A sequential split-and-conquer approach for the analysis of big dependent data in computer experiments

Massive correlated data with many inputs are often generated from computer experiments to study complex systems. The Gaussian process (GP) model is a widely used tool for the analysis of computer experiments. Although GPs provide a simple and effective approximation to computer experiments, two critical issues remain unresolved. One is the computational issue in GP estimation and prediction where intensive manipulations of a large correlation matrix are required. For a large sample size and with a large number of variables, this task is often unstable or infeasible. The other issue is how to improve the naive plug-in predictive distribution which is known to underestimate the uncertainty. In this article, we introduce a unified framework that can tackle both issues simultaneously. It consists of a sequential split-and-conquer procedure, an information combining technique using confidence distributions (CD), and a frequentist predictive distribution based on the combined CD. It is shown that the proposed method maintains the same asymptotic efficiency as the conventional likelihood inference under mild conditions, but dramatically reduces the computation in both estimation and prediction. The predictive distribution contains comprehensive information for inference and provides a better quantification of predictive uncertainty as compared with the plug-in approach. Simulations are conducted to compare the estimation and prediction accuracy with some existing methods, and the computational advantage of the proposed method is also illustrated. The proposed method is demonstrated by a real data example based on tens of thousands of computer experiments generated from a computational fluid dynamic simulator.     

Control Charts for the Variance and Coefficient of Variation Based on Their Predictive Distribution

This article develops a control chart for the variance of a normal distribution and, equivalently, the coefficient of variation of a log-normal distribution. A Bayesian approach is used to incorporate parameter uncertainty, and the control limits are obtained from the predictive distribution for the variance. We evaluate this control chart by examining its performance for various values of the process variance.     

A bayesian approach to prediction in analysis of variance settings

This paper takes the results of Lindley and smith ( 1972 ) one step further, by finding the predictive distribution of an observation y^* whose distribution is normal, and centred at A^* ₁θ₁ We then apply this distribution to the case of prediction based on data obtained in one and two wau ANOVA situations. For Example, it turns out that for two way ANOVA with interaction, the predictive mean, (which we would use as the predictor) is a weighted combination of sample main effects and interaction effects     

On estimating the distribution of data envelopment analysis efficiency scores: an application to nursing homes’ care planning process

Data envelopment analysis (DEA) is a deterministic econometric model for calculating efficiency by using data from an observed set of decision-making units (DMUs). We propose a method for calculating the distribution of efficiency scores. Our framework relies on estimating data from an unobserved set of DMUs. The model provides posterior predictive data for the unobserved DMUs to augment the frontier in the DEA that provides a posterior predictive distribution for the efficiency scores. We explore the method on a multiple-input and multiple-output DEA model. The data for the example are from a comprehensive examination of how nursing homes complete a standardized mandatory assessment of residents.     

Bayesian predictive power: choice of prior and some recommendations for its use as probability of success in drug development

Bayesian predictive power, the expectation of the power function with respect to a prior distribution for the true underlying effect size, is routinely used in drug development to quantify the probability of success of a clinical trial. Choosing the prior is crucial for the properties and interpretability of Bayesian predictive power. We review recommendations on the choice of prior for Bayesian predictive power and explore its features as a function of the prior. The density of power values induced by a given prior is derived analytically and its shape characterized. We find that for a typical clinical trial scenario, this density has a u‐shape very similar, but not equal, to a β‐distribution. Alternative priors are discussed, and practical recommendations to assess the sensitivity of Bayesian predictive power to its input parameters are provided. Copyright © 2016 John Wiley & Sons, Ltd.     

Estimation of random coefficient regression models

Linear regression models with coefficients across individual units regarded as random samples from some population are studied in this article from a Bayesian viewpoint. A prior distribution of the secondary parameters is derived following the Jeffreys rule. Posterior distribution of the primary and secondary parameters, and the predictive distribution of the future value are then examined. Computations of the parameter estimates are found to be rather straightforward. Data from a performance test on pigs is analysed and discussed. We also discuss the difficulties involved in using a Lindley and Smith (1972) prior in this problem.     

Bayesian inferences and forecasts with moving averages processes

This paper will develop Bayesian inferential and forecasting techniques which can be used with any moving average process. By employing the conditional likelihood function, at-approximation to the predictive distribution and the marginal posterior distribution of the moving average parameters is developed. Several examples demonstrate posterior and predictive inferences.     

The General Segmented Distribution

We develop a distribution supported on a bounded interval with a probability density function that is constructed from any finite number of linear segments. With an increasing number of segments, the distribution can approach any continuous density function of arbitrary form. The flexibility of the distribution makes it a useful tool for various modeling purposes. We further demonstrate that it is capable of fitting data with considerable precision—outperforming distributions recommended by previous studies. We suggest that this distribution is particularly effective in fitting data with sufficient observations that are skewed and multimodal.     

Planning a Bayesian early-phase phase I/II study for human vaccines in HER2 carcinomas

Recent innovative statistical approaches for phase I/II clinical trials allow one to jointly model the toxicity and efficacy of a new treatment, taking into account the information gathered during the trial. Prior probabilities are then updated with interim data and thus predictive probabilities become more accurate as the trial progresses. In this study, prior distribution elicited from a physician's opinion on the available dose levels planned for a vaccination dose-finding trial, with human DNA in patients with HER2-positive tumours in terms of toxicity and therapeutic response is presented and discussed. A simulation study was conducted in order to quantify the impact of the choice of prior on study results, i.e. the recommended dose level at the end of the trial.     

On nonparametric Bayesian inference for the distribution of a random sample

The nonparametric Bayesian approach for inference regarding the unknown distribution of a random sample customarily assumes that this distribution is random and arises through Dirichlet-process mixing. Previous work within this setting has focused on the mean of the posterior distribution of this random distribution, which is the predictive distribution of a future observation given the sample. Our interest here is in learning about other features of this posterior distribution as well as about posteriors associated with functionals of the distribution of the data. We indicate how to do this in the case of linear functionals. An illustration, with a sample from a Gamma distribution, utilizes Dirichlet-process mixtures of normals to recover this distribution and its features.     

Criticism of the predictive distribution

The predictive distribution is a mixture of the original distribution model and is used for predicting a future observation. Therein, the mixing distribution is the posterior distribution of the distribution parameters in the Bayesian inference. The mixture can also be computed for the frequentist inference because the Bayesian posterior distribution has the same meaning as a frequentist confidence interval. I present arguments against the concept of predictive distribution. Examples illustrate these. The most important argument is that the predictive distribution can depend on the parameterization. An improvement of the theory of the predictive distribution is recommended.     

A note on predictive inference for multivariate elliptically contoured distributions

The prediction problem is considered for the multivariate regression model with an elliptically contoured error distribution. We show that the predictive distribution under elliptical errors assumption is the same as that obtained under normally distributed error in both the Bayesian approach using an im-proper prior and the classical approach. This gives inference robustness with respect to departures from the reference case of independent sampling from the normal distribution.     

Application of a predictive distribution formula to Bayesian computation for incomplete data models

We consider exact and approximate Bayesian computation in the presence of latent variables or missing data. Specifically we explore the application of a posterior predictive distribution formula derived in Sweeting And Kharroubi (2003), which is a particular form of Laplace approximation, both as an importance function and a proposal distribution. We show that this formula provides a stable importance function for use within poor man’s data augmentation schemes and that it can also be used as a proposal distribution within a Metropolis-Hastings algorithm for models that are not analytically tractable. We illustrate both uses in the case of a censored regression model and a normal hierarchical model, with both normal and Student t distributed random effects. Although the predictive distribution formula is motivated by regular asymptotic theory, it is not necessary that the likelihood has a closed form or that it possesses a local maximum.     

Calibrating predictive distributions

This paper concerns prediction from the frequentist point of view. The aim is to define a well-calibrated predictive distribution giving prediction intervals, and in particular prediction limits, with coverage probability equal or close to the target nominal value. This predictive distribution can be considered in a number of situations, including discrete data and non-regular cases, and it is founded on the idea of calibrating prediction limits to control the associated coverage probability. Whenever the computation of the proposed distribution is not feasible, this can be approximated using a suitable bootstrap simulation procedure or by considering high-order asymptotic expansions, giving predictive distributions already known in the literature. Examples and applications of the results to different contexts show the wide applicability and the very good performance of the proposed predictive distribution.     

Bayesian Point Estimation Using the Predictive Distribution

There is considerable question about how a Bayesian might provide a point estimate for a parameter when no loss function is specified. The mean, median, and mode of the posterior distribution have all been suggested. This article considers a natural Bayesian estimator based on the predictive distribution of future observations. It is shown that for the set of parameters that admit an unbiased estimate, this predictive estimate coincides with the posterior mean of the parameter. It is argued that this result provides some justification for use of the posterior mean as a Bayesian point estimate when there is no loss structure.     

Dropping observations without affecting posterior and predictive distributions

Suppose in a distribution problem, the sample information W is split into two pieces W ₁ and W ₂, and the parameters involved are split into two sets, π containing the parameters of interest, and θ containing nuisance parameters. It is shown that, under certain conditions, the posterior distribution of π does not depend on the data W ₂, which can thus be ignored. This also has consequences for the predictive distribution of future (or missing) observations. In fact, under similar conditions, the predictive distributions using W or just W ₁ are identical.     

Combined Exponentially Weighted Moving Average Charts for the Mean and Variance Based on the Predictive Distribution

This article develops combined exponentially weighted moving average (EWMA) charts for the mean and variance of a normal distribution. A Bayesian approach is used to incorporate parameter uncertainty. We first use a Bayesian predictive distribution to construct the control chart, and we then use a sampling theory approach to evaluate it under various hypothetical specifications for the data generation model. Simulations are used to compare the proposed charts for different values of both the weighing constant for the exponentially weighted moving averages and for the size of the calibration sample that is used to estimate the in-statistical-control process parameters. We also examine the separate performance of the EWMA chart for the variance.     

Modeling Financial Return Dynamics via Decomposition

While the predictability of excess stock returns is detected by traditional predictive regressions as statistically small, the direction-of-change and volatility of returns exhibit a substantially larger degree of dependence over time. We capitalize on this observation and decompose the returns into a product of sign and absolute value components whose joint distribution is obtained by combining a multiplicative error model for absolute values, a dynamic binary choice model for signs, and a copula for their interaction. Our decomposition model is able to incorporate important nonlinearities in excess return dynamics that cannot be captured in the standard predictive regression setup. The empirical analysis of U.S. stock return data shows statistically and economically significant forecasting gains of the decomposition model over the conventional predictive regression.     

Implementation of a robust bayesian method

In this work we study robustness in Bayesian models through a generalization of the Normal distribution. We show new appropriate techniques in order to deal with this distribution in Bayesian inference. Then we propose two approaches to decide, in some applications, if we should replace the usual Normal model by this generalization. First, we pose this dilemma as a model rejection problem, using diagnostic measures. In the second approach we evaluate the model's predictive efficiency. We illustrate those perspectives with a simulation study, a non linear model and a longitudinal data model.     

On Parametric Bootstrapping and Bayesian Prediction

Abstract.  We investigate bootstrapping and Bayesian methods for prediction. The observations and the variable being predicted are distributed according to different distributions. Many important problems can be formulated in this setting. This type of prediction problem appears when we deal with a Poisson process. Regression problems can also be formulated in this setting. First, we show that bootstrap predictive distributions are equivalent to Bayesian predictive distributions in the second-order expansion when some conditions are satisfied. Next, the performance of predictive distributions is compared with that of a plug-in distribution with an estimator. The accuracy of prediction is evaluated by using the Kullback–Leibler divergence. Finally, we give some examples.