Dani Gamerman

Sampling from the posterior distribution in generalized linear mixed models

Generalized linear mixed models provide a unified framework for treatment of exponential family regression models, overdispersed data and longitudinal studies. These problems typically involve the presence of random effects and this paper presents a new methodology for making Bayesian inference about them. The approach is simulation-based and involves the use of Markov chain Monte Carlo techniques. The usual iterative weighted least squares algorithm is extended to include a sampling step based on the Metropolis–Hastings algorithm thus providing a unified iterative scheme. Non-normal prior distributions for the regression coefficients and for the random effects distribution are considered. Random effect structures with nesting required by longitudinal studies are also considered. Particular interests concern the significance of regression coefficients and assessment of the form of the random effects. Extensions to unknown scale parameters, unknown link functions, survival and frailty models are outlined.

Bayesian analysis of extreme events with threshold estimation

The aim of this paper is to analyse extremal events using generalized Pareto distributions (GPD), considering explicitly the uncertainty about the threshold. Current practice empirically determines this quantity and proceeds by estimating the GPD parameters on the basis of data beyond it, discarding all the information available below the threshold. We introduce a mixture model that combines a parametric form for the center and a GPD for the tail of the distributions and uses all observations for inference about the unknown parameters from both distributions, the threshold included. Prior distributions for the parameters are indirectly obtained through experts quantiles elicitation. Posterior inference is available through Markov chain Monte Carlo methods. Simulations are carried out in order to analyse the performance of our proposed model under a wide range of scenarios. Those scenarios approximate realistic situations found in the literature. We also apply the proposed model to a real dataset, Nasdaq 100, an index of the financial market that presents many extreme events. Important issues such as predictive analysis and model selection are considered along with possible modeling extensions.

Spatial dynamic factor analysis

A new class of space-time models derived from standard dynamic factor models is proposed. The temporal dependence is modeled by latent factors while the spatial dependence is modeled by the factor loadings. Factor analytic arguments are used to help identify temporal components that summarize most of the spatial variation of a given region. The temporal evolution of the factors is described in a number of forms to account for different aspects of time variation such as trend and seasonality. The spatial dependence is incorporated into the factor loadings by a combination of deterministic and stochastic elements thus giving them more flexibility and generalizing previous approaches. The new structure implies nonseparable space-time variation to observables, despite its conditionally independent nature, while reducing the overall dimensionality, and hence complexity, of the problem. The number of factors is treated as another unknown parameter and fully Bayesian inference is performed via a reversible jump Markov Chain Monte Carlo algorithm. The new class of models is tested against one synthetic dataset and applied to pollution data obtained from the Clean Air Status and Trends Network (CASTNet). Our factor model exhibited better predictive performance when compared to benchmark models, while capturing important aspects of spatial and temporal behavior of the data.

Generalized spatial dynamic factor models

This paper introduces a new class of spatio-temporal models for measurements belonging to the exponential family of distributions. In this new class, the spatial and temporal components are conditionally independently modeled via a latent factor analysis structure for the (canonical) transformation of the measurements mean function. The factor loadings matrix is responsible for modeling spatial variation, while the common factors are responsible for modeling the temporal variation. One of the main advantages of our model with spatially structured loadings is the possibility of detecting similar regions associated to distinct dynamic factors. We also show that the new class outperforms a large class of spatial-temporal models that are commonly used in the literature. Posterior inference for fixed parameters and dynamic latent factors is performed via a custom tailored Markov chain Monte Carlo scheme for multivariate dynamic systems that combines extended Kalman filter-based Metropolis-Hastings proposal densities with block-sampling schemes. Factor model uncertainty is also fully addressed by a reversible jump Markov chain Monte Carlo algorithm designed to learn about the number of common factors. Three applications, two based on synthetic Gamma and Bernoulli data and one based on real Bernoulli data, are presented in order to illustrate the flexibility and generality of the new class of models, as well as to discuss features of the proposed MCMC algorithm.

An Integrated Bayesian Model for DIF Analysis

In this article, an integrated bayesian model for differential item functioning (DIF) analysis is proposed. The model is integrated in the sense of modeling the responses along with the DIF analysis. This approach allows DIF detection and explanation in a simultaneous setup. Previous empirical studies and/or subjective beliefs about the item parameters, including differential functioning behavior, may be conveniently expressed in terms of prior distributions. Values of indicator variables are estimated in the model, indicating which items have DIF and which do not; as a result, the data analyst may not be required to specify an “anchor set” of items that do not exhibit DIF a priori to identify the model. It reduces the iterative procedures that are commonly used for proficiency purification and DIF detection and explanation. Examples demonstrate the efficiency of this method in simulated and real situations.

A Non‐Gaussian Family of State‐Space Models with Exact Marginal Likelihood

The Gaussian assumption generally employed in many state‐space models is usually not satisfied for real time series. Thus, in this work, a broad family of non‐Gaussian models is defined by integrating and expanding previous work in the literature. The expansion is obtained at two levels: at the observational level, it allows for many distributions not previously considered, and at the latent state level, it involves an expanded specification for the system evolution. The class retains analytical availability of the marginal likelihood function, uncommon outside Gaussianity. This expansion considerably increases the applicability of the models and solves many previously existing problems such as long‐term prediction, missing values and irregular temporal spacing. Inference about the state components can be performed because of the introduction of a new and exact smoothing procedure, in addition to filtered distributions. Inference for the hyperparameters is presented from the classical and Bayesian perspectives. The results seem to indicate competitive results of the models when compared with other non‐Gaussian state‐space models available. The methodology is applied to Gaussian and non‐Gaussian dynamic linear models with time‐varying means and variances and provides a computationally simple solution to inference in these models. The methodology is illustrated in a number of examples.

A semiparametric Bayesian approach to extreme value estimation

This paper is concerned with extreme value density estimation. The generalized Pareto distribution (GPD) beyond a given threshold is combined with a nonparametric estimation approach below the threshold. This semiparametric setup is shown to generalize a few existing approaches and enables density estimation over the complete sample space. Estimation is performed via the Bayesian paradigm, which helps identify model components. Estimation of all model parameters, including the threshold and higher quantiles, and prediction for future observations is provided. Simulation studies suggest a few useful guidelines to evaluate the relevance of the proposed procedures. They also provide empirical evidence about the improvement of the proposed methodology over existing approaches. Models are then applied to environmental data sets. The paper is concluded with a few directions for future work.

Bayesian spatiotemporal model of fMRI data

This research describes a new Bayesian spatiotemporal model to analyse block-design BOLD fMRI studies. In the temporal dimension, we parameterise the hemodynamic response functions (HRF) shape with a potential increase of signal and a subsequent exponential decay. In the spatial dimension, we use Gaussian Markov random fields (GMRF) priors on activation characteristics parameters (location and magnitude) that embody our prior knowledge that evoked responses are spatially contiguous and locally homogeneous. The result is a spatiotemporal model with a small number of parameters, all of them interpretable. Simulations from the model are performed in order to ascertain the performance of the sampling scheme and the ability of the posterior to estimate model parameters, as well as to check the model sensitivity to signal to noise ratio. Results are shown on synthetic data and on real data from a block-design fMRI experiment.

Transfer functions in dynamic generalized linear models

In a time series analysis it is sometimes necessary to assume that the effect of a regressor does not have only immediate impact on the mean response, but that its effects somehow propagate to future times. We adopt, in this work, transfer functions to model such impacts, represented by structural blocks present in dynamic generalized linear models. All the inference is carried under the Bayesian paradigm. Two sources of difficulties emerge for the analytical derivation of posterior distributions: non-Gaussian nature of the response, associated to non-conjugate priors and also non-linearity of the predictor on auto regressive parameters present in transfer functions. The purpose of this work is to produce full Bayesian inference on dynamic generalized linear models with transfer functions, using Markov chain Monte Carlo methods to build samples of the posterior joint distribution of the parameters involved in such models. Several transfer structures are specified, associated to Poisson, Binomial, Gamma and inverse Gaussian responses. Simulated data are analyzed under the resulting models in order to assess their performance. Finally, two applications to real data concerning environmental sciences are made under different model formulations.

Dynamic hierarchical models: an extension to matrix-variate observations

Abstract In this paper we propose a dynamic hierarchical model formulation in an environment where the observations are matrix normal random variables. First, we present the model assuming known variance–covariance matrices, except for a common scale factor matrix. With such an assumption one can perform conjugate analysis of the model by specifying a Normal-inverted-Wishart prior distribution. We proceed by extending the model to the case where all variance–covariance matrices are unknown and propose an approach based on the Gibbs sampler to estimate all model parameters. A sampling-based approach to obtain forecasts is also presented. Examples with artificial and real data are presented.