Abstract: This paper considers Bayesian counterparts of the classical tests for goodness of fit and their use in judging the fit of a single Bayesian model to the observed data. We focus on posterior predictive assessment, in a framework that also includes conditioning on auxiliary statistics. The Bayesian formulation facilitates the construction and calculation of a meaningful reference distribution not only for any (classical) statistic, but also for any parameter-dependent “statistic ” or discrepancy. The latter allows us to propose the realized discrepancy assessment of model fitness, which directly measures the true discrepancy between data and the posited model, for any aspect of the model which we want to explore. The computation required for the realized discrepancy assessment is a straightforward byproduct of the posterior simulation used for the original Bayesian analysis. We illustrate with three applied examples. The first example, which serves mainly to motivate the work, illustrates the difficulty of classical tests in assessing the fitness of a Poisson model to a positron emission tomography image that is constrained to be nonnegative. The second and third examples illustrate the details of the posterior predictive approach in two problems: estimation in a model with inequality constraints on the parameters, and estimation in a mixture model. In all three examples, standard test statistics (either a χ 2 or a likelihood ratio) are not pivotal: the difficulty is not just how to compute the reference distribution for the test, but that in the classical framework no such distribution exists, independent of the unknown model parameters. Key words and phrases: Bayesian p-value, χ 2 test, discrepancy, graphical assessment, mixture model, model criticism, posterior predictive p-value, prior predictive

this article then is to develop methodology for modeling the nonnormality of the ut, the vt, or both. A second departure from the model specification ( 1 ) is to allow for unknown variances in the state or observational equation, as well as for unknown parameters in the transition matrices Ft and Ht. As a third generalization we allow for nonlinear model structures; that is, X t = ft(Xt-l) q- Ut, and Yt = ht(xt) + vt, t = 1, ..., n, (2) whereft( ) and ht(. ) are given, but perhaps also depend on some unknown parameters. The experimenter may wish to entertain a variety of error distributions. Our goal throughout the article is an analysis for general state-space models that does not resort to convenient assumptions at the expense of model adequacy

rapolation techniques, especially exponential smoothing and exponentially weighted moving average methods ([20, 71]). Developments of smoothing and discounting techniques in stock control and production planning areas led to formalisms in terms of linear, state-space models for time series with time-varying trends and seasonal patterns, and eventually to the associated Bayesian formalism of methods of inference and prediction. From the early 1960s, practical Bayesian forecasting systems in this context involved the combination of formal time series models and historical data analysis together with methods for subjective intervention and forecast monitoring, so that complete forecasting systems, rather than just routine and automatic data analysis and extrapolation, were in use at that time ([19, 22]). Methods developed in those early days are still in use now in some companies in sales forecasting and stock control areas. There have been major developments in models and methods since t

In this paper we develop particle learning (PL) methods for state filtering, sequential parameter learning and smoothing in a general class of nonlinear state space models. The approach extends existing particle methods by incorporating static parameters and utilizing sufficient statistics for the parameters and/or the states as particles. State smoothing with parameter uncertainty is also solved as a by product of particle learning. In a number of applications, we show that our algorithms outperform existing particle filtering algorithms as well as MCMC.

A state space model for multivariate longitudinal count data driven by a latent gamma Markov process is proposed, the observed counts being conditionally independent and Poisson distributed given the latent process. We consider regression analysis for this model with time-varying covariates entering either via the Poisson model or via the latent gamma process. We develop the Kalman filter and smoother and investigate estimation based on the EM algorithm with the E-step approximated by the Kalman smoother. We also consider analysis of residuals from both the Poisson model and the gamma process. Key words: EM algorithm; Estimation function; Generalized linear model; Kalman filter; Kalman smoother; Latent process; Mixed Poisson distribution; Overdispersion; Random effects; Regression model; Residual analysis; Time-varying covariates. 1 Introduction An important problem in longitudinal data analysis is the development of models for nonnormal response variables. We consider a k-dimensiona...

This dissertation introduces new simulation-based analysis approaches, including both sequential and off-line learning algorithms, for various Bayesian time series models. We provide a Markov Chain Monte Carlo (MCMC) method for an autoregressive (AR) model with innovations following exponential power distributions using the fact that an exponential power distribution is a scale mixture of normals. This model has application in signal processing, specifically image processing, with orthogonal wave...

This research investigates a method that attempts to represent a database of expert decisions as an independent probability distribution. To implement this representation, our method searches for a simple probability model that exhibits maximum independence property and preserves a given set of inequality constraints. We will show that finding such a representation can be formulated as a search problem over a log probability space with a representational complexity in a linear order of the number of variables. We will show the search can be achieved by employing linear programming technique in combination with a greedy (best first) search algorithm.

No. DMS-0112069. Any opinions, findings, and conclusions or recommendations expressed in this

Bayesian time series and forecasting is a very broad field and any attempt at