Conditional Posterior Distributions Research Articles

Bayesian analysis of joint quantile regression for multi-response longitudinal data with application to primary biliary cirrhosis sequential cohort study.

This article proposes a Bayesian approach for jointly estimating marginal conditional quantiles of multi-response longitudinal data with multivariate mixed effects model. The multivariate asymmetric Laplace distribution is employed to construct the working likelihood of the considered model. Penalization priors on regression parameters are incorporated into the working likelihood to conduct Bayesian high-dimensional inference. Markov chain Monte Carlo algorithm is used to obtain the fully conditional posterior distributions of all parameters and latent variables. Monte Carlo simulations are conducted to evaluate the sample performance of the proposed joint quantile regression approach. Finally, we analyze a longitudinal medical dataset of the primary biliary cirrhosis sequential cohort study to illustrate the real application of the proposed modeling method.

Virtual sensing based on Hierarchical Bayesian Modeling framework using a Laplace-based Gibbs sampler

This paper presents a Bayesian-based approach to realize virtual strain sensing and quantify the uncertainty in these estimates using sparse output-only measurements. This method relies on output-only measurements to identify the modal coordinates of the structure and realizes the virtual sensing according to the modal superposition method. A robust Hierarchical Bayesian modeling (HBM) framework is developed to fully account for the uncertainties arising from modeling errors and measurement noise. Moreover, the HBM framework has the ability to quantify the uncertainty of parameters in the statistical model by introducing an additional layer of the prior distribution. The Gibbs sampler is adopted to obtain the marginal posterior distributions of the hyper-parameters, with the implementation of Gibbs methods relying on the derived analytical expressions of the full conditional posterior distributions. The Laplace asymptotic approximation simplified the form of the full conditional posterior distributions, thus improving the computational efficiency. Given the hyper-parameters’ marginal posterior distribution, the dynamic parameters’ marginal posterior distribution can be calculated according to the total probability theorem. Finally, one can propagate the uncertainty to make predictions of quantities of interest (QoI). Three case studies, including two numerical examples and a laboratory model test, are carried out to verify the accuracy and efficiency of the proposed algorithm. The results show that the estimates of the parameters are close to the true values. Besides, the prediction results of QoI also show high accuracy, verifying the effectiveness of the proposed HBM method in virtual sensing.

Bayesian Subset Selection of Seasonal Autoregressive Models

Seasonal autoregressive (SAR) models have many applications in different fields, such as economics and finance. It is well known in the literature that these models are nonlinear in their coefficients and that their Bayesian analysis is complicated. Accordingly, choosing the best subset of these models is a challenging task. Therefore, in this paper, we tackled this problem by introducing a Bayesian method for selecting the most promising subset of the SAR models. In particular, we introduced latent variables for the SAR model lags, assumed model errors to be normally distributed, and adopted and modified the stochastic search variable selection (SSVS) procedure for the SAR models. Thus, we derived full conditional posterior distributions of the SAR model parameters in the closed form, and we then introduced the Gibbs sampler, along with SSVS, to present an efficient algorithm for the Bayesian subset selection of the SAR models. In this work, we employed mixture–normal, inverse gamma, and Bernoulli priors for the SAR model coefficients, variance, and latent variables, respectively. Moreover, we introduced a simulation study and a real-world application to evaluate the accuracy of the proposed algorithm.

Full Bayesian analysis of double seasonal autoregressive models with real applications

We present a full Bayesian analysis of multiplicative double seasonal autoregressive (DSAR) models in a unified way, considering identification (best subset selection), estimation, and prediction problems. We assume that the DSAR model errors are normally distributed and introduce latent variables for the model lags, and then we embed the DSAR model in a hierarchical Bayes normal mixture structure. By employing the Bernoulli prior for each latent variable and the mixture normal and inverse gamma priors for the DSAR model coefficients and variance, respectively, we derive the full conditional posterior and predictive distributions in closed form. Using these derived conditional posterior and predictive distributions, we present the full Bayesian analysis of DSAR models by proposing the Gibbs sampling algorithm to approximate the posterior and predictive distributions and provide multi-step-ahead predictions. We evaluate the efficiency of the proposed full Bayesian analysis of DSAR models using an extensive simulation study, and we then apply our work to several real-world hourly electricity load time series datasets in 16 European countries.

BAYESIAN ANALYSIS OF TOBIT QUANTILE REGRESSION WITH ADAPTIVE LASSO PENALTY IN HOUSEHOLD EXPENDITURE FOR CIGARETTE CONSUMPTION

Tobit Quantile Regression with Adaptive Lasso Penalty is a quantile regression model on censored data that adds Lasso's adaptive penalty to its parameter estimation. The estimation of the regression parameters is solved by Bayesian analysis. Parameters are assumed to follow a certain distribution called the prior distribution. Using the sample information along with the prior distribution, the conditional posterior distribution is searched using the Box-Tiao rule. Computational solutions are solved by the MCMC Gibbs Sampling algorithm. Gibbs Sampling can generate samples based on the conditional posterior distribution of each parameter in order to obtain a posterior joint distribution. Tobit Quantile Regression with Adaptive Lasso Penalty was applied to data on Household Expenditure for Cigarette Consumption in 2011. As a comparison for data analysis, Tobit Quantile Regression was used. The results of data analysis show that the Tobit Quantile Regression model with Adaptive Lasso Penalty is better than the Tobit Quantile Regression.

Functional principal component models for sparse and irregularly spaced data by Bayesian inference

The area of functional principal component analysis (FPCA) has seen relatively few contributions from the Bayesian inference. A Bayesian method in FPCA is developed under the cases of continuous and binary observations for sparse and irregularly spaced data. In the proposed Markov chain Monte Carlo (MCMC) method, Gibbs sampler approach is adopted to update the different variables based on their conditional posterior distributions. In FPCA, a set of eigenfunctions is suggested under Stiefel manifold, and samples are drawn from a Langevin–Bingham matrix variate distribution. Penalized splines are used to model mean trajectory and eigenfunction trajectories in generalized functional mixed models; and the proposed model is casted into a mixed-effects model framework for Bayesian inference. To determine the number of principal components, reversible jump Markov chain Monte Carlo (RJ-MCMC) algorithm is implemented. Four different simulation settings are conducted to demonstrate competitive performance against non-Bayesian approaches in FPCA. Finally, the proposed method is illustrated to the analysis of body mass index (BMI) data by gender and ethnicity.

Study of Bayesian variable selection method on mixed linear regression models.

Variable selection has always been an important issue in statistics. When a linear regression model is used to fit data, selecting appropriate explanatory variables that strongly impact the response variables has a significant effect on the model prediction accuracy and interpretation effect. redThis study introduces the Bayesian adaptive group Lasso method to solve the variable selection problem under a mixed linear regression model with a hidden state and explanatory variables with a grouping structure. First, the definition of the implicit state mixed linear regression model is presented. Thereafter, the Bayesian adaptive group Lasso method is used to determine the penalty function and parameters, after which each parameter's specific form of the fully conditional posterior distribution is calculated. Moreover, the Gibbs algorithm design is outlined. Simulation experiments are conducted to compare the variable selection and parameter estimation effects in different states. Finally, a dataset of Alzheimer's Disease is used for application analysis. The results demonstrate that the proposed method can identify the observation from different hidden states, but the results of the variable selection in different states are obviously different.

Bayesian analysis of seasonally cointegrated VAR models

The aim is to develop a Bayesian seasonally cointegrated model for quarterly data. Relevant prior structure is proposed, and the set of full conditional posterior distributions is derived, enabling us to employ the Gibbs sampler for posterior inference. The identification of cointegrating spaces is obtained by orthonormality restrictions imposed on vectors spanning them. The point estimation of the cointegrating spaces is also discussed. In the presence of a seasonal pattern with one cycle per year, the cointegrating vectors belong to the complex space, which should be taken into account in the identification scheme. The methodology is illustrated by the analysis of money and prices in the Polish economy.

Modularized Bayesian analyses and cutting feedback in likelihood-free inference

There has been much recent interest in modifying Bayesian inference for misspecified models so that it is useful for specific purposes. One popular modified Bayesian inference method is “cutting feedback” which can be used when the model consists of a number of coupled modules, with only some of the modules being misspecified. Cutting feedback methods represent the full posterior distribution in terms of conditional and sequential components, and then modify some terms in such a representation based on the modular structure for specification or computation of a modified posterior distribution. The main goal of this is to avoid contamination of inferences for parameters of interest by misspecified modules. Computation for cut posterior distributions is challenging, and here we consider cutting feedback for likelihood-free inference based on Gaussian mixture approximations to the joint distribution of parameters and data summary statistics. We exploit the fact that marginal and conditional distributions of a Gaussian mixture are Gaussian mixtures to give explicit approximations to marginal or conditional posterior distributions so that we can easily approximate cut posterior analyses. The mixture approach allows repeated approximation of posterior distributions for different data based on a single mixture fit. This is important for model checks which aid in the decision of whether to “cut”. A semi-modular approach to likelihood-free inference where feedback is partially cut is also developed. The benefits of the method are illustrated on two challenging examples, a collective cell spreading model and a continuous time model for asset returns with jumps.

Off-Grid Channel Estimation With Sparse Bayesian Learning for OTFS Systems

This paper proposes an off-grid channel estimation scheme for orthogonal time-frequency space (OTFS) systems adopting the sparse Bayesian learning (SBL) framework. To avoid channel spreading caused by the fractional delay and Doppler shifts and to fully exploit the channel sparsity in the delay-Doppler (DD) domain, we estimate the original DD domain channel response rather than the effective DD domain channel response as commonly adopted in the literature. OTFS channel estimation is firstly formulated as a one-dimensional (1D) off-grid sparse signal recovery (SSR) problem based on a virtual sampling grid defined in the DD space, where the on-grid and off-grid components of the delay and Doppler shifts are separated for estimation. In particular, the on-grid components of the delay and Doppler shifts are jointly determined by the entry indices with significant values in the recovered sparse vector. Then, the corresponding off-grid components are modeled as hyper-parameters in the proposed SBL framework, which can be estimated via the expectation-maximization method. To strike a balance between channel estimation performance and computational complexity, we further propose a two-dimensional (2D) off-grid SSR problem via decoupling the delay and Doppler shift estimations. In our developed 1D and 2D off-grid SBL-based channel estimation algorithms, the hyper-parameters are updated alternatively for computing the conditional posterior distribution of channels, which can be exploited to reconstruct the effective DD domain channel. Compared with the 1D method, the proposed 2D method enjoys a much lower computational complexity while only suffers a slight performance degradation. Simulation results verify the superior performance of the proposed channel estimation schemes over state-of-the-art schemes.

Statistical Inference of Sound Speed and Attenuation Dispersion of a Fine-Grained Marine Sediment

Acoustic recordings of signals in the 1.5–4.0-kHz band were analyzed for information about the sound speed and attenuation frequency dispersion of a fine-grained sediment found in the New England Mudpatch. Analysis of piston cores established prior bounds for a geophysical parameterization of a seabed model that predicts Kramers–Kronig dispersion relations. Sediment layers are described by the Buckingham viscous grain shearing (VGS) model that accounts for the effects of overburden pressure of compressional and shear speeds and attenuations. A statistical inverse problem was solved by using multiple samples of received levels recorded on two vertical line arrays as a function time and hydrophone depth for six frequencies in the 1.5–4.0-kHz band. A statistical inference model that assumed both model parameters and data samples are random variables quantified information content from marginalization of a conditional posterior probability distribution for the geophysical parameters that characterize the mud layer. From the inferred geophysical parameter point estimates the sediment sound speed and attenuation frequency dispersion are predicted and compared to previously reported direct measurements. Also, the predicted sound-speed gradient in the mud sediment from the VGS model is compared to a previous inference that utilized explosive sources.

Bayesian analysis of multiple break-points threshold ARMA model with exogenous inputs

In this paper, we introduce a Bayesian statistical inference approach for multiple break-points threshold autoregressive moving average model with exogenous inputs (MB-TARMAX) which change in state space and time domain. Based on the appropriate prior information of parameters, we give the full conditional posterior distribution of parameters including the thresholds and break-points. In order to obtain the estimates of parameters, we employ the Markov chain Monte Carlo (MCMC) method via Gibbs sampler with Metropolis-Hastings algorithm. Compared with Metropolis-Hastings algorithm, we apply Hamiltonian Monte Carlo algorithm to avoid the slow space exploration from simple random walk and improve the sampling efficiency. As applications, we demonstrate the effectiveness of our method from simulation experiments and a real example.

Gibbs sampling for Bayesian estimation of triple seasonal autoregressive models

In last few decades some researchers have extended existing time series models to fit high frequency time series characterized by exhibiting multiple seasonalities. In the Bayesian framework, most of the work in this direction is only at the level of double seasonality. Therefore, in this paper we have the objective to fill part of this gap by extending autoregressive models to fit time series that have three layers of seasonality, i.e., triple seasonal autoregressive (TSAR) models, and introducing the Bayesian estimation for these models using the Gibbs sampling algorithm. In order to achieve this objective, we first assume the model errors are normally distributed and employ the normal-gamma and g priors for the model parameters. We then derive the full conditional posterior distributions to be multivariate normal for the model coefficients and inverse gamma for the model variance. Using these closed-form conditional posterior distributions, we propose the Gibbs sampling algorithm to approximate empirically the joint posterior of the TSAR model parameters, enabling us easily to carry out the Bayesian estimation of TSAR models. By employing the g prior for the model parameters, we execute an extensive simulation study to evaluate the efficiency of the proposed Bayesian estimation of TSAR models, and we then apply our work to real-world hourly electricity load time series datasets in six European countries.

Conditional Gaussian mixture model for warranty claims forecasting

Forecasting warranty claims for complex products is a reliability challenge for most manufacturers. Several factors increase the complexity of warranty claims forecasting, including, the limited number of claims reported at the early stage of launch, reporting delays, dynamic change in the fleet size, and design/manufacturing adjustments for the production line. The aggregated effect of those complexities is often referred to as the “warranty data maturation” effect. Unfortunately, most of the existing models for warranty claims forecasting fail to explicitly consider warranty data maturation. This work address warranty data maturation by proposing the Conditional Gaussian Mixture Model (CGMM). CGMM uses historical warranty data from similar products to develop a robust prior joint Gaussian mixture distribution of warranty trends at both, the current and future maturation levels. CGMM then utilizes Bayesian theories to estimate the conditional posterior distribution of the warranty claims at the future maturation level conditional on the warranty data available at the current maturation level. The CGMM identifies non-parametric temporal warranty trends and automatically clusters products into latent groups to establish (learn) an effective prior joint distribution. The CGMM is validated on an extensive automotive warranty claims dataset comprising of four model years and >15,000 different components from >10 million vehicles.

Bayesian maximum entropy for geoacoustic inversion using ship-radiated noise

Several acoustic signals from merchant ships, also known as ships of opportunity (SOOs), are employed for geacoustic inversion of the seabed on the New England Mud Patch Area using a Bayesian maximum entropy (BME) approach. The SOOs were recorded by three vertical line arrays deployed at different locations in the Seabed Characterization Experiment in 2017 (SBCEX2017). An effective seabed composed of a mud layer on top of a sandy layer is used for this work. For each SOO, the space of possible hypotheses H is uniformly sampled using Monte Carlo. For each hypothesis in H containing a set of geoacoustic parameters, a SOO spectrogram is simulated using the range independent normal mode model ORCA where the squared error between the measured and simulated SOO is calculated. BME is then applied to compute the marginal conditional posterior probability distribution of the geoacoustic parameters where the probability density function (PDF) provides the current state of knowledge. The inferred PDFs are consistent across all the measured SOOs used for this research and agree with previous geoacoustic inversions in the SBCEX2017 [Work supported by ONR Contract No. N00014-19-C-2001.]

Using time-varying volatility for identification in Vector Autoregressions: An application to endogenous uncertainty

We develop a structural vector autoregression with stochastic volatility in which one of the variables can impact both the mean and the variance of the other variables. We provide conditional posterior distributions for this model, develop an MCMC algorithm for estimation, and show how stochastic volatility can be used to provide useful restrictions for the identification of structural shocks. We then use the model with US data to show that some variables have a significant contemporaneous feedback effect on macroeconomic uncertainty, and overlooking this channel can lead to distortions in the estimated effects of uncertainty on the economy.

Spatial-temporal multivariate semi-Bayesian hierarchical framework for extreme precipitation frequency analysis

We present a semi-Bayesian hierarchical modeling framework for conducting space–time frequency analysis of precipitation extremes over a large domain. In this framework, the data layer, the precipitation extreme – i.e., seasonal maximum precipitation, at each station in each year is modeled using a generalized extreme value (GEV) distribution with temporally varying parameters, which are decomposed as linear functions of covariates. The coefficients of the covariates are estimated via maximum likelihood (ML). In the process layer, the estimated ML coefficients of each of the covariates across the stations are spatially modeled with a Gaussian multivariate process which enables capturing the spatial structure and correlation between the spatial model parameters. Suitable priors are used for the spatial model hyperparameters to complete the Bayesian formulation. Since the Bayesian formulation is only at the second level, our model is semi-Bayesian and thus, the posteriors are conditional posterior distributions. With the conditional posterior distribution of spatial fields of the GEV parameters for each time, conditional posterior distribution of the nonstationary space–time return levels of the precipitation extremes are obtained. We demonstrate this framework by application to summer precipitation extreme at 73 stations covering a large domain of Southwest US consisting of Arizona, New Mexico, Colorado, and Utah. The results from fitting and cross-validation indicate that our model captures the historical variability at the stations very well. Conditional posterior distributions of return levels are simulated on a grid over the domain, which will be of immense utility in management of natural resources and infrastructure.

Did Noise Pollution Really Improve during COVID-19? Evidence from Taiwan

Background and objectives: The impacts of COVID-19 are like two sides of one coin. During 2020, there were many research papers that proved our environmental and climate conditions were improving due to lockdown or large-scale restriction regulations. In contrast, the economic conditions deteriorated due to disruption in industry business activities and most people stayed at home and worked from home, which probably reduced the noise pollution. Methods: To assess whether there were differences in noise pollution before and during COVID-19. In this paper, we use various statistical methods following odds ratios, Wilcoxon and Fisher’s tests and Bayesian Markov chain Monte Carlo (MCMC) with various comparisons of prior selection. The outcome of interest for a parameter in Bayesian inference is complete posterior distribution. Roughly, the mean of the posterior will be clear with point approximation. That being said, the median is an available choice. Findings: To make the Bayesian MCMC work, we ran the sampling from the conditional posterior distributions. It is straightforward to draw random samples from these distributions if they have regular shapes using MCMC. The case of over-standard noise per time frame, number of noise petition cases, number of industry petition cases, number of motorcycles, number of cars and density of vehicles are significant at α = 5%. In line with this, we prove that there were differences of noise pollution before and during COVID-19 in Taiwan. Meanwhile, the decreased noise pollution in Taiwan can improve quality of life.

Bayesian Estimation of the Polynomial Time Trend AR(1) Model through Spline Function

In this paper, we develop an estimation procedure for an autoregressive model with polynomial time trend approximated by a spline function. Spline function has the advantage of approximating the non-linear time series in an appropriate degree of polynomial time trend model. For Bayesian parameter estimation, the conditional posterior distribution is obtained under two symmetric loss functions. Due to the complex form of the conditional posterior distribution, Markov Chain Monte Carlo (MCMC) approach is used to estimate the Bayes estimators. The performance of Bayes estimators is compared with that of the corresponding maximum likelihood estimators (MLEs) in terms of mean squared error (MSE) and average absolute bias (AB) via a simulation study. To illustrate the proposed study, import series of Brazil, Russia, India, China, and South Africa (BRICS) countries are analyzed.

The Odd Lindley Burr XII Model: Bayesian Analysis, Classical Inference and Characterizations

In this work, we study the odd Lindley Burr XII model initially introduced by Silva et al. [29]. This model has the advantage of being capable of modeling various shapes of aging and failure criteria. Some of its statistical structural properties including ordinary and incomplete moments, quantile and generating function and order statistics are derived. The odd Lindley Burr XII density can be expressed as a simple linear mixture of BurrXII densities. Useful characterizations are presented. The maximum likelihood method is used to estimate the model parameters. Simulation results to assess the performance of the maximum likelihood estimators are discussed. We prove empirically the importance and flexibility of the new model in modeling various types of data. Bayesian estimation is performed by obtaining the posterior marginal distributions as well as using the simulation method of Markov Chain Monte Carlo (MCMC) by the Metropolis-Hastings algorithm in each step of Gibbs algorithm. The trace plots and estimated conditional posterior distributions are also presented.