Approximated sensitivity analysis in posterior predictive distribution

In this article we present a Bayesian prediction of multiplicative seasonal autoregressive moving average (SARMA) processes using the Gibbs sampling algorithm. First, we estimate the unobserved errors using the nonlinear least squares (NLS) method to approximate the likelihood function. Second, we employ conjugate priors on the model parameters and initial values and assume the model errors are normally distributed to derive the conditional posterior and predictive distributions. In particular, we show that the conditional posterior distribution of the model parameters and the variance are multivariate normal and inverse gamma respectively, and the conditional predictive distribution of the future observations is a multivariate normal. Finally, we use these closed-form conditional posterior and predictive distributions to apply the Gibbs sampling algorithm to approximate empirically the marginal posterior and predictive distributions, enabling us easily to carry out multiple-step ahead predictions. We evaluate our proposed Bayesian method using simulation study and real-world time series datasets.

In this paper we use the Gibbs sampling algorithm to present a Bayesian analysis to multiplicative double seasonal autoregressive (DSAR) models, considering both estimation and prediction problems. Assuming the model errors are normally distributed and using natural conjugate and g priors on the initial values and model parameters, we show that the conditional posterior distributions of the model parameters and variance are multivariate normal and inverse gamma respectively, and the conditional predictive distribution of the future observations is a multivariate normal. Using these closed-form conditional posterior and predictive distributions, we apply the Gibbs sampling to approximate empirically the marginal posterior and predictive distributions, enabling us easily to carry out multiple-step ahead predictions. The proposed Bayesian method is evaluated using simulation study and real-world time series dataset.

At the start of new operation, a production process is commonly unstable. Then, a process condition is into being stable gradually over time. For the purpose of shifting to mass production, the stability in a process should be evaluated. This paper proposes a method of evaluating the process stability based on Bayesian statistics and information theory. In Bayesian statistics, the knowledge for a process is renewed by deriving posterior distribution based on current prior distribution and observations. Here, equivalency between prior and posterior distributions could be considered to be a criterion for the stability in a process. It is needed to define difference between prior and posterior distributions in order to evaluate the equivalency between prior and posterior distributions. We first formulate the relation between the prior distribution and the posterior distribution for process mean using Bayesian statistics. Secondly, we evaluate the difference between the both distributions based on Kullback‐Leibler (K‐L) divergence in information theory. Finally, some numerical examples in the method of evaluating the stability in process mean are illustrated. Also, we discuss about a decision rule for the stability in process mean through the numerical investigation.

We discuss the problem of model criticism, with emphasis on developing summary diagnostic measures. We approach model criticism by identifying possible troublesome features of the currently entertained model, embedding the model in an elaborated model, and measuring the value of elaborating. This requires three elements: a model elaboration, a prior distribution, and a utility function. Each triplet generates a different diagnostic measure. We focus primarily on the measure given by a Kullback—Leibler divergence between the marginal prior and posterior distributions on the elaboration parameter. We also develop a linearized version of this diagnostic and use it to show that our procedure is related to other tools commonly used for model diagnostics, such as Bayes factors and the score function. One attraction of this approach is that it allows model criticism to be performed jointly with parameter inference and prediction. Also, this diagnostic approach aims at maintaining an exploratory nature t...

A crucial decision in designing a spatial sample for soil survey is the number of sampling locations required to answer, with sufficient accuracy and precision, the questions posed by decision makers at different levels of geographic aggregation. In the Indian Soil Health Card (SHC) scheme, many thousands of locations are sampled per district. In this paper the SHC data are used to estimate the mean of a soil property within a defined study area, e.g., a district, or the areal fraction of the study area where some condition is satisfied, e.g., exceedence of a critical level. The central question is whether this large sample size is needed for this aim. The sample size required for a given maximum length of a confidence interval can be computed with formulas from classical sampling theory, using a prior estimate of the variance of the property of interest within the study area. Similarly, for the areal fraction a prior estimate of this fraction is required. In practice we are uncertain about these prior estimates, and our uncertainty is not accounted for in classical sample size determination (SSD). This deficiency can be overcome with a Bayesian approach, in which the prior estimate of the variance or areal fraction is replaced by a prior distribution. Once new data from the sample are available, this prior distribution is updated to a posterior distribution using Bayes’ rule. The apparent problem with a Bayesian approach prior to a sampling campaign is that the data are not yet available. This dilemma can be solved by computing, for a given sample size, the predictive distribution of the data, given a prior distribution on the population and design parameter. Thus we do not have a single vector with data values, but a finite or infinite set of possible data vectors. As a consequence, we have as many posterior distribution functions as we have data vectors. This leads to a probability distribution of lengths or coverages of Bayesian credible intervals, from which various criteria for SSD can be derived. Besides the fully Bayesian approach, a mixed Bayesian-likelihood approach for SSD is available. This is of interest when, after the data have been collected, we prefer to estimate the mean from these data only, using the frequentist approach, ignoring the prior distribution. The fully Bayesian and mixed Bayesian-likelihood approach are illustrated for estimating the mean of log-transformed Zn and the areal fraction with Zn-deficiency, defined as Zn concentration <0.9 mg kg −1, in the thirteen districts of Andhra Pradesh state. The SHC data from 2015–2017 are used to derive prior distributions. For all districts the Bayesian and mixed Bayesian-likelihood sample sizes are much smaller than the current sample sizes. The hyperparameters of the prior distributions have a strong effect on the sample sizes. We discuss methods to deal with this. Even at the mandal (sub-district) level the sample size can almost always be reduced substantially. Clearly SHC over-sampled, and here we show how to reduce the effort while still providing information required for decision-making. R scripts for SSD are provided as supplementary material.

SummaryScattered reports of multiple maxima in posterior distributions or likelihoods for mixed linear models appear throughout the literature. Less scrutinised is the restricted likelihood, which is the posterior distribution for a specific prior distribution. This paper surveys existing literature and proposes a unifying framework for understanding multiple maxima. For those problems with covariance structures that are diagonalisable in a specific sense, the restricted likelihood can be viewed as a generalised linear model with gamma errors, identity link and a prior distribution on the error variance. The generalised linear model portion of the restricted likelihood can be made to conflict with the portion of the restricted likelihood that functions like a prior distribution on the error variance, giving two local maxima in the restricted likelihood. Applying in addition an explicit conjugate prior distribution to variance parameters permits a second local maximum in the marginal posterior distribution even if the likelihood contribution has a single maximum. Moreover, reparameterisation from variance to precision can change the posterior modality; the converse also is true. Modellers should beware of these potential pitfalls when selecting prior distributions or using peak‐finding algorithms to estimate parameters.

Computed tomography (CT) angiography (CTA) is a non-invasive imaging method used to detect arteries and examine various brain diseases. When CTA is performed for follow-up or postoperative evaluation, reproducibility of vessel delineation is required. A reproducible and stable contrast enhancement can be achieved by manipulating the factors affecting it. Previous studies have investigated several factors that alter the contrast enhancement of arteries. However, no reports establishing the effect of different operators on contrast enhancement exist. To assess the differences between inter-operator arterial contrast enhancement in cerebral CTA using Bayesian statistical modeling. Image data were obtained using a multistage sampling method from the cerebral CTA scans of patients who underwent the process between January 2015 and December 2018. Several Bayesian statistical models were developed, and the objective variable was the mean CT number of the bilateral internal carotid arteries after contrast enhancement. The explanatory variables were sex, age, fractional dose (FD), and the operator's information. The posterior distributions of the parameters were computed via Bayesian inference using the Markov chain Monte Carlo (MCMC) method, with the Hamiltonian Monte Carlo method employed as the algorithm. The posterior predictive distributions were computed using the posterior distributions of the parameters. Finally, the differences between inter-operator arterial contrast enhancement on the CT number in cerebral CTA were estimated. The posterior distributions showed that all parameters representing the difference between operators included zero at the 95% credible intervals (CIs). The maximum mean difference between inter-operator CT number in the posterior predictive distribution was only 12.59 Hounsfield units (HUs). The Bayesian statistical modeling results suggest that contrast enhancement of cerebral CTA examination between operator-to-operator differences in postcontrast CT number was small compared to those within-operator differences resulting from factors not considered in the model.

We have discussed various statistical models and their implementation so far. However, the ways we used the MCMC samples from the posterior distributions and predictive distributions were only kept on very basic levels, such as computing the intervals and visualizations. In this chapter, we will introduce more advanced usages of the MCMC sample. They would be helpful in practice because it is very common to encounter the situation where we need to extract more information from the MCMC sample.

: Certain best population problems adopting a Bayesian approach were investigated. One main advantage of such an approach is that satisfactory decision procedures can be obtained in the presence of nuisance parameters. The use of Bayes' theorem allows one to analyze best population problems from many points of view. A general description of best population problems is outlined. The criterion for 'bestness' is regarded as a 'utility' function of the statistician. The decision procedure then adopted is based upon the principle of maximizing posterior expected utility. The application of this procedure when sampling from normal populations and exponential populations was discussed. The criterion defining the best population is taken to be the coverage of the population considered in a certain given interval. The procedures are shown to be consistent. The decision analysis by considering the pos terior distribution of the criterion is presented. This extension enables other decision procedures to be proposed which may be more appropriate in certain situations than that resulting from the principle of maximizing posterior expected utility.

BackgroundTranslating a known metabolic network into a dynamic model requires reasonable guesses of all enzyme parameters. In Bayesian parameter estimation, model parameters are described by a posterior probability distribution, which scores the potential parameter sets, showing how well each of them agrees with the data and with the prior assumptions made.ResultsWe compute posterior distributions of kinetic parameters within a Bayesian framework, based on integration of kinetic, thermodynamic, metabolic, and proteomic data. The structure of the metabolic system (i.e., stoichiometries and enzyme regulation) needs to be known, and the reactions are modelled by convenience kinetics with thermodynamically independent parameters. The parameter posterior is computed in two separate steps: a first posterior summarises the available data on enzyme kinetic parameters; an improved second posterior is obtained by integrating metabolic fluxes, concentrations, and enzyme concentrations for one or more steady states. The data can be heterogenous, incomplete, and uncertain, and the posterior is approximated by a multivariate log-normal distribution. We apply the method to a model of the threonine synthesis pathway: the integration of metabolic data has little effect on the marginal posterior distributions of individual model parameters. Nevertheless, it leads to strong correlations between the parameters in the joint posterior distribution, which greatly improve the model predictions by the following Monte-Carlo simulations.ConclusionWe present a standardised method to translate metabolic networks into dynamic models. To determine the model parameters, evidence from various experimental data is combined and weighted using Bayesian parameter estimation. The resulting posterior parameter distribution describes a statistical ensemble of parameter sets; the parameter variances and correlations can account for missing knowledge, measurement uncertainties, or biological variability. The posterior distribution can be used to sample model instances and to obtain probabilistic statements about the model's dynamic behaviour.

Based on left type II censored samples from a Gumbel type II distribution, the Bayes estimators and corresponding risks of the unknown parameter were obtained under different asymmetric loss functions, assuming different informative and non-informative priors. Elicitation of hyper-parameters through prior predictive approach has also been discussed. The expressions for the credible intervals and posterior predictive distributions have been derived. Comparisons of these estimators are made through simulation study using numerical and graphical methods.

The paper takes up posterior analysis of a linear regression model with disturbances from a logarithmic Birnbaum–Saunders distribution. The posterior distribution cannot be analyzed using analytical techniques and, therefore, posterior simulation methods are proposed. Computations are organized around Markov Chain Monte Carlo methods. Associated methods to compute posterior odds and posterior predictive distributions are also discussed.

Left-censored datasets of virus density in wastewater samples make it difficult to evaluate the virus removal efficiency in wastewater treatment processes. In the present study, we modeled the probabilistic distribution of virus removal efficiency in a wastewater treatment process with a Bayesian approach, and investigated how many detect samples in influent and effluent are necessary for accurate estimation. One hundred left-censored data of virus density in wastewater (influent and effluent) were artificially generated based on assumed log-normal distributions and the posterior predictive distribution of virus density, and the log-ratio distribution were estimated. The estimation accuracy of distributions was quantified by Bhattacharyya coefficient. When it is assumed that the accurate estimation of posterior predictive distributions is possible when a 100% positive rate is obtained for 12 pairs of influent and effluent, 11 out of 144, 60 out of 324, and 201 out of 576 combinations of detect samples gave an accurate estimation at the significant level of 0.01 in a Kruskal-Wallis test when the total sample number was 12, 18, and 24, respectively. The combinations with the minimum number of detect samples were (12, 9), (16, 10), and (21, 8) when the total sample number was 12, 18, and 24, respectively.

PurposeThe purpose of this paper is to present an economic–statistical design (ESD) for the Bayesian control chart based on predictive distribution with two types of informative and noninformative prior distributions.Design/methodology/approachThe design used in this study is based on determining the control chart of the predictive distribution and then its ESD. The new proposed cost model is presented by considering the conjugate and Jeffrey's prior distribution in calculating the expected total cycle time and expected cost per cycle, and finally, the optimal design parameters and related costs are compared with the fixed ratio sampling (FRS) mode.FindingsNumerical results show decreases in costs in this Bayesian approach with both Jeffrey's and conjugate prior distribution compared to the FRS mode. This result shows that the Bayesian approach which is based on predictive density works better than the classical approach. Also, for the Bayesian approach, however, there is no significant difference between the results of using Jeffrey's and conjugate prior distributions. Using sensitivity analysis, the effect of cost parameters and shock model parameters and deviation from the mean on the optimal values of design parameters and related costs have been investigated and discussed.Practical implicationsThis research adds to the body of knowledge related to quality control of process monitoring systems. This paper may be of particular interest to quality system practitioners for whom the effect of the prior distribution of parameters on the quality characteristic distribution is important.Originality/valueeconomic statistical design (ESD) of Bayesian control charts based on predictive distribution is presented for the first time.

The article proposes a Bayesian methodology to implement complete and cohesive analysis of vector time series. Assuming the series are generated by vector autoregressive moving average process, the identification, diagnostic checking, estimation and forecasting phases of time series analysis are done by referring to the appropriate posterior or predictive distributions. The identification phase is based on approximating the posterior distribution of the coefficients of the largest possible model by a matric-variate generalization of the t-distribution (matrix t-distribution). Then the insignificant coefficients are eliminated by a sequence of F or Chi-square tests using a procedure similar to the backward elimination technique used in regression analysis. The diagnostic checking phase is done using overfitting tests, the estimation phase is done using the matrix t and Wishart distributions, and the forecasting phase is performed using multivariate t-distribution. Using the proposed posterior distributions, the article proposes the machinery necessary to implement the four phases of multivariate time series analysis. The proposed Bayesian methodology has been illustrated and checked by a simulated data used by distinguished researchers. The Initial examination of the numerical results supports the adequacy of using the proposed methodology.