Bayesian Probability Distribution Research Articles

Bayesian non-parametric modeling by mixture of Kibble-Pólya tree to detect Low-Activity uranium contamination

Accurate detection of low-level radioactivity is critical for decommissioning projects in nuclear facilities, particularly in the design of radiation monitoring systems with a low false alarm rate. Utilizing a non-parametric Bayesian continuous probability distribution enables reliable mapping of potential contamination. Our method introduces a statistical test based on a Pólya tree prior, applied to radiation detection. The detection efficiency of this proposed Bayesian test is quantified using receiver-operating characteristic (ROC) curves and compared to a Bayesian test based on the Kibble bivariate gamma distribution developed for the same purpose. The results demonstrate that the new Bayesian test generally outperforms the previous method in terms of detection performance under very low signal-to-noise ratios, with improvements ranging from 3% to 28% against both stationary and non-stationary radiological backgrounds, respectively. This superiority is further reaffirmed through comparisons with alternative Bayesian and frequentist hypothesis tests, with gains estimated at 52% and 4%, respectively.

Bayesian geochemical correlation and tomography

To accurately reconstruct palaeoenvironmental change through time it is important to determine which rock samples were deposited contemporaneously at different sites or transects, as erroneous correlation may lead to incorrectly inferred processes and rates. To correlate samples, current practice interpolates geological age between datable units along each transect, then temporal signatures observed in geochemical logs are matched between transects. Unfortunately spatiotemporally variable and unknown rates of sedimentary deposition create highly nonlinear space-time transforms, significantly altering apparent geochemical signatures. The resulting correlational hypotheses are also untestable against independent transects, because correlations have no spatially-predictive power. Here we use geological process information stored within neural networks to correlate spatially offset logs nonlinearly and geologically. The same method creates tomographic images of geological age and geochemical signature across intervening rock volumes. Posterior tomographic images closely resemble the true depositional age throughout the inter-transect volume, even for scenarios with long hiatuses in preserved geochemical signals. Bayesian probability distributions describe data-consistent variations in the results, showing that centred summary statistics such as mean and variance do not adequately describe correlational uncertainties. Tomographic images demonstrate spatially predictive power away from geochemical transects, creating novel hypotheses attributable to each geochemical correlation which are testable against independent data.

The hazards of dealing with response time outliers.

The presence of outliers in response times can affect statistical analyses and lead to incorrect interpretation of the outcome of a study. Therefore, it is a widely accepted practice to try to minimize the effect of outliers by preprocessing the raw data. There exist numerous methods for handling outliers and researchers are free to choose among them. In this article, we use computer simulations to show that serious problems arise from this flexibility. Choosing between alternative ways for handling outliers can result in the inflation of p-values and the distortion of confidence intervals and measures of effect size. Using Bayesian parameter estimation and probability distributions with heavier tails eliminates the need to deal with response times outliers, but at the expense of opening another source of flexibility.

Experimentally characterizing the spatially varying anisotropic mechanical property of cancellous bone via a Bayesian calibration method

Traditional experimental tests for characterizing bone's mechanical properties usually hypothesize a uniaxial stress condition without quantitatively evaluating the influence of spatially varying principal material orientations, which cannot accurately predict the mechanical properties distribution of bones in vivo environment. In this study, a Bayesian calibrating procedure was developed using quantified multiaxial stress to investigate cancellous bone's local anisotropic elastic performance around joints as the spatial variation of main bearing orientations. First, the bone cube specimens from the distal femur of sheep are prepared using traditional anatomical axes. The multiaxial stress state of each bone specimen is calibrated using the actual principal material orientations derived from fabric tensor at different anatomical locations. Based on the calibrated multiaxial stress state, the process of identifying mechanical properties is described as an inverse problem. Then, a Bayesian calibration procedure based on a surrogate constitutive model was developed via multiaxial stress correction to identify the anisotropic material parameters. Finally, a comparison between the experiment and simulation results is discussed by applying the optimal model parameters obtained from the Bayesian probability distribution. Compared to traditional uniaxial methods, our results prove that the calibration based on the spatial variation of the main bearing orientations can significantly improve the accuracy of characterizing regional anisotropic mechanical responses. Moreover, we determine that the actual mechanical property distribution is influenced by complicated mechanical stimulation. This study provides a novel method to evaluate the spatially varying mechanical properties of bone tissues enduring complex mechanical loading accurately and effectively. It is expected to provide more realistic mechanical design targets in vivo for a personalized artificial bone prosthesis in clinical treatment.

Construction and Simulation of a Strategic HR Decision Model Based on Recurrent Neural Network

In this paper, RNN (Recurrent Neural Network) algorithm is used to conduct an in‐depth analysis of HR strategic decision‐making and an HR strategic decision model is constructed for simulation. The four evaluation dimensions of index screening are extracted, the generalized gray correlation analysis is applied to screen the initial selection indexes of HR strategic decision‐making, and then the HR strategic decision‐making index system of power supply enterprises is constructed. On this basis, the applicability of the BP neural network in HR strategic decision‐making is analysed and demonstrated, a BP neural network‐based HR strategic decision‐making model for power supply enterprises is constructed, and the rationality of the model is illustrated through the model training and testing. Finally, an empirical study is conducted with S power supply company as an example to illustrate the operability of the BP network model for HR strategic decision‐making. The results of this paper provide a scientific basis for human resource decision‐making in power supply enterprises and provide theoretical support for promoting the healthy development of power supply enterprises. Improving the level of human resource management can make fuller use of human resources and realize the strategic goals of the enterprise. A strategy for optimizing the training of hybrid convolutional neural networks is proposed, using an exponential linear unit activation function to solve the problem of neuron dead zones, a hybrid pooling strategy is used to improve the problem of feature information loss in maximum pooling processing, and an improved cross‐entropy loss function is used to solve the problem of insufficient learning of difficult classification samples. The optimization of the model training process is finally completed, and the recommendation quality is improved. The Bayesian probability distribution table is learned to be filled in the expert data set; finally, the accuracy and effectiveness of the model are tested on the experimental platform. The results of the experiments show that the established Bayesian model beats the platform’s built‐in intelligent method with a 78.2% win rate. That is, the model can make intelligent recommendations for strategies in staffed and unmanned platform decision‐making and command‐and‐control combat units to execute tactical actions to achieve the best operational effectiveness.

Analyzing Uncertainty in Release Planning: A Method and Experiment for Fixed-Date Release Cycles

Release planning—deciding what features to implement in upcoming releases of a software system—is a critical activity in iterative software development. Many release planning methods exist, but most ignore the inevitable uncertainty in estimating software development effort and business value. The article’s objective is to study whether analyzing uncertainty during release planning generates better release plans than if uncertainty is ignored. To study this question, we have developed a novel release planning method under uncertainty, called BEARS, that models uncertainty using Bayesian probability distributions and recommends release plans that maximize expected net present value and expected punctuality. We then compare release plans recommended by BEARS to those recommended by methods that ignore uncertainty on 32 release planning problems. The experiment shows that BEARS recommends release plans with higher expected net present value and expected punctuality than methods that ignore uncertainty, thereby indicating the harmful effects of ignoring uncertainty during release planning. These results highlight the importance of eliciting and analyzing uncertainty in software effort and value estimations and call for increased research in these areas.

Bayesian parameter and joint probability distribution estimation for a hysteretic constitutive model of reinforcing steel

With the structural design paradigm shift since the early 2000′s from the traditional approach to performance-based design (PBD), there has been a growing need for reliable nonlinear finite element (FE) models that can accurately predict the response of structures when subjected to extreme loads, such as earthquakes. In the case of reinforced concrete (RC) structures, a proper representation of the hysteretic nonlinear behavior of reinforcing steel becomes crucial in order to carry out nonlinear time history analyses. The Giuffrè-Menegotto-Pinto (GMP) uniaxial steel constitutive law has been widely used by researchers and practitioners to model reinforcing steel bars. Despite the widespread in its implementation, a limited number of studies have proposed well-calibrated parameter values for this model. In addition, low identifiability of its governing parameters and the high cost of generating reliable experimental data have prevented a thorough probabilistic characterization of the GMP model parameters. Usually, only default parameter values from the early development of the model tend to be used. This paper uses experimental data from cyclic tests conducted on 36 reinforcing steel coupons manufactured in accordance to ASTM A615 and A706 Grade 60 reinforcing steel and proposes a joint probability density function (PDF) for the most influential parameters of the GMP material model. First, a local sensitivity analysis (LSA) is conducted to provide insight into the influence of each parameter in the model response. Also, global sensitivity analysis (GSA) is used to have a deep understanding of the composition of the variability in the model response due to parameter uncertainty and the level of interactions among parameters. The Bayesian approach is combined with the information obtained from GSA and LSA as input, to estimate model parameters and quantify the estimation uncertainties and propagate them to the material stress response. Uncertainty in model predictions obtained with the proposed PDF is assessed, and the impact of considering parameter correlations on the material response is investigated.

X-ray computed tomography using partially coherent Fresnel diffraction with application to an optical fiber.

A reconstruction algorithm for partially coherent x-ray computed tomography (XCT) including Fresnel diffraction is developed and applied to an optical fiber. The algorithm is applicable to a high-resolution tube-based laboratory-scale x-ray tomography instrument. The computing time is only a few times longer than the projective counterpart. The algorithm is used to reconstruct, with projections and diffraction, a tilt series acquired at the micrometer scale of a graded-index optical fiber using maximum likelihood and a Bayesian method based on the work of Bouman and Sauer. The inclusion of Fresnel diffraction removes some reconstruction artifacts and use of a Bayesian prior probability distribution removes others, resulting in a substantially more accurate reconstruction.

Carbon oxides emissions from lithium-ion batteries under thermal runaway from measurements and predictive model

A concentration test device was set up to study and analyze the thermal runaway gas production process in lithium-ion batteries. The experimental conditions for the research were determined by the influence of different external environments and gas flow rates on the thermal runaway process. Gas measurement and analysis of CO and CO2 generated in thermal runaway process under different states of charge (SOCs), different ambient temperatures and different electric heating power were conducted. The following results were obtained. Under different SOCs, the concentrations of CO and CO2 decreased as the charge capacity of the lithium-ion batteries decreased. Under different ambient temperatures, the concentration of CO2 decreased with the decrease of the ambient temperature. However, the concentration of CO at the ambient temperature of 180 °C was higher than that at the ambient temperature of 220 °C. Under different electric heating power, the concentrations of CO and CO2 decreased with the decrease of the electric heating power. Besides, Bayesian prior probability distribution theory was adopted to analyze the variations of CO concentration relative to time in the thermal runaway process, the results of which can predict thermal runaway state of lithium-ion batteries. The findings can serve to provide reference for accident prevention and control of thermal runaway of lithium-ion batteries.

Improving Orbit Estimates for Incomplete Orbits with a New Approach to Priors: with Applications from Black Holes to Planets

Abstract We propose a new approach to Bayesian prior probability distributions (priors) that can improve orbital solutions for low-phase-coverage orbits, where data cover less than ∼40% of an orbit. In instances of low phase coverage—such as with stellar orbits in the Galactic center or with directly imaged exoplanets—data have low constraining power and thus priors can bias parameter estimates and produce underestimated confidence intervals. Uniform priors, which are commonly assumed in orbit fitting, are notorious for this. We propose a new observable-based prior paradigm that is based on uniformity in observables. We compare performance of this observable-based prior and of commonly assumed uniform priors using Galactic center and directly imaged exoplanet (HR 8799) data. The observable-based prior can reduce biases in model parameters by a factor of two and helps avoid underestimation of confidence intervals for simulations with less than ∼40% phase coverage. Above this threshold, orbital solutions for objects with sufficient phase coverage—such as S0-2, a short-period star at the Galactic center with full phase coverage—are consistent with previously published results. Below this threshold, the observable-based prior limits prior influence in regions of prior dominance and increases data influence. Using the observable-based prior, HR 8799 orbital analyses favor low-eccentricity orbits and provide stronger evidence that the four planets have a consistent inclination of ∼30° to within 1σ. This analysis also allows for the possibility of coplanarity. We present metrics to quantify improvements in orbital estimates with different priors so that observable-based prior frameworks can be tested and implemented for other low-phase-coverage orbits.

Why results from Bayesian statistical analyses of clinical trials with a strong prior and small sample sizes may be misleading The case of the NICHD Neonatal Research Network Late Hypothermia Trial.

Why results from Bayesian statistical analyses of clinical trials with a strong prior and small sample sizes may be misleading The case of the NICHD Neonatal Research Network Late Hypothermia Trial.

Bayesian Weight Decay on Bounded Approximation for Deep Convolutional Neural Networks.

This paper determines the weight decay parameter value of a deep convolutional neural network (CNN) that yields a good generalization. To obtain such a CNN in practice, numerical trials with different weight decay values are needed. However, the larger the CNN architecture is, the higher is the computational cost of the trials. To address this problem, this paper formulates an analytical solution for the decay parameter through a proposed objective function in conjunction with Bayesian probability distributions. For computational efficiency, a novel method to approximate this solution is suggested. This method uses a small amount of information in the Hessian matrix. Theoretically, the approximate solution is guaranteed by a provable bound and is obtained by a proposed algorithm, where its time complexity is linear in terms of both the depth and width of the CNN. The bound provides a consistent result for the proposed learning scheme. By reducing the computational cost of determining the decay value, the approximation allows for the fast investigation of a deep CNN (DCNN) which yields a small generalization error. Experimental results show that our assumption verified with different DCNNs is suitable for real-world image data sets. In addition, the proposed method significantly reduces the time cost of learning with setting the weight decay parameter while achieving good classification performances.

Risk-Stratified Incidence of Renal Replacement Therapy Initiation: A Longitudinal Analysis Using Medical Claims and Health Checkup Data.

Intervention for higher-risk participants of health checkups especially with diabetes has been started in Japan to prevent renal replacement therapy (RRT) initiation, but evidence about RRT initiation risk among checkup participants has been scarce. To estimate the incidence by risk factors, we conducted a retrospective cohort study using medical claims and checkup data of a community-based insurance scheme in Japan. Beneficiaries who participated in the checkup in 2012-2013 were included and followed up for about five years. We estimated the incidence of RRT initiation by the subject characteristics, followed by investigation for risk factors in bivariate analyses and multivariable regression analyses with Bayesian prior probability distributions. As a result, among 49,252 participants, 37 initiated dialysis (0.21/1,000 person-years); no kidney transplantation was performed during the period. Baseline estimated glomerular filtration rate was strongly associated with dialysis initiation. No dialysis was initiated among those without baseline hypertension; cumulative incidence by hypertension status was significantly different (p < 0.001). Diabetes was significantly associated with dialysis initiation in bivariate analysis, but the association was not significant in multivariable regression analysis [reference: no diabetes; incidence rate ratio (IRR) for diabetes without medication, 3.30 (95% credible interval, 0.48-15.56); IRR for diabetes with medication, 1.69 (95% credible interval, 0.68-3.47)]. In conclusion, potential risk factors for RRT initiation include male sex, comorbid hypertension, and current smoking status, in addition to advanced chronic kidney disease, proteinuria, and diabetes. New initiatives should consider these factors to increase the efficacy of the programs at the population level.

Ensemble variational assimilation as a probabilistic estimator – Part 1: The linear and weak non-linear case

Abstract. Data assimilation is considered as a problem in Bayesian estimation, viz. determine the probability distribution for the state of the observed system, conditioned by the available data. In the linear and additive Gaussian case, a Monte Carlo sample of the Bayesian probability distribution (which is Gaussian and known explicitly) can be obtained by a simple procedure: perturb the data according to the probability distribution of their own errors, and perform an assimilation on the perturbed data. The performance of that approach, called here ensemble variational assimilation (EnsVAR), also known as ensemble of data assimilations (EDA), is studied in this two-part paper on the non-linear low-dimensional Lorenz-96 chaotic system, with the assimilation being performed by the standard variational procedure. In this first part, EnsVAR is implemented first, for reference, in a linear and Gaussian case, and then in a weakly non-linear case (assimilation over 5 days of the system). The performances of the algorithm, considered either as a probabilistic or a deterministic estimator, are very similar in the two cases. Additional comparison shows that the performance of EnsVAR is better, both in the assimilation and forecast phases, than that of standard algorithms for the ensemble Kalman filter (EnKF) and particle filter (PF), although at a higher cost. Globally similar results are obtained with the Kuramoto–Sivashinsky (K–S) equation.

A Practical Pricing Approach to Smart Grid Demand Response Based on Load Classification

Appropriate pricing schemes are of great importance in terms of managing large-scale and distributed controllable loads in smart grids. In this paper, a practical pricing scheme is proposed to encourage different consumers to participate in demand response by providing them with a list of price plans. First, a classification algorithm is employed to divide consumers into different categories based on their individual information, such as marginal cost, upper limit of load adjustment, and elasticity coefficient. Afterwards, from the perspective of load service entity, a pricing model is formulated as a nonlinear programming problem, aiming to minimize the overall operation cost. Moreover, the Bayesian discrete probability distribution function is adopted to tackle the uncertainty of consumers’ choosing behavior. Meanwhile, an incentive compatible constraint is added to capture the real private information of controllable loads, e.g., inducing them to bid as close as their marginal costs. According to the case studies on the IEEE 30-bus and 118-bus systems, the proposed pricing scheme is an effective demand response approach to managing controllable loads and achieving lower system cost.

Optimal Belief Approximation

In Bayesian statistics probability distributions express beliefs. However, for many problems the beliefs cannot be computed analytically and approximations of beliefs are needed. We seek a loss function that quantifies how “embarrassing” it is to communicate a given approximation. We reproduce and discuss an old proof showing that there is only one ranking under the requirements that (1) the best ranked approximation is the non-approximated belief and (2) that the ranking judges approximations only by their predictions for actual outcomes. The loss function that is obtained in the derivation is equal to the Kullback-Leibler divergence when normalized. This loss function is frequently used in the literature. However, there seems to be confusion about the correct order in which its functional arguments—the approximated and non-approximated beliefs—should be used. The correct order ensures that the recipient of a communication is only deprived of the minimal amount of information. We hope that the elementary derivation settles the apparent confusion. For example when approximating beliefs with Gaussian distributions the optimal approximation is given by moment matching. This is in contrast to many suggested computational schemes.

Integrated B2B‐NMPC control strategy for batch/semibatch crystallization processes

The uncertainty in crystallization kinetics is of major concern in manufacturing processes, which can result in deterioration of most model‐based control strategies. In this study, uncertainties in crystallization kinetic parameters were characterized by Bayesian probability distributions. An integrated B2B‐NMPC control strategy was proposed to first update the kinetic parameters from batch to batch using a multiway partial least‐squares (MPLS) model, which described the variances of kinetic parameters from that of process variables and batch‐end product qualities. The process model with updated kinetic parameters was then incorporated into an NMPC design, the extended prediction self‐adaptive control (EPSAC), for online control of the final product qualities. Promising performance of the proposed integrated strategy was demonstrated in a simulated semibatch pH‐shift reactive crystallization process to handle major crystallization kinetic uncertainties of L‐glutamic acid, wherein smoother and faster convergences than the conventional B2B control were observed when process dynamics were shifted among three scenarios of kinetic uncertainties. © 2017 American Institute of Chemical Engineers AIChE J, 2017

Approximate Bayesian MLP regularization for regression in the presence of noise

We present a novel regularization method for a multilayer perceptron (MLP) that learns a regression function in the presence of noise regardless of how smooth the function is. Unlike general MLP regularization methods assuming that a regression function is smooth, the proposed regularization method is also valid when a regression function has discontinuities (non-smoothness). Since a true regression function to be learned is unknown, we examine a training set with our Bayesian approach that identifies non-smooth data, analyzing discontinuities in a regression function. The use of a Bayesian probability distribution identifies the non-smooth data. These identified data is used in a proposed objective function to fit an MLP response to the desired regression function regardless of its smoothness and noise. Experimental simulations show that the MLP with our presented training method yields more accurate fits to non-smooth functions than other MLP training methods. Further, we show that the suggested training methodology can be incorporated with deep learning models.

Quantifying truncation errors in effective field theory

Bayesian procedures designed to quantify truncation errors in perturbative calculations of quantum chromodynamics observables are adapted to expansions in effective field theory (EFT). In the Bayesian approach, such truncation errors are derived from degree-of-belief (DOB) intervals for EFT predictions. Computation of these intervals requires specification of prior probability distributions ("priors") for the expansion coefficients. By encoding expectations about the naturalness of these coefficients, this framework provides a statistical interpretation of the standard EFT procedure where truncation errors are estimated using the order-by-order convergence of the expansion. It also permits exploration of the ways in which such error bars are, and are not, sensitive to assumptions about EFT-coefficient naturalness. We first demonstrate the calculation of Bayesian probability distributions for the EFT truncation error in some representative examples, and then focus on the application of chiral EFT to neutron-proton scattering. Epelbaum, Krebs, and Mei{\ss}ner recently articulated explicit rules for estimating truncation errors in such EFT calculations of few-nucleon-system properties. We find that their basic procedure emerges generically from one class of naturalness priors considered, and that all such priors result in consistent quantitative predictions for 68% DOB intervals. We then explore several methods by which the convergence properties of the EFT for a set of observables may be used to check the statistical consistency of the EFT expansion parameter.

Applications of the robust Bayesian regression analysis

The paper introduces a mathematical algorithm of curve-to-data best fit with robust Bayesian approach. Using Gaussian distribution as a background, one can find an outlier-resistant dedicated Bayesian probability distribution. With the use of a computational iterative method it can then be applied to find the best fit to experimental data points, including vertical and horizontal asymmetric uncertainties. The presented method was tested for constant fitting (so called Bayesian constant), linear regression as well as parabolic fitting for a number of outliers. Practical applications are also presented, e.g., the curves' fitting in radiation cytogenetic biodosimetry, in the financial quotations' trend detection as well as in the ecological study of human cancer mortality. The paper focuses also on other topics, including uncertainties of fitting parameters and the joint Gaussian and Bayesian probability distribution. This paper is an easy and useful tutorial (or technical note) dedicated for robust data analysis in both mathematical and non-mathematical interdisciplinary sciences.