Joint Posterior Distribution Research Articles

A Bayesian Approach for Modeling and Forecasting Solar Photovoltaic Power Generation.

In this paper, we propose a Bayesian approach to estimate the curve of a function f(·) that models the solar power generated at k moments per day for n days and to forecast the curve for the (n+1)th day by using the history of recorded values. We assume that f(·) is an unknown function and adopt a Bayesian model with a Gaussian-process prior on the vector of values f(t)=f(1),…, f(k). An advantage of this approach is that we may estimate the curves of f(·) and fn+1(·) as "smooth functions" obtained by interpolating between the points generated from a k-variate normal distribution with appropriate mean vector and covariance matrix. Since the joint posterior distribution for the parameters of interest does not have a known mathematical form, we describe how to implement a Gibbs sampling algorithm to obtain estimates for the parameters. The good performance of the proposed approach is illustrated using two simulation studies and an application to a real dataset. As performance measures, we calculate the absolute percentage error, the mean absolute percentage error (MAPE), and the root-mean-square error (RMSE). In all simulated cases and in the application to real-world data, the MAPE and RMSE values were all near 0, indicating the very good performance of the proposed approach.

Analyzing Burr-X competing risk model using adaptive progressive Type-II censored binomial removal data with application to electrodes and electronics

The aim of this paper is to investigate the Burr-X competing risks model in the context of adaptive progressively Type-II censored samples. In this scenario, the removal pattern is assumed to be a random variable that follows the binomial distribution, which is a more realistic assumption compared to assuming a fixed removal pattern. In this study, we explore both classical and Bayesian estimation approaches to estimate the parameters of the Burr-X competing risks model, as well as the reliability parameter and the parameter of the binomial distribution. The interval ranges of different parameters are determined by utilizing the asymptotic normality of the maximum likelihood estimators. Furthermore, the Bayes credible intervals are calculated by sampling from the joint posterior distribution using the Markov Chain Monte Carlo procedure. To assess the efficiency of the acquired estimators, a comprehensive simulation study that considered various types of experimental designs is conducted. Finally, two applications are considered by analyzing data sets of electrodes and electronics.

Gaussian-inverse gamma mixture distribution–based extended Kalman observer for dynamic positioning control system of offshore platform

Under different working conditions and control objectives, fuzzy model predictive control is used to optimize the control effect by adjusting the weight to achieve stable and accurate control for the dynamic positioning control of offshore platforms. Also, aiming at the situation where the measurement noise of dynamic positioning control may be time-varying and non-Gaussian and the statistics of the noise are not exact, this paper proposes a variational Bayesian extended Kalman filter observer. In order to model non-Gaussian noise more reasonably, the joint posterior distribution of non-Gaussian noise with state and time-varying covariance is described as the product of Gaussian and independent inverse gamma distributions. Then, the variational Bayesian (VB) method was used to simultaneously estimate the system state, the intermediate variable values of the new distribution, and the noise parameters through fixed-point iteration. The fuzzy algorithm is applied to the model predictive control, and the weight parameters of the model predictive control are adaptively changed to better adapt to the changing working conditions in the control process. The simulation results show that the variational Bayesian extended Kalman filter observer has better performance than the existing algorithms when dealing with the time-varying non-Gaussian observation noise, whose statistics are not exact. The variational Bayesian extended Kalman filter observer can significantly improve the accuracy of the control when it is used in fuzzy model predictive control.

Bayesian modelling of multiple plasma diagnostics at Wendelstein 7-X

Abstract Inference of electron density and temperature has been performed using multiple, diverse sets of plasma diagnostic data at Wendelstein 7-X. Predictive models for the interferometer, Thomson scattering and helium beam emission spectroscopy (He-BES) systems have been developed within the Minerva framework and integrated into a unified model. Electron density and temperature profiles are modelled using Gaussian processes. Calibration factors for the Thomson scattering system and predictive uncertainties are considered as additional unknown parameters. The joint posterior probability distribution for the electron density and temperature profiles as well as Gaussian process hyperparameters and model parameters is explored through a Markov chain Monte Carlo algorithm. Samples from this distribution are numerically marginalised over the hyperparameters and model parameters to yield marginal posterior distributions for the electron density and temperature profiles. The profile inferences incorporate various data combinations from the interferometer and Thomson scattering as well as constraints at the limiter/divertor positions through virtual observations or edge data from He-BES. Additionally, the integration of x-ray imaging crystal spectrometer data into the model for ion temperature profiles is presented. All profiles presented in this study are inferred with optimally selected hyperparameters and model parameters by exploring the joint posterior distribution, inherently applying Bayesian Occam’s razor.

Analysis of Xgamma distribution using adaptive Type-I progressively censored competing risks data with applications

This paper considers the competing risks model with two causes of death to analyze time-to-event data for a group of male mice exposed to three hundred roentgen radiation for 5–6 weeks. The analysis is based on the assumption that the parent distribution is the Xgamma distribution and the data are gathered using an adaptive Type-I progressively censored sample. Two estimation approaches are considered to complete the analysis: maximum likelihood and Bayesian methods. Besides acquiring the estimations of the model parameters, the estimations of the reliability and failure rate are also discussed. Both point and interval estimates using both estimation approaches are studied. In Bayesian estimations, the squared error loss function is used and the Markov Chain Monte Carlo technique is proposed to get samples from the joint posterior distribution. The various methods are compared using simulation studies to compare their performance. The mentioned radiation data set is investigated and the analysis showed the suitability of the competing risks model with Xgamma distribution to analyze the data and to estimate the reliability metrics.

BNP-Track: a framework for superresolved tracking

Superresolution tools, such as PALM and STORM, provide nanoscale localization accuracy by relying on rare photophysical events, limiting these methods to static samples. By contrast, here, we extend superresolution to dynamics without relying on photodynamics by simultaneously determining emitter numbers and their tracks (localization and linking) with the same localization accuracy per frame as widefield superresolution on immobilized emitters under similar imaging conditions (≈50 nm). We demonstrate our Bayesian nonparametric track (BNP-Track) framework on both in cellulo and synthetic data. BNP-Track develops a joint (posterior) distribution that learns and quantifies uncertainty over emitter numbers and their associated tracks propagated from shot noise, camera artifacts, pixelation, background and out-of-focus motion. In doing so, we integrate spatiotemporal information into our distribution, which is otherwise compromised by modularly determining emitter numbers and localizing and linking emitter positions across frames. For this reason, BNP-Track remains accurate in crowding regimens beyond those accessible to other single-particle tracking tools.

Robust Kalman filters based on the sub-Gaussian [formula omitted]-stable distribution

Motivated by filtering tasks under a linear system with non-Gaussian heavy-tailed noise, various robust Kalman filters (RKFs) based on different heavy-tailed distributions have been proposed. Although the sub-Gaussian α-stable (SGαS) distribution captures heavy tails well and is applicable in various scenarios, its potential has not yet been explored for RKFs. The main hindrance is that there is no closed-form expression of its mixing density. This paper proposes a novel RKF framework, RKF-SGαS, where the process noise is assumed to be Gaussian and the heavy-tailed measurement noise is modelled by the SGαS distribution. The corresponding joint posterior distribution of the state vector and auxiliary random variables is approximated by the Variational Bayesian approach. Also, four different minimum mean square error (MMSE) estimators of the scale function are presented. The first two methods are based on the Importance Sampling (IS) and Gauss–Laguerre quadrature (GLQ), respectively. In contrast, the last two estimators combine a proposed Gamma series (GS) based method with the IS and GLQ estimators and hence are called GSIS and GSGL. Besides, the RKF-SGαS is compared with the state-of-the-art RKFs under three kinds of heavy-tailed measurement noises, and the simulation results demonstrate its estimation accuracy and efficiency.

Emulator-Based Bayesian Calibration of the CISNET Colorectal Cancer Models.

To calibrate Cancer Intervention and Surveillance Modeling Network (CISNET)'s SimCRC, MISCAN-Colon, and CRC-SPIN simulation models of the natural history colorectal cancer (CRC) with an emulator-based Bayesian algorithm and internally validate the model-predicted outcomes to calibration targets. We used Latin hypercube sampling to sample up to 50,000 parameter sets for each CISNET-CRC model and generated the corresponding outputs. We trained multilayer perceptron artificial neural networks (ANNs) as emulators using the input and output samples for each CISNET-CRC model. We selected ANN structures with corresponding hyperparameters (i.e., number of hidden layers, nodes, activation functions, epochs, and optimizer) that minimize the predicted mean square error on the validation sample. We implemented the ANN emulators in a probabilistic programming language and calibrated the input parameters with Hamiltonian Monte Carlo-based algorithms to obtain the joint posterior distributions of the CISNET-CRC models' parameters. We internally validated each calibrated emulator by comparing the model-predicted posterior outputs against the calibration targets. The optimal ANN for SimCRC had 4 hidden layers and 360 hidden nodes, MISCAN-Colon had 4 hidden layers and 114 hidden nodes, and CRC-SPIN had 1 hidden layer and 140 hidden nodes. The total time for training and calibrating the emulators was 7.3, 4.0, and 0.66 h for SimCRC, MISCAN-Colon, and CRC-SPIN, respectively. The mean of the model-predicted outputs fell within the 95% confidence intervals of the calibration targets in 98 of 110 for SimCRC, 65 of 93 for MISCAN, and 31 of 41 targets for CRC-SPIN. Using ANN emulators is a practical solution to reduce the computational burden and complexity for Bayesian calibration of individual-level simulation models used for policy analysis, such as the CISNET CRC models. In this work, we present a step-by-step guide to constructing emulators for calibrating 3 realistic CRC individual-level models using a Bayesian approach. We use artificial neural networks (ANNs) to build emulators that surrogate complex individual-based models to reduce the computational burden in the Bayesian calibration process.ANNs showed good performance in emulating the CISNET-CRC microsimulation models, despite having many input parameters and outputs.Using ANN emulators is a practical solution to reduce the computational burden and complexity for Bayesian calibration of individual-level simulation models used for policy analysis.This work aims to support health decision scientists who want to quantify the uncertainty of calibrated parameters of computationally intensive simulation models under a Bayesian framework.

Improving crop yield estimation by unified model parameters and state variable with Bayesian inference

Data assimilation techniques integrating crop growth models and remote sensing technologies offer a feasible approach for large-scale crop yield estimation. Previous research has primarily focused on either recalibrate the uncertain model parameters or updating model state variables independently using remotely sensed observations. In this study, we developed a two-step inference algorithm that couples the parameter inference and the state update, to solve the joint posterior distribution of uncertain parameters and state variables given the remote sensing observations. An Observing Simulation System (OSS) experiment was first performed based on the WOFOST crop model to validate the effectiveness of the parameter inference method. The results indicate that, the parameter inference method successfully improved the estimation of different types of model parameters and enhanced yield estimation. Furthermore, leaf area index (LAI) retrieved from Sentinel-2 was assimilated into the WOFOST model to simulate winter wheat yield at the plot scale in the northeastern part of Henan Province. The results demonstrated that the proposed two-step inference algorithm can more effectively correct model simulation biases and improve winter wheat yield estimation accuracy (R²=0.58, MAPE=12.75 %, and RMSE=1112 kg·ha⁻¹), outperforming the standard EnKF algorithm (R²=0.51, MAPE=14.62 %, RMSE=1328 kg·ha⁻¹). Overall, attributed to its unified approach to estimating both model parameters and state variables, the proposed two-step inference algorithm shows promising application prospects for data assimilation-based crop yield estimation.

Posterior Representations for Bayesian Context Trees: Sampling, Estimation and Convergence

We revisit the Bayesian Context Trees (BCT) modelling framework for discrete time series, which was recently found to be very effective in numerous tasks including model selection, estimation and prediction. A novel representation of the induced posterior distribution on model space is derived in terms of a simple branching process, and several consequences of this are explored in theory and in practice. First, it is shown that the branching process representation leads to a simple variable-dimensional Monte Carlo sampler for the joint posterior distribution on models and parameters, which can efficiently produce independent samples. This sampler is found to be more efficient than earlier MCMC samplers for the same tasks. Then, the branching process representation is used to establish the asymptotic consistency of the BCT posterior, including the derivation of an almost-sure convergence rate. Finally, an extensive study is carried out on the performance of the induced Bayesian entropy estimator. Its utility is illustrated through both simulation experiments and real-world applications, where it is found to outperform several state-of-the-art methods.

Enabling identification of component processes in perceptual learning with nonparametric hierarchical Bayesian modeling.

Perceptual learning is a multifaceted process, encompassing general learning, between-session forgetting or consolidation, and within-session fast relearning and deterioration. The learning curve constructed from threshold estimates in blocks or sessions, based on tens or hundreds of trials, may obscure component processes; high temporal resolution is necessary. We developed two nonparametric inference procedures: a Bayesian inference procedure (BIP) to estimate the posterior distribution of contrast threshold in each learning block for each learner independently and a hierarchical Bayesian model (HBM) that computes the joint posterior distribution of contrast threshold across all learning blocks at the population, subject, and test levels via the covariance of contrast thresholds across blocks. We applied the procedures to the data from two studies that investigated the interaction between feedback and training accuracy in Gabor orientation identification over 1920 trials across six sessions and estimated learning curve with block sizes L = 10, 20, 40, 80, 160, and 320 trials. The HBM generated significantly better fits to the data, smaller standard deviations, and more precise estimates, compared to the BIP across all block sizes. In addition, the HBM generated unbiased estimates, whereas the BIP only generated unbiased estimates with large block sizes but exhibited increased bias with small block sizes. With L = 10, 20, and 40, we were able to consistently identify general learning, between-session forgetting, and rapid relearning and adaptation within sessions. The nonparametric HBM provides a general framework for fine-grained assessment of the learning curve and enables identification of component processes in perceptual learning.

Time-Varying Zero-Adjusted Poisson Distribution for Modeling Count Time Series

Many studies have used extensions of ARMA models for the analysis of non-Gaussian time series. One of them is the Generalized Autoregressive Moving Average, GARMA, enabling the modeling of count time series with distributions such as Poisson. The GARMA class is being expanded to accommodate other distributions, aiming to capture the typical characteristics of count data, including under or overdispersion and excess zeros. This study aims to propose an approach based on the GARMA class in order to analyze count time series with excess zeros, assuming a time-varying zero-adjusted Poisson distribution. This approach allows for capturing serial correlation, forecasting the future values, and estimating the future probability of zeros. For inference, a Bayesian analysis was adopted using the Hamiltonian Monte Carlo (HMC) algorithm for sampling from the joint posterior distribution. We conducted a simulation study and presented an application to influenza mortality reported in Brazil. Our findings demonstrated the usefulness of the model in estimating the probability of non-occurrence and the number of counts in future periods.

Bayesian analysis on a natural conjugate prior for the nonhomogeneous Poisson process with a power-law intensity under time-truncated sampling

The core content in this paper discusses the Bayesian approach, which essentially estimates and predicts the reliability of a repairable system during the actual testing process. As specified in previous literature, the nonhomogeneous Poisson process (NHPP) with a power-law intensity function has often been employed as a model for describing the failure process of repairable systems since it provides a reasonable way of presenting the rate of change for the reliability of a system as a function of time. This research proposed the same time-truncated sampling assumption to mainly ensure that the joint posterior distribution of the power-law failure process has a considerably closed-form representation. This proposed approach facilitates the analysis of the actual performance, with which analysts can provide the prior information about the authentic means and variations of the shape and scale parameters of the power-law intensity. The corresponding posterior means and variations can then be significantly and easily obtained without any complex integral arithmetic. Using Monte Carlo simulation for comparison, the Bayesian estimation with the proposed prior outperforms the maximum likelihood estimation, particularly when the prior aging of the repairable is underestimated.

Genetic algorithm with a Bayesian approach for multiple change-point detection in time series of counting exceedances for specific thresholds

Although the applications of Non-Homogeneous Poisson Processes (NHPP) to model and study the threshold overshoots of interest in different time series of measurements have proven to provide good results, they needed to be complemented with an efficient and automatic diagnostic technique to establish the location of the change-points, which, when taken into account, make the estimated model fit poorly in regards of the information contained in the real one. Because of this, a new method is proposed to solve the segmentation uncertainty of the time series of measurements, where the generating distribution of exceedances of a specific threshold is the focus of investigation. One of the great contributions of the present algorithm is that all the days that trespassed are candidates to be a change-point, so all the possible configurations of overflow days under the heuristics of a genetic algorithm are the possible chromosomes, which will unite to produce new solutions. Also, such methods will be guarantee to non-local and the best possible one solution, reducing wasted machine time evaluating the least likely chromosomes to be a feasible solution. The analytical evaluation technique will be by means of the Minimum Description Length (MDL) as the objective function, which is the joint posterior distribution function of the parameters of the NHPP of each regime and the change-points that determines them and which account as well for the influence of the presence of said times. Thus, one of the practical implications of the present work comes in terms of overcoming the need of modeling the time series of measurements, where the distributions of exceedances of certain thresholds, or where the counting of certain events involving abrupt changes, is the main focus with applications in phenomena such as climate change, information security and epidemiology, to name a few.

Bayesian Estimation Using Product of Spacing for Modified Kies Exponential Progressively Censored Data

In life testing and reliability studies, most researchers have used the maximum likelihood estimation method to estimate unknown parameters, even though it has been proven that the maximum product of spacing method has properties as good as the maximum likelihood estimation method and sometimes even better. In this study, we aim to estimate the unknown parameters of the modified Kies exponential distribution along with the reliability and hazard rate functions under progressive type-II censoring scheme. The maximum likelihood and maximum product of spacing methods are considered in order to find the point estimates and approximate confidence intervals of the various parameters. Moreover, Bayesian estimations based on the likelihood function and the product of the spacing function of the unknown parameters are obtained using the squared error loss function with independent gamma priors. It is observed that the joint posterior distributions have complicated forms. Because of this, Lindley’s approximation and the Markov chain Monte Carlo technique are used to obtain the Bayesian estimates and highest posterior credible intervals. Monte Carlo simulations are performed in order to evaluate the performance of the proposed estimation methods. Two real datasets are studied to demonstrate the efficacy of the offered methodologies and highlight how simple and applicable it might be to apply them in practical fields.

Inferences of inverted gompertz parameters from an adaptive type-ii progressive hybrid censoring

A new two-parameter inverted Gompertz distribution with an upside-down bathtub-shaped failure rate is investigated in the presence of incomplete (censored) data. Reliability experimenters favor employing censoring strategies to collect data in order to strike a compromise between the time needed to complete the test, the required sample size, and the cost. Recently, an adaptive Type-II hybrid progressively censoring strategy has been proposed to improve the effectiveness of statistical inference. Therefore, using this strategy, this study explores the classical and Bayesian estimates of the inverted Gompertz distribution. The distribution parameters, reliability, and hazard functions are estimated using maximum likelihood and Bayesian estimation methods. On the presumption that the gamma priors are independent, symmetric and asymmetric loss functions are used to create the Bayesian estimation. The Markov chain Monte Carlo approach is used to collect samples from the entire conditional distributions and then acquire the Bayes estimates since the joint posterior distribution has a difficult shape. The highest posterior density and asymptotic confidence intervals are also obtained. Through simulated research, the efficacy of the various recommended strategies is contrasted. The best progressive censoring schemes are also shown, and various optimality criteria are investigated. Two real data sets, representing the lifetimes of mechanical components and the Boeing 720 jet airplane, are also looked at to show how the recommended point and interval estimators may be used. When the experimenter’s main concern is the number of failures, the results of the simulation research and data analysis showed that the proposed scheme is adaptive and highly useful in concluding the experiment.

Penalty parameter selection and asymmetry corrections to Laplace approximations in Bayesian P-splines models

Laplace P-splines (LPS) combine the P-splines smoother and the Laplace approximation in a unifying framework for fast and flexible inference under the Bayesian paradigm. The Gaussian Markov random field prior assumed for penalized parameters and the Bernstein-von Mises theorem typically ensure a razor-sharp accuracy of the Laplace approximation to the posterior distribution of these quantities. This accuracy can be seriously compromised for some unpenalized parameters, especially when the information synthesized by the prior and the likelihood is sparse. Therefore, we propose a refined version of the LPS methodology by splitting the parameter space in two subsets. The first set involves parameters for which the joint posterior distribution is approached from a non-Gaussian perspective with an approximation scheme tailored to capture asymmetric patterns, while the posterior distribution for the penalized parameters in the complementary set undergoes the LPS treatment with Laplace approximations. As such, the dichotomization of the parameter space provides the necessary structure for a separate treatment of model parameters, yielding improved estimation accuracy as compared to a setting where posterior quantities are uniformly handled with Laplace. In addition, the proposed enriched version of LPS remains entirely sampling-free, so that it operates at a computing speed that is far from reach to any existing Markov chain Monte Carlo approach. The methodology is illustrated on the additive proportional odds model with an application on ordinal survey data.

Statistical inference of reliability distributions based on progressively hybrid censoring data

This paper studies the statistical inferences of the unknown parameters of some reliability models using progressively hybrid censoring of type-I and type-II. We apply the maximum likelihood and Bayesian approaches to perform the statistical inferences of the parameters when the data follow Weibull, power Lindley, and Sarhan–Tadj–Hamilton distributions. Bayes method is implemented under the squared error loss function when the parameters are independent and follow gamma prior distributions with known hyperparameters. The maximum likelihood estimates cannot be obtained in an analytic form. Therefore we use the R function “optim” to find those estimates. Also, the joint posterior distribution of the unknown parameters cannot be obtained in closed form. Therefore, we will use the Markov Chain Monte Carlo method to approximate the Bayesian analysis. We constructed the highest posterior density intervals of the model parameters. We analyzed three real data sets for illustration and comparison purposes. We found out that Sarhan–Tadj–Hamilton distribution fits those three real data sets better than Weibull and Power Lindley distribution using both complete and progressively hybrid censored samples. We used expected experimentation time to discuss the optimal test plans. Simulation studies are performed to investigate the proposed methods. Based on the simulation results, we could conclude that Bayesian method does not provide better estimations than the maximum likelihood method.

Inflationary interpretation of the stochastic gravitational wave background signal detected by pulsar timing array experiments

Various pulsar timing array (PTA) experiments (NANOGrav, EPTA, PPTA, CPTA, including data from InPTA) very recently reported evidence for excess red common-spectrum signals in their latest datasets, with inter-pulsar correlations following the Hellings-Downs pattern, pointing to a stochastic gravitational wave background (SGWB) origin. Focusing for concreteness on the NANOGrav signal (given that all signals are in good agreement between each other), I inspect whether it supports an inflationary SGWB explanation, finding that such an interpretation calls for an extremely blue tensor spectrum, with spectral index nT≃1.8±0.3, while Big Bang Nucleosynthesis limits require a very low reheating scale, Trh≲10GeV. While not impossible, an inflationary origin for the PTA signal is barely tenable: within well-motivated inflationary models it is hard to achieve such a blue tilt, whereas models who do tend to predict sizeable non-Gaussianities, excluded by observations. Intriguingly, ekpyrotic models naturally predict a SGWB with spectral index nT=2, although with an amplitude too suppressed to be able to explain the signal detected by PTA experiments. Finally, I provide explicit expressions for a bivariate Gaussian approximation to the joint posterior distribution for the intrinsic-noise amplitude and spectral index of the NANOGrav signal, which can facilitate extending similar analyses to different theoretical signals.

Collective endpoint of visual acuity and contrast sensitivity function from hierarchical Bayesian joint modeling.

Clinical trials typically analyze multiple endpoints for signals of efficacy. To improve signal detection for treatment effects using the high-dimensional data collected in trials, we developed a hierarchical Bayesian joint model (HBJM) to compute a five-dimensional collective endpoint (CE5D) of contrast sensitivity function (CSF) and visual acuity (VA). The HBJM analyzes row-by-row CSF and VA data across multiple conditions, and describes visual functions across a hierarchy of population, individuals, and tests. It generates joint posterior distributions of CE5D that combines CSF (peak gain, peak frequency, and bandwidth) and VA (threshold and range) parameters. The HBJM was applied to an existing dataset of 14 eyes, each tested with the quantitative VA and quantitative CSF procedures in four Bangerter foil conditions. The HBJM recovered strong correlations among CE5D components at all levels. With 15 qVA and 25 qCSF rows, it reduced the variance of the estimated components by 72% on average. Combining signals from VA and CSF and reducing noises, CE5D exhibited significantly higher sensitivity and accuracy in discriminating performance differences between foil conditions at both the group and test levels than the original tests. The HBJM extracts valuable information about covariance of CSF and VA parameters, improves precision of the estimated parameters, and increases the statistical power in detecting vision changes. By combining signals and reducing noise from multiple tests for detecting vision changes, the HBJM framework exhibits potential to increase statistical power for combining multi-modality data in ophthalmic trials.