracy virtual models. In this paper, a Bayesian updating framework based on Transitional Markov Chain Monte Carlo (TMCMC) is proposed to construct the desired virtual model of monopile OWTs to address this difficult problem. The likelihood function for Bayesian inference was created using measurements from physical entity monitoring as well as natural frequencies and damping ratios from the monopile OWT finite element (FE) model. Then, through a series of relatively simple intermediate likelihood functions, it gradually converges to the target function. During this time, the artificial neural network (ANN) surrogate model calculates the intermediate likelihood function of each sample, and a parallel computing strategy is used to enhance the computational efficiency. The results show that the posterior distribution of the uncertain parameters can be successfully estimated from the Markov chain using this method to update the finite element model. In addition, the proposed Bayesian updating method is resistant to measurement noise.

Mapping the risk of Rift Valley fever in Uganda using national seroprevalence data from cattle, sheep and goats.

Uganda has had repeated outbreaks of Rift Valley fever (RVF) since March 2016 when human and livestock cases were reported in Kabale after a long interval. The disease has a complex and poorly described transmission patterns which involves several mosquito vectors and mammalian hosts (including humans). We conducted a national serosurvey in livestock to determine RVF virus (RVFV) seroprevalence, risk factors, and to develop a risk map that could be used to guide risk-based surveillance and control measures. A total of 3,253 animals from 175 herds were sampled. Serum samples collected were screened at the National Animal Disease Diagnostics and Epidemiology Centre (NADDEC) using a competition multispecies anti-RVF IgG ELISA kit. Data obtained were analyzed using a Bayesian model that utilizes integrated nested Laplace approximation (INLA) and stochastic partial differential equation (SPDE) approaches to estimate posterior distributions of model parameters, and account for spatial autocorrelation. Variables considered included animal level factors (age, sex, species) and multiple environmental data including meteorological factors, soil types, and altitude. A risk map was generated by projecting fitted (mean) values, from a final model that had environmental factors onto a spatial grid that covered the entire domain. The overall RVFV seroprevalence was 11.3% (95% confidence interval: 10.2-12.3%). Higher RVFV seroprevalences were observed in older animals compared to the young, and cattle compared to sheep and goats. RVFV seroprevalence was also higher in areas that had (i) lower precipitation seasonality, (ii) haplic planosols, and (iii) lower cattle density. The risk map generated demonstrated that RVF virus was endemic in several regions including those that have not reported clinical outbreaks in the northeastern part of the country. This work has improved our understanding on spatial distribution of RVFV risk in the country as well as the expected burden of the disease in livestock.

Bayesian Updating and Model Class Selection for Magnitude Estimation in Earthquake Early Warning: Application to Earthquake Sequences in Sichuan Region, China

ABSTRACT A key component of earthquake early warning system is the development of an accurate and robust predictive model relating the recorded waveform to seismicity and ground-shaking characteristics. This article presents a probabilistic Bayesian inference methodology to address relevant sources of uncertainty in the development of such predictive models. We investigate the calibration of models for the earthquake magnitude based on the maximum predominant period, considering the Sichuan region of Southwestern China as a case study. Established approaches for developing predictive models in this context adopt deterministic tools for some aspects of this development. They consider a linear regression calibrated typically through least squares optimization and frequently utilize the mean observations for each event averaging across the data available from different stations. The proposed Bayesian learning accommodates the following improvements: a model class selection is established, comparing across different candidate models to promote the most appropriate from accuracy and robustness perspectives; the full posterior distribution of the model parameters is identified, quantifying relevant uncertainties in their values; a heteroscedastic model is considered for the estimation error variance; and the observations are separately considered at the calibration stage. Each of these improvements ultimately addresses a different source of uncertainty impacting the predictive model development. We utilize transitional Markov chain Monte Carlo for obtaining samples from the posterior and for calculating the evidence to perform the model class selection. Different regression models are examined, and the Bayesian-based model identification is compared against the common least squares identification approach. Results show the value added by comparing across the different models and by considering a heteroskedastic variance model, offering insights into the advantages of Bayesian-based predictive models in earthquake early warning applications.

Uncertainty quantification in three‐dimensional magnetohydrodynamic equilibrium reconstruction via surrogate‐assisted Bayesian inference

AbstractIn three‐dimensional (3D) equilibrium, reconstruction defining parameters of an ideal magneto‐hydrodynamic equilibrium are inferred from a set of plasma diagnostic measurements. For the reconstructed parameters, various forms of uncertainty estimates exist within common 3D reconstruction frameworks. These estimates often assume a Gaussian posterior distribution. The validity of this assumption is not obvious in such highly nonlinear inverse problems, and therefore the accuracy of the estimates cannot be guaranteed. In this work, we formulate the problem of 3D equilibrium reconstruction in a Bayesian sense and explore the posterior distribution of reconstruction parameters via Markov chain Monte Carlo (MCMC) sampling. The target reconstruction parameters, that is, shape and scaling factors of the pressure and toroidal current‐density profiles as well as the total toroidal flux, are taken from a reduced subspace of Wendelstein 7‐X equilibrium configurations. Since the corresponding forward model evaluations are computationally demanding, we replace the forward model via a polynomial chaos expansion surrogate. We compare the posterior distribution obtained via MCMC sampling to Laplace's approximation, which assumes a Gaussian posterior. We find that the two approaches provide similar results in regimes where the uncertainty on the plasma diagnostic signals is low. However, discrepancies between the posteriors are observed in cases of higher diagnostic signal uncertainty. Therefore, we provide further validation of the commonly used Laplace's approximation as a method of uncertainty quantification in 3D equilibrium reconstruction and showcase some of its limitations.

Bayesian composite quantile regression for the single-index model.

By using a Gaussian process prior and a location-scale mixture representation of the asymmetric Laplace distribution, we develop a Bayesian analysis for the composite quantile single-index regression model. The posterior distributions for the unknown parameters are derived, and the Markov chain Monte Carlo sampling algorithms are also given. The proposed method is illustrated by three simulation examples and a real dataset.

A deep learning synthetic likelihood approximation of a non-stationary spatial model for extreme streamflow forecasting

Extreme streamflow is a key indicator of flood risk, and quantifying the changes in its distribution under non-stationary climate conditions is key to mitigating the impact of flooding events. We propose a non-stationary process mixture model (NPMM) for annual streamflow maxima over the central US (CUS) which uses downscaled climate model precipitation projections to forecast extremal streamflow. Spatial dependence for the model is specified as a convex combination of transformed Gaussian and max-stable processes, indexed by a weight parameter which identifies the asymptotic regime of the process. The weight parameter is modeled as a function of the annual precipitation for each of the two hydrologic regions within the CUS, introducing spatio-temporal non-stationarity within the model. The NPMM is flexible with desirable tail dependence properties, but yields an intractable likelihood. To address this, we embed a neural network within a density regression model which is used to learn a synthetic likelihood function using simulations from the NPMM with different parameter settings. Our model is fitted using observational data for 1972–2021, and inference carried out in a Bayesian framework. The two regions within the CUS are estimated to be in different asymptotic regimes based on the posterior distribution of the weight parameter. Annual streamflow maxima estimates based on global climate models for two representative climate pathway scenarios suggest an overall increase in the frequency and magnitude of extreme streamflow for 2006–2035 compared to the historical period of 1972–2005.

Longitudinal Data Analysis Based on Bayesian Semiparametric Method

A Bayesian semiparametric model framework is proposed to analyze multivariate longitudinal data. The new framework leads to simple explicit posterior distributions of model parameters. It results in easy implementation of the MCMC algorithm for estimation of model parameters and demonstrates fast convergence. The proposed model framework associated with the MCMC algorithm is validated by four covariance structures and a real-life dataset. A simple Monte Carlo study of the model under four covariance structures and an analysis of the real dataset show that the new model framework and its associated Bayesian posterior inferential method through the MCMC algorithm perform fairly well in the sense of easy implementation, fast convergence, and smaller root mean square errors compared with the same model without the specified autoregression structure.

Analytical marginalization over photometric redshift uncertainties in cosmic shear analyses

ABSTRACT As the statistical power of imaging surveys grows, it is crucial to account for all systematic uncertainties. This is normally done by constructing a model of these uncertainties and then marginalizing over the additional model parameters. The resulting high dimensionality of the total parameter spaces makes inferring the cosmological parameters significantly more costly using traditional Monte Carlo sampling methods. A particularly relevant example is the redshift distribution, p($z$ ), of the source samples, which may require tens of parameters to describe fully. However, relatively tight priors can be usually placed on these parameters through calibration of the associated systematics. In this paper, we show, quantitatively, that a linearization of the theoretical prediction with respect to these calibrated systematic parameters allows us to analytically marginalize over these extra parameters, leading to a factor of ∼30 reduction in the time needed for parameter inference, while accurately recovering the same posterior distributions for the cosmological parameters that would be obtained through a full numerical marginalization over 160 p($z$ ) parameters. We demonstrate that this is feasible not only with current data and current achievable calibration priors but also for future Stage-IV data sets.

Bayesian instrumental variable estimation in linear measurement error models

AbstractIn this article, we study the problem of parameter estimation for measurement error models by combining the Bayes method with the instrumental variable approach, deriving the posterior distribution of parameters under different priors with known and unknown variance parameters, respectively, and calculating the Bayes estimator (BE) of the parameters under quadratic loss. However, it is difficult to obtain an explicit expression for BE because of the complex multiple integrals involved. Therefore, we adopt the linear Bayes method, which does not specify the form of the prior and avoids these complicated integral calculations, to obtain an expression for the linear Bayes estimator (LBE) for different priors. We prove that this LBE is superior to the two‐stage least squares estimator under the mean squared error matrix criterion. Numerical simulations show that our LBE is very close to the real parameter whether the variance parameters are known or unknown, and it gradually approaches BE as the sample size increases. Our results indicate that this instrumental variable approach is valid for measurement error models.

Hybrid AI-Bayesian-based demand models and fragility estimates for tall buildings against multi-hazard of earthquakes and winds

The uncertainty coming from various sources will have a considerable influence on the credibility of the structural performance evaluation of tall buildings. This paper proposes a hybrid AI-Bayesian-based methodology for fragility estimates of tall buildings subjected to simultaneous earthquake and wind events that is feasible to incorporate into both the epistemic and aleatory uncertainties. The main contributions and concept of this proposed methodology include (1) The Back Propagation (BP) Artificial Neural Network (ANN) technique is applied to train a surrogate model to replace finite element (FE) analysis of tall buildings under concurrent seismic and wind excitations, which can highly reduce computing time of nonlinear dynamic analyses and is beneficial for quantifying the aleatory uncertainty with material properties and structural characteristics. (2) A physics-based demand model is developed for fragility estimates, and the Bayesian statistics method is utilized to obtain the posterior probability distributions of the unknown parameters in the demand model, which is beneficial for quantifying the epistemic uncertainty in the fragility estimates. Finally, this proposed method is implemented in a representative composite tall building. The application highlights the importance of incorporating both the epistemic and aleatory uncertainties into the fragility estimates of tall buildings under multiple hazards. This AI-Bayesian-based method provides great help for quantifying various uncertainties and improving resilience assessment reliability of tall buildings under multiple hazards.

A General Integrative Neurocognitive Modeling Framework to Jointly Describe EEG and Decision-making on Single Trials

Despite advances in techniques for exploring reciprocity in brain-behavior relations, few studies focus on building neurocognitive models that describe both human EEG and behavioral modalities at the single-trial level. Here, we introduce a new integrative joint modeling framework for the simultaneous description of single-trial EEG measures and cognitive modeling parameters of decision-making. As specific examples, we formalized how single-trial N200 latencies and centro-parietal positivities (CPPs) are predicted by changing single-trial parameters of various drift-diffusion models (DDMs). We trained deep neural networks to learn Bayesian posterior distributions of unobserved neurocognitive parameters based on model simulations. These models do not have closed-form likelihoods and are not easy to fit using Markov chain Monte Carlo (MCMC) methods because nuisance parameters on single trials are shared in both behavior and neural activity. We then used parameter recovery assessment and model misspecification to ascertain how robustly the models’ parameters can be estimated. Moreover, we fit the models to three different real datasets to test their applicability. Finally, we provide some evidence that single-trial integrative joint models are superior to traditional integrative models. The current single-trial paradigm and the simulation-based (likelihood-free) approach for parameter recovery can inspire scientists and modelers to conveniently develop new neurocognitive models for other neural measures and to evaluate them appropriately.

Quantifying the Magnitude and Longevity of the Effect of Repetitive Head Impacts in Adolescent Soccer Players: Deleterious Effect of Long Headers Extend Beyond a Month.

There is growing interest in the effects of sports-related repetitive head impacts (RHIs) on athletes' cognitive capabilities. This study examines the effect of RHIs in data collected from adolescent athletes to estimate the magnitude and longevity of RHIs on sensorimotor and cognitive performance. A non-linear regression model estimated the longevity of RHI effects by adding a half-life parameter embedded in an exponential decay function. A model estimate of this parameter allows the possibility of RHI effects to attenuate over time and introduces a mechanism to study the cumulative effect of RHIs. The posterior distribution of the half-life parameter associated with short-distance headers (<30 m) is centered around 6 days, whereas the posterior distribution of the half-life parameter associated with long-distance headers extends beyond a month. Additionally, the magnitude of the effect of each short header is around 3 times smaller than that of a long header. The results indicate that, on both tasks, response time (RT) changes after long headers are bigger in magnitude and last longer compared to the effects of short headers. Most important, we demonstrate that deleterious effects of long headers extend beyond 1 month. Although estimates are based on data from a relatively short-duration study with a relatively small sample size, the proposed model provides a mechanism to estimate long-term behavioral slowing from RHIs, which may be helpful to reduce the risk of additional injury. Finally, differences in the longevity of the effects of short and long RHIs may help to explain the large variance found between biomechanical input and clinical outcome in studies of concussion tolerance.