Marginal Posterior Distributions Research Articles

THE UNINFORMATIVE PRIOR OF JEFFREYS’ DISTRIBUTION IN BAYESIAN GEOGRAPHICALLY WEIGHTED REGRESSION

When using the Bayesian method for estimating parameters in a geographically weighted regression model, the choice of the prior distribution directly impacts the posterior distribution. The distribution known as the Jeffreys prior is an uninformative type of prior distribution and is invariant to reparameterization. In cases where information about the parameter is not available, the Jeffreys' prior is utilized. The data was fitted with an uninformative Jeffreys' prior distribution, which yielded a posterior distribution that was utilized for estimating parameters. This study aims to derive the prior and marginal posterior distributions of the Jeffreys' and in Bayesian geographically weighted regression (BGWR). The marginal posterior distributions of and can be obtained by integrating the other parameters of a common posterior distribution. Based on the results and discussion, the Jeffreys prior in BGWR with the likelihood function is . On the other hand, the marginal posterior distribution of follows a normal multivariate distribution, that is, , while the marginal posterior distribution of follows an inverse gamma distribution, that is, . As further research, it is necessary to follow up on several limitations of the results of this research, namely numerical simulations and application to a particular case that related to the results of the analytical studies that we have carried out.

Bayesian Optimal Designs for Multi-Arm Multi-Stage Phase II Randomized Clinical Trials with Multiple Endpoints

There is a growing need to evaluate of multiple competing drugs in phase II trials where the number of patients is often limited, and simultaneous assessment of both efficacy and toxicity is crucial. To avoid the waste of research resources, it is indeed more efficient to screen multiple drugs at once in a platform phase II setting. We aim to adapt the Bayesian optimal phase II (BOP2) design to multi-arm trials for both uncontrolled and controlled settings. The binary efficacy and toxicity endpoints are modeled by a Dirichlet distribution as a vector of four outcomes. Posterior marginal distributions at each analysis are used to derive the monitoring threshold that varies during the trial. We control the family-wise Type I error rate for multiple comparison against a common reference value or a shared control. We conduct simulation studies under both uncontrolled and controlled settings to evaluate the operating characteristics of the proposed design. Our simulations demonstrate that the design exhibits better operating characteristics compared to a design using a constant threshold and is less sensitive to changes in accrual rate relative to what was planned. The design had promising operating characteristics and could be used in phase II oncology clinical trials for evaluating multiple drugs at a time.

Bayesian modeling of quantiles of body mass index among under-five children in Ethiopia

BackgroundBody Mass Index (BMI) is a measurement of nutritional status, which is a vital pre-condition for good health. The prevalence of childhood malnutrition and the potential long-term health risks associated with obesity in Ethiopia have recently increased globally. The main objective of this study was to investigate the factors associated with the quantiles of under-five children’s BMI in Ethiopia.MethodsData on 5,323 children, aged between 0-59 months from March 21, 2019, to June 28, 2019, were obtained from the Ethiopian Mini Demographic Health Survey (EMDHS, 2019), based on the standards set by the World Health Organization. The study used a Bayesian quantile regression model to investigate the association of factors with the quantiles of under-five children’s body mass index. Markov Chain Monte Carlo (MCMC) with Gibbs sampling was used to estimate the country-specific marginal posterior distribution estimates of model parameters, using the Brq R package.ResultsOut of a total of 5323 children included in this study, 5.09% were underweight (less than 12.92 BMI), 10.05% were overweight (BMI: 17.06 – 18.27), and 5.02% were obese (greater than or equal to 18.27 BMI) children’s. The result of the Bayesian quantile regression model, including marginal posterior credible intervals (CIs), showed that for the prediction of the 0.05 quantile of BMI, the current age of children [β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upbeta$$\\end{document}= -0.007, 95% CI :(-0.01, -0.004)], the region Afar [β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upbeta$$\\end{document} = - 0.32, 95% CI: (-0.57, -0.08)] and Somalia[β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upbeta$$\\end{document} = -0.72, 95% CI: (-0.96, -0.49)] were negatively associated with body mass index while maternal age [β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upbeta$$\\end{document} = 0.01, 95% CI: (0.005, 0.02)], mothers primary education [β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upbeta$$\\end{document}= 0.19, 95% CI: (0.08, 0.29)], secondary and above [β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upbeta$$\\end{document} = 0.44, 95% CI: (0.29, 0.58)], and family follows protestant [β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upbeta$$\\end{document} = 0.22, 95% CI: (0.07, 0.37)] were positively associated with body mass index. In the prediction of the 0.95 (or 0.85?) quantile of BMI, in the upper quantile, still breastfeeding [β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upbeta$$\\end{document} = -0.25, 95% CI: (-0.41, -0.10)], being female [β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upbeta$$\\end{document} = -0.13, 95% CI: (-0.23, -0.03)] were negatively related while wealth index [β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upbeta$$\\end{document} = 0.436, 95% CI: (0.25, 0.62)] was positively associated with under-five children’s BMI.ConclusionsIn conclusion, the research findings indicate that the percentage of lower and higher BMI for under-five children in Ethiopia is high. Factors such as the current age of children, sex of children, maternal age, religion of the family, region and wealth index were found to have a significant impact on the BMI of under-five children both at lower and upper quantile levels. Thus, these findings highlight the need for administrators and policymakers to devise and implement strategies aimed at enhancing the normal or healthy weight status among under-five children in Ethiopia.

Parameters Estimation of the Exponentiated Chen Distribution Based on Upper Record Values

This article discusses the Bayesian and frequentist inferences for the exponentiated Chen distribution assuming upper record values. Due to unavailability of the compact form of marginal posterior distributions, a Markov Chain Monte Carlo algorithm is designed to compute the posterior summaries. Prediction of future record values under Bayesian and frequentist methods is also discussed mathematically and numerically. Further, a sensitivity analysis to assess the effect of prior on the estimated parameters is also a part of this study. Besides the simulation studies, the importance of the present study is illustrated with the help of a real data example. It is noted that the Bayes estimates outperform the frequentist inference.

Spatially Dependent Bayesian Modeling of Geostatistics Data and Its Application for Tuberculosis (TB) in China

Geostatistics data in regions always have highly spatial heterogeneous, yet the regional features of the data itself cannot be ignored. In this paper, a novel latent Bayesian spatial model is proposed, which incorporates the spatial dependence of different subjects and the spatial random effects to further analysis the influence of spatial effect. The model is verified to be compatible with the integrated nested Laplace approximation (INLA) framework and is fitted using INLA and stochastic partial differential equation (SPDE). The posterior marginal distribution of parameters is estimated with high precision. Additionally, a practical application of the model is presented using tuberculosis (TB) incidence data in China from 2015 to 2019. The results show that the fitting accuracy of our model is higher than the general Bayesian spatial model using INLA-SPDE.

Adaptive MCMC for Bayesian Variable Selection in Generalised Linear Models and Survival Models.

Developing an efficient computational scheme for high-dimensional Bayesian variable selection in generalised linear models and survival models has always been a challenging problem due to the absence of closed-form solutions to the marginal likelihood. The Reversible Jump Markov Chain Monte Carlo (RJMCMC) approach can be employed to jointly sample models and coefficients, but the effective design of the trans-dimensional jumps of RJMCMC can be challenging, making it hard to implement. Alternatively, the marginal likelihood can be derived conditional on latent variables using a data-augmentation scheme (e.g., Pólya-gamma data augmentation for logistic regression) or using other estimation methods. However, suitable data-augmentation schemes are not available for every generalised linear model and survival model, and estimating the marginal likelihood using a Laplace approximation or a correlated pseudo-marginal method can be computationally expensive. In this paper, three main contributions are presented. Firstly, we present an extended Point-wise implementation of Adaptive Random Neighbourhood Informed proposal (PARNI) to efficiently sample models directly from the marginal posterior distributions of generalised linear models and survival models. Secondly, in light of the recently proposed approximate Laplace approximation, we describe an efficient and accurate estimation method for marginal likelihood that involves adaptive parameters. Additionally, we describe a new method to adapt the algorithmic tuning parameters of the PARNI proposal by replacing Rao-Blackwellised estimates with the combination of a warm-start estimate and the ergodic average. We present numerous numerical results from simulated data and eight high-dimensional genetic mapping data-sets to showcase the efficiency of the novel PARNI proposal compared with the baseline add-delete-swap proposal.

Bayesian Identification Procedure for Triple Seasonal Autoregressive Models

Triple seasonal autoregressive (TSAR) models have been introduced to model time series date with three layers of seasonality; however, the Bayesian identification problem of these models has not been tackled in the literature. Therefore, in this paper, we have the objective of filling this gap by presenting a Bayesian procedure to identify the best order of TSAR models. Assuming that the TSAR model errors are normally distributed along with employing three priors, i.e., normal-gamma, Jeffreys’ and g priors, on the model parameters, we derive the marginal posterior distributions of the TSAR model parameters. In particular, we show that the marginal posteriors are multivariate t and gamma distributions for the TSAR model coefficients vector and precision, respectively. Using the marginal posterior distribution of the TSAR model coefficients vector, we present an identification procedure for the TSAR models based on a sequence of t-test of significance. We evaluate the accuracy of the proposed Bayesian identification procedure by conducting an extensive simulation study, followed by a real application to hourly electricity load datasets in six European countries.

Quantification of Empirical Determinacy: The Impact of Likelihood Weighting on Posterior Location and Spread in Bayesian Meta-Analysis Estimated with JAGS and INLA

The popular Bayesian meta-analysis expressed by Bayesian normal-normal hierarchical model (NNHM) synthesizes knowledge from several studies and is highly relevant in practice. Moreover, NNHM is the simplest Bayesian hierarchical model (BHM), which illustrates problems typical in more complex BHMs. Until now, it has been unclear to what extent the data determines the marginal posterior distributions of the parameters in NNHM. To address this issue we computed the second derivative of the Bhattacharyya coefficient with respect to the weighted likelihood, defined the total empirical determinacy (TED), the proportion of the empirical determinacy of location to TED (pEDL), and the proportion of the empirical determinacy of spread to TED (pEDS). We implemented this method in the R package \texttt{ed4bhm} and considered two case studies and one simulation study. We quantified TED, pEDL and pEDS under different modeling conditions such as model parametrization, the primary outcome, and the prior. This clarified to what extent the location and spread of the marginal posterior distributions of the parameters are determined by the data. Although these investigations focused on Bayesian NNHM, the method proposed is applicable more generally to complex BHMs.

Probabilistic radwaste characterization: Findings of a multi-method multi-mockup exercise using interpolation-based surrogate efficiencies

In this paper, we present the lessons learned from a multi-method, multi-mockup probabilistic radiological characterization exercise conducted within the EU project CHANCE “Characterization of Conditioned Nuclear Waste for its Safe Disposal in Europe”. A Bayesian approach that accounts for uncertainty in the measurement efficiencies is used to interpret combinations of (i) open geometry gamma spectrometry (OGGS), (ii) passive neutron coincidence counting (PNCC) and (iii) large-volume calorimetry (LVC) measurements, performed on five different combinations of mockup drum and source position. The used approach treats uncertainty in the measurement efficiencies by doing multilinear interpolation between reference efficiencies representing potential “end-member” matrices with respect to both composition and density, for the considered drum. The end-member proportions are then jointly inferred with the other unknowns. Our results indicate that the success of this method critically depends on the accuracy of the set of reference end-member efficiencies obtained from physics-based Monte Carlo particle transport modeling. We find that Bayesian inversion of OGGS data alone can recover reasonably accurate nuclide masses, with relatively high precision and low bias. Furthermore, when little prior information about the drum is available then jointly inverting OGGS and PNCC data is found to allow for a much higher precision compared to inverting OGGS data alone. In addition, calorimetry appears to be a promising method and brings some added value to OGGS as well in certain cases. In some setups and for some nuclides, errors in the reference efficiencies cause the marginal posterior distribution to be fairly biased. We thus design a simple approach to increase posterior uncertainty (that is, decrease precision) such that the true mass values are included in the bulk of the posterior distribution, in a controlled way. The approach consists of multiplying the interpolated efficiencies with a series of multipliers, that are jointly inferred under a relatively strong prior.

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.

Incorporation of feeding behaviour traits to increase the genetic gain of feed efficiency in Pietrain pigs.

Improved feed efficiency is an essential goal for the sustainability of pig production in economic and environmental terms. Traits such as feed conversion rate (FCR), residual feed intake (RFI), residual body weight gain (RG) and feeding behaviour, such as duration (TPV) and feeding rate per visit (FR) can now be measured by automatic feeding systems. The aim of this study was to evaluate the benefits of incorporating feeding behaviour traits into a selection index to improve feed efficiency in a nucleus of purebred Pietrain pigs. Data on body weight, feed intake and duration were recorded at each visit in 1608 animals. The information contained in 843,605 visits was grouped by animal ID to obtain a set of feed efficiency and feeding behaviour traits. These traits were obtained in three periods (first, second and total period). Bayesian models were built to estimate the posterior marginal distribution of the variance components. The heritabilities were between 0.44 and 0.59 for feeding behaviour traits and between 0.31 and 0.49 for feed efficiency traits. The FCR and RFI showed a considerable genetic correlation with daily feed intake (~0.65). FCR showed a genetic correlation with feeding behaviour traits, such as feed intake per visit (FPV) (0.44) and FR (0.33). Furthermore, the fast-eating pigs were less efficient. This was due to the positive genetic correlation found between the FR and the FCR (0.33) and the RFI (0.23), and the negative correlation found with the RG (-0.28). On the other hand, the inclusion of the feeding behaviour traits into a selection index slightly increased the selection response for FCR (4%) and RFI (1.8%). However, there was an increase of up to 19% in the selection response for RG and an improvement in accuracy from 0.59 to 0.70. Therefore, we concluded that it would be interesting to include feeding behaviour traits in a selection index to improve the selection response and accuracy of feed efficiency traits.

Bayesian Inference and Data Analysis of the Unit–Power Burr X Distribution

The unit–power Burr X distribution (UPBXD), a bounded version of the power Burr X distribution, is presented. The UPBXD is produced through the inverse exponential transformation of the power Burr X distribution, which is also beneficial for modelling data on the unit interval. Comprehensive analysis of its key characteristics is performed, including shape analysis of the primary functions, analytical expression for moments, quantile function, incomplete moments, stochastic ordering, and stress–strength reliability. Rényi, Havrda and Charvat, and d-generalized entropies, which are measures of uncertainty, are also obtained. The model’s parameters are estimated using a Bayesian estimation approach via symmetric and asymmetric loss functions. The Bayesian credible intervals are constructed based on the marginal posterior distribution. Monte Carlo simulation research is intended to test the accuracy of various estimators based on certain measures, in accordance with the complex forms of Bayesian estimators. Finally, we show that the new distribution is more appropriate than certain other competing models, according to their application for COVID-19 in Saudi Arabia and the United Kingdom.

Multiple‐model generalised labelled multi‐Bernoulli filter with distributed sensors for tracking manoeuvring targets using belief propagation

AbstractThe multi‐sensor multiple‐model generalised labelled multi‐Bernoulli filter (MS‐MM‐GLMB) is presented for tracking multiple manoeuvring targets. And we develop efficient implementation for computing multi‐target posterior. The proposed implementation of the MS‐MM‐GLMB filter is composed of two main parts including centralised fusion with distributed sensors and posterior calculation using belief propagation. Specifically, the centralised fusion with distributed sensors is performed based on common predicted information to enable a full parallelisation. This fusion strategy yields the exact solution along with independence of the sensor ordering. More importantly, this fusion allows multiple sensor fusion with the MM‐GLMB filters for iterative update step for synchronous measurements. Subsequently, the belief propagation algorithm is adopted to implement the MS‐MM‐GLMB filter by efficiently calculating the marginal posterior distributions of GLMB form with superior tracking performance. The belief propagation solution to solving data association is presented for the computationally efficient centralised tracking of the manoeuvring targets. Compared with Gibbs‐based implementation, the proposed belief‐propagation‐based method performs well or better, especially in more challenging scenario.

The Bayesian Estimate of Vector Autoregressive Model Parameters Adopt Informative Prior Information

This research included the bayesian estimate for vector Autoregressive model with rank (p) in addition to statistical tests and predict Bayesian when the random error of model followed generalized multivariate modified Bessel distribution. The prior information about the parameters of model is represented by probability distributions belong to conjugate families. It found that the posterior marginal probability distribution for parameters matrix (Φ) is a Matrix t distribution, The posterior marginal probability distribution of covariance matrix (Σ) is uncommon and the predictive probability distribution of future observations vector is multivariate t distribution.

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.

Use of the reversible jump Markov chain Monte Carlo algorithm to select multiplicative terms in the AMMI-Bayesian model.

The model selection stage has become a central theme in applying the additive main effects and multiplicative interaction (AMMI) model to determine the optimal number of bilinear components to be retained to describe the genotype-by-environment interaction (GEI). In the Bayesian context, this problem has been addressed by using information criteria and the Bayes factor. However, these procedures are computationally intensive, making their application unfeasible when the model's parametric space is large. A Bayesian analysis of the AMMI model was conducted using the Reversible Jump algorithm (RJMCMC) to determine the number of multiplicative terms needed to explain the GEI pattern. Three a priori distributions were assigned for the singular value scale parameter under different justifications, namely: i) the insufficient reason principle (uniform); ii) the invariance principle (Jeffreys' prior) and iii) the maximum entropy principle. Simulated and real data were used to exemplify the method. An evaluation of the predictive ability of models for simulated data was conducted and indicated that the AMMI analysis, in general, was robust, and models adjusted by the Reversible Jump method were superior to those in which sampling was performed only by the Gibbs sampler. In addition, the RJMCMC showed greater feasibility since the selection and estimation of parameters are carried out concurrently in the same sampling algorithm, being more attractive in terms of computational time. The use of the maximum entropy principle makes the analysis more flexible, avoiding the use of procedures for correcting prior degrees of freedom and obtaining improper posterior marginal distributions.

Normalized power priors always discount historical data

Power priors are used for incorporating historical data in Bayesian analyses by taking the likelihood of the historical data raised to the power as the prior distribution for the model parameters. The power parameter is typically unknown and assigned a prior distribution, most commonly a beta distribution. Here, we give a novel theoretical result on the resulting marginal posterior distribution of in case of the normal and binomial model. Counterintuitively, when the current data perfectly mirror the historical data and the sample sizes from both data sets become arbitrarily large, the marginal posterior of does not converge to a point mass at but approaches a distribution that hardly differs from the prior. The result implies that a complete pooling of historical and current data is impossible if a power prior with beta prior for is used.

Detecting renewal states in chains of variable length via intrinsic Bayes factors

Markov chains with variable length are useful parsimonious stochastic models able to generate most stationary sequence of discrete symbols. The idea is to identify the suffixes of the past, called contexts, that are relevant to predict the future symbol. Sometimes a single state is a context, and looking at the past and finding this specific state makes the further past irrelevant. States with such property are called renewal states and they can be used to split the chain into independent and identically distributed blocks. In order to identify renewal states for chains with variable length, we propose the use of Intrinsic Bayes Factor to evaluate the hypothesis that some particular state is a renewal state. In this case, the difficulty lies in integrating the marginal posterior distribution for the random context trees for general prior distribution on the space of context trees, with Dirichlet prior for the transition probabilities, and Monte Carlo methods are applied. To show the strength of our method, we analyzed artificial datasets generated from different models and one example coming from the field of Linguistics.

Bayesian Inference of Triple Seasonal Autoregressive Models

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

Gaussian Orthogonal Latent Factor Processes for Large Incomplete Matrices of Correlated Data

We introduce Gaussian orthogonal latent factor processes for modeling and predicting large correlated data. To handle the computational challenge, we first decompose the likelihood function of the Gaussian random field with a multi-dimensional input domain into a product of densities at the orthogonal components with lower-dimensional inputs. The continuous-time Kalman filter is implemented to compute the likelihood function efficiently without making approximations. We also show that the posterior distribution of the factor processes is independent, as a consequence of prior independence of factor processes and orthogonal factor loading matrix. For studies with large sample sizes, we propose a flexible way to model the mean, and we derive the marginal posterior distribution to solve identifiability issues in sampling these parameters. Both simulated and real data applications confirm the outstanding performance of this method.