A Bayesian growth and yield model for slash pine plantations

We formulate a traditional growth and yield model as a Bayes model. We attempt to introduce as few new assumptions as possible. Zellner's Bayesian method of moments procedure is used, because the published model did not include any distributional assumptions. We generate predictive posterior samples for a number of stand variables using the Gibbs sampler. The means of the samples compare favorably with the predictions from the published model. In addition, our model delivers distributions of outcomes, from which it is easy to establish measures of uncertainty, such as highest posterior density regions.     

Posterior Predictive p-values in Bayesian Hierarchical Models

Abstract.  The present work focuses on extensions of the posterior predictive p -value (ppp-value) for models with hierarchical structure, designed for testing assumptions made on underlying processes. The ppp-values are popular as tools for model criticism, yet their lack of a common interpretation limit their practical use. We discuss different extensions of ppp-values to hierarchical models, allowing for discrepancy measures that can be used for checking properties of the model at all stages. Through analytical derivations and simulation studies on simple models, we show that similar to the standard ppp-values, these extensions are typically far from uniformly distributed under the model assumptions and can give poor power in a hypothesis testing framework. We propose a calibration of the p -values, making the resulting calibrated p -values uniformly distributed under the model conditions. Illustrations are made through a real example of multinomial regression to age distributions of fish.     

A bayesian wls approach to generalized linear models

We use a Bayesian approach to fitting a linear regression model to transformations of the natural parameter for the exponential class of distributions. The usual Bayesian approach is to assume that a linear model exactly describes the relationship among the natural parameters. We assume only that a linear model is approximately in force. We approximate the theta-links by using a linear model obtained by minimizing the posterior expectation of a loss function.While some posterior results can be obtained analytically considerable generality follows from an exact Monte Carlo method for obtaining random samples of parameter values or functions of parameter values from their respective posterior distributions. The approach that is presented is justified for small samples, requires only one-dimensional numerical integrations, and allows for the use of regression matrices with less than full column rank. Two numerical examples are provided.     

A Bayesian nonparametric Markovian model for non-stationary time series

Stationary time series models built from parametric distributions are, in general, limited in scope due to the assumptions imposed on the residual distribution and autoregression relationship. We present a modeling approach for univariate time series data, which makes no assumptions of stationarity, and can accommodate complex dynamics and capture non-standard distributions. The model for the transition density arises from the conditional distribution implied by a Bayesian nonparametric mixture of bivariate normals. This results in a flexible autoregressive form for the conditional transition density, defining a time-homogeneous, non-stationary Markovian model for real-valued data indexed in discrete time. To obtain a computationally tractable algorithm for posterior inference, we utilize a square-root-free Cholesky decomposition of the mixture kernel covariance matrix. Results from simulated data suggest that the model is able to recover challenging transition densities and non-linear dynamic relationships. We also illustrate the model on time intervals between eruptions of the Old Faithful geyser. Extensions to accommodate higher order structure and to develop a state-space model are also discussed.     

Consistency of Bernstein polynomial posteriors

A Bernstein prior is a probability measure on the space of all the distribution functions on [0, 1]. Under very general assumptions, it selects absolutely continuous distribution functions, whose densities are mixtures of known beta densities. The Bernstein prior is of interest in Bayesian nonparametric inference with continuous data. We study the consistency of the posterior from a Bernstein prior. We first show that, under mild assumptions, the posterior is weakly consistent for any distribution function P ₀ on [0, 1] with continuous and bounded Lebesgue density. With slightly stronger assumptions on the prior, the posterior is also Hellinger consistent. This implies that the predictive density from a Bernstein prior, which is a Bayesian density estimate, converges in the Hellinger sense to the true density (assuming that it is continuous and bounded). We also study a sieve maximum likelihood version of the density estimator and show that it is also Hellinger consistent under weak assumptions. When the order of the Bernstein polynomial, i.e. the number of components in the beta distribution mixture, is truncated, we show that under mild restrictions the posterior concentrates on the set of pseudotrue densities. Finally, we study the behaviour of the predictive density numerically and we also study a hybrid Bayes–maximum likelihood density estimator.     

A note on fiducial model averaging as an alternative to checking Bayesian and frequentist models

Just as frequentist hypothesis tests have been developed to check model assumptions, prior predictive p-values and other Bayesian p-values check prior distributions as well as other model assumptions. These model checks not only suffer from the usual threshold dependence of p-values, but also from the suppression of model uncertainty in subsequent inference. One solution is to transform Bayesian and frequentist p-values for model assessment into a fiducial distribution across the models. Averaging the Bayesian or frequentist posterior distributions with respect to the fiducial distribution can reproduce results from Bayesian model averaging or classical fiducial inference.     

A note on dealing with missing standard errors in meta-analyses of continuous outcome measures in WinBUGS

A meta-analysis of a continuous outcome measure may involve missing standard errors. This is not a problem depending on assumptions made about the population standard deviation. Multiple imputation can be used to impute missing values while allowing for uncertainty in the imputation. Markov chain Monte Carlo simulation is a multiple imputation technique for generating posterior predictive distributions for missing data. We present an example of imputing missing variances using WinBUGS. The example highlights the importance of checking model assumptions, whether for missing or observed data.     

A non-homogeneous skew-Gaussian Bayesian spatial model

In spatial statistics, models are often constructed based on some common, but possible restrictive assumptions for the underlying spatial process, including Gaussianity as well as stationarity and isotropy. However, these assumptions are frequently violated in applied problems. In order to simultaneously handle skewness and non-homogeneity (i.e., non-stationarity and anisotropy), we develop the fixed rank kriging model through the use of skew-normal distribution for its non-spatial latent variables. Our approach to spatial modeling is easy to implement and also provides a great flexibility in adjusting to skewed and large datasets with heterogeneous correlation structures. We adopt a Bayesian framework for our analysis, and describe a simple MCMC algorithm for sampling from the posterior distribution of the model parameters and performing spatial prediction. Through a simulation study, we demonstrate that the proposed model could detect departures from normality and, for illustration, we analyze a synthetic dataset of CO\(_2\) measurements. Finally, to deal with multivariate spatial data showing some degree of skewness, a multivariate extension of the model is also provided.     

A Bayesian analysis of clustered discrete and continuous outcomes

Many study designs yield a variety of outcomes from each subject clustered within an experimental unit. When these outcomes are of mixed data types, it is challenging to jointly model the effects of covariates on the responses using traditional methods. In this paper, we develop a Bayesian approach for a joint regression model of the different outcome variables and show that the fully conditional posterior distributions obtained under the model assumptions allow for estimation of posterior distributions using Gibbs sampling algorithm.     

Alternative posterior consistency results in nonparametric binary regression using Gaussian process priors

We establish consistency of posterior distribution when a Gaussian process prior is used as a prior distribution for the unknown binary regression function. Specifically, we take the work of Ghosal and Roy [2006. Posterior consistency of Gaussian process prior for nonparametric binary regression. Ann. Statist. 34, 2413–2429] as our starting point, and then weaken their assumptions on the smoothness of the Gaussian process kernel while retaining a stronger yet applicable condition about design points. Furthermore, we extend their results to multi-dimensional covariates under a weaker smoothness condition on the Gaussian process. Finally, we study the extent to which posterior consistency can be achieved under a general model where, when additional hyperparameters in the covariance function of a Gaussian process are involved.     

A non-iterative Bayesian sampling algorithm for censored Student-t linear regression models

ABSTRACT

In this paper, we consider an effective Bayesian inference for censored Student-t linear regression model, which is a robust alternative to the usual censored Normal linear regression model. Based on the mixture representation of the Student-t distribution, we propose a non-iterative Bayesian sampling procedure to obtain independently and identically distributed samples approximately from the observed posterior distributions, which is different from the iterative Markov Chain Monte Carlo algorithm. We conduct model selection and influential analysis using the posterior samples to choose the best fitted model and to detect latent outliers. We illustrate the performance of the procedure through simulation studies, and finally, we apply the procedure to two real data sets, one is the insulation life data with right censoring and the other is the wage rates data with left censoring, and we get some interesting results.     

Diagnostic tools for approximate Bayesian computation using the coverage property

Approximate Bayesian computation (ABC) is an approach to sampling from an approximate posterior distribution in the presence of a computationally intractable likelihood function. A common implementation is based on simulating model, parameter and dataset triples from the prior, and then accepting as samples from the approximate posterior, those model and parameter pairs for which the corresponding dataset, or a summary of that dataset, is ‘close’ to the observed data. Closeness is typically determined though a distance measure and a kernel scale parameter. Appropriate choice of that parameter is important in producing a good quality approximation. This paper proposes diagnostic tools for the choice of the kernel scale parameter based on assessing the coverage property, which asserts that credible intervals have the correct coverage levels in appropriately designed simulation settings. We provide theoretical results on coverage for both model and parameter inference, and adapt these into diagnostics for the ABC context. We re‐analyse a study on human demographic history to determine whether the adopted posterior approximation was appropriate. Code implementing the proposed methodology is freely available in the R package abctools .     

Robust linear mixed models for Small Area Estimation

Hierarchical models are popular in many applied statistics fields including Small Area Estimation. One well known model employed in this particular field is the Fay–Herriot model, in which unobservable parameters are assumed to be Gaussian. In Hierarchical models assumptions about unobservable quantities are difficult to check. For a special case of the Fay–Herriot model, Sinharay and Stern [2003. Posterior predictive model checking in Hierarchical models. J. Statist. Plann. Inference 111, 209–221] showed that violations of the assumptions about the random effects are difficult to detect using posterior predictive checks. In this present paper we consider two extensions of the Fay–Herriot model in which the random effects are assumed to be distributed according to either an exponential power (EP) distribution or a skewed EP distribution. We aim to explore the robustness of the Fay–Herriot model for the estimation of individual area means as well as the empirical distribution function of their ‘ensemble’. Our findings, which are based on a simulation experiment, are largely consistent with those of Sinharay and Stern as far as the efficient estimation of individual small area parameters is concerned. However, when estimating the empirical distribution function of the ‘ensemble’ of small area parameters, results are more sensitive to the failure of distributional assumptions.     

A Bayesian nonparametric causal model

Typically, in the practice of causal inference from observational studies, a parametric model is assumed for the joint population density of potential outcomes and treatment assignments, and possibly this is accompanied by the assumption of no hidden bias. However, both assumptions are questionable for real data, the accuracy of causal inference is compromised when the data violates either assumption, and the parametric assumption precludes capturing a more general range of density shapes (e.g., heavier tail behavior and possible multi-modalities). We introduce a flexible, Bayesian nonparametric causal model to provide more accurate causal inferences. The model makes use of a stick-breaking prior, which has the flexibility to capture any multi-modalities, skewness and heavier tail behavior in this joint population density, while accounting for hidden bias. We prove the asymptotic consistency of the posterior distribution of the model, and illustrate our causal model through the analysis of small and large observational data sets.     

Scalable Bayes under Informative Sampling

Bayesian hierarchical formulations are utilized by the U.S. Bureau of Labor Statistics (BLS) with respondent‐level data for missing item imputation because these formulations are readily parameterized to capture correlation structures. BLS collects survey data under informative sampling designs that assign probabilities of inclusion to be correlated with the response on which sampling‐weighted pseudo posterior distributions are estimated for asymptotically unbiased inference about population model parameters. Computation is expensive and does not support BLS production schedules. We propose a new method to scale the computation that divides the data into smaller subsets, estimates a sampling‐weighted pseudo posterior distribution, in parallel, for every subset and combines the pseudo posterior parameter samples from all the subsets through their mean in the Wasserstein space of order 2. We construct conditions on a class of sampling designs where posterior consistency of the proposed method is achieved. We demonstrate on both synthetic data and in application to the Current Employment Statistics survey that our method produces results of similar accuracy as the usual approach while offering substantially faster computation.     

Alternatives to post‐processing posterior predictive p values

The posterior predictive p value (ppp) was invented as a Bayesian counterpart to classical p values. The methodology can be applied to discrepancy measures involving both data and parameters and can, hence, be targeted to check for various modeling assumptions. The interpretation can, however, be difficult since the distribution of the ppp value under modeling assumptions varies substantially between cases. A calibration procedure has been suggested, treating the ppp value as a test statistic in a prior predictive test. In this paper, we suggest that a prior predictive test may instead be based on the expected posterior discrepancy, which is somewhat simpler, both conceptually and computationally. Since both these methods require the simulation of a large posterior parameter sample for each of an equally large prior predictive data sample, we furthermore suggest to look for ways to match the given discrepancy by a computation‐saving conflict measure. This approach is also based on simulations but only requires sampling from two different distributions representing two contrasting information sources about a model parameter. The conflict measure methodology is also more flexible in that it handles non‐informative priors without difficulty. We compare the different approaches theoretically in some simple models and in a more complex applied example.     

PubPredict: Prediction of progression and survival in oncology leveraging publications and early efficacy data

In oncology/hematology early phase clinical trials, efficacies were often observed in terms of response rate, depth, timing, and duration. However, the true clinical benefits that eventually support registrational purpose are progression-free survival (PFS) and/or overall survival (OS), the follow-up of which are typically not long enough in early phase trials. This gap imposes challenges in strategies for late phase drug development. In this article, we tackle the question by leveraging published study to establish a quantitative link between early efficacy outcomes and late phase efficacy endpoints. We used solid tumor cancer as disease model. We modeled the disease course of a RECISTv1.1 assessed solid tumor with a continuous Markov chain (CMC) model. We parameterize the transition intensity matrix of a CMC model based on published aggregate-level summary statistics, and then simulate subject-level time-to-event data. The simulated data is shown to have good approximation to published studies. PFS and/or OS could be predicted with the transition intensity matrix modified given clinical knowledge to reflect various assumptions on response rate, depth, timing, and duration. The authors have built a R shiny application named PubPredict, the tool implements the algorithm described above and allows customized features including multiple response levels, treatment crossover and varying follow-up duration. This toolset has been applied to advise phase 3 trial design when only early efficacy data are available from phase 1 or 2 studies.     

Quantifying uncertainty in transdimensional Markov chain Monte Carlo using discrete Markov models

Bayesian analysis often concerns an evaluation of models with different dimensionality as is necessary in, for example, model selection or mixture models. To facilitate this evaluation, transdimensional Markov chain Monte Carlo (MCMC) relies on sampling a discrete indexing variable to estimate the posterior model probabilities. However, little attention has been paid to the precision of these estimates. If only few switches occur between the models in the transdimensional MCMC output, precision may be low and assessment based on the assumption of independent samples misleading. Here, we propose a new method to estimate the precision based on the observed transition matrix of the model-indexing variable. Assuming a first-order Markov model, the method samples from the posterior of the stationary distribution. This allows assessment of the uncertainty in the estimated posterior model probabilities, model ranks, and Bayes factors. Moreover, the method provides an estimate for the effective sample size of the MCMC output. In two model selection examples, we show that the proposed approach provides a good assessment of the uncertainty associated with the estimated posterior model probabilities.

     

Exact and efficient Bayesian inference for multiple changepoint problems

We demonstrate how to perform direct simulation from the posterior distribution of a class of multiple changepoint models where the number of changepoints is unknown. The class of models assumes independence between the posterior distribution of the parameters associated with segments of data between successive changepoints. This approach is based on the use of recursions, and is related to work on product partition models. The computational complexity of the approach is quadratic in the number of observations, but an approximate version, which introduces negligible error, and whose computational cost is roughly linear in the number of observations, is also possible. Our approach can be useful, for example within an MCMC algorithm, even when the independence assumptions do not hold. We demonstrate our approach on coal-mining disaster data and on well-log data. Our method can cope with a range of models, and exact simulation from the posterior distribution is possible in a matter of minutes.     

A BAYESIAN INTERPRETATION OF MULTIPLE POINT ESTIMATES

Consider a large number of econometric investigations using different estimation techniques and/or different subsets of all available data to estimate a fixed set of parameters. The resulting empirical distribution of point estimates can be shown - under suitable conditions - to coincide with a Bayesian posterior measure on the parameter space induced by a minimum information procedure. This Bayesian interpretation makes it easier to combine the results of various empirical exercises for statistical decision making. The collection of estimators may be generated by one investigator to ensure the satisfaction of our conditions, or they may be collected from published works, where behavioral assumptions need to be made regarding the dependence structure of econometric studies.