An algorithm, the bootstrap filter, is proposed for implementing recursive Bayesian filters. The required density of the state vector is represented as a set of random samples, which are updated and propagated by the algorithm. The method is not restricted by assumptions of linear- ity or Gaussian noise: it may be applied to any state transition or measurement model. A simula- tion example of the bearings only tracking problem is presented. This simulation includes schemes for improving the efficiency of the basic algorithm. For this example, the performance of the bootstrap filter is greatly superior to the standard extended Kalman filter.

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Novel approach to nonlinear/non-Gaussian Bayesian state estimation

Monte Carlo methods are revolutionizing the on-line analysis of data in fields as diverse as financial modeling, target tracking and computer vision. These methods, appearing under the names of bootstrap filters, condensation, optimal Monte Carlo filters, particle filters and survival of the fittest, have made it possible to solve numerically many complex, non-standard problems that were previously intractable. This book presents the first comprehensive treatment of these techniques, including convergence results and applications to tracking, guidance, automated target recognition, aircraft navigation, robot navigation, econometrics, financial modeling, neural networks, optimal control, optimal filtering, communications, reinforcement learning, signal enhancement, model averaging and selection, computer vision, semiconductor design, population biology, dynamic Bayesian networks, and time series analysis. This will be of great value to students, researchers and practitioners, who have some basic knowledge of probability. Arnaud Doucet received the Ph. D. degree from the University of Paris-XI Orsay in 1997. From 1998 to 2000, he conducted research at the Signal Processing Group of Cambridge University, UK. He is currently an assistant professor at the Department of Electrical Engineering of Melbourne University, Australia. His research interests include Bayesian statistics, dynamic models and Monte Carlo methods. Nando de Freitas obtained a Ph.D. degree in information engineering from Cambridge University in 1999. He is presently a research associate with the artificial intelligence group of the University of California at Berkeley. His main research interests are in Bayesian statistics and the application of on-line and batch Monte Carlo methods to machine learning. Neil Gordon obtained a Ph.D. in Statistics from Imperial College, University of London in 1993. He is with the Pattern and Information Processing group at the Defence Evaluation and Research Agency in the United Kingdom. His research interests are in time series, statistical data analysis, and pattern recognition with a particular emphasis on target tracking and missile guidance.

/pdf/sequential-monte-carlo-methods-in-practice-3kfxeebn8e.pdf

Sequential Monte Carlo methods in practice

Statistics for Spatial Data.

Statistical approaches to overdispersion, correlated errors, shrinkage estimation, and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model (GLMM). Given an unobserved vector of random effects, observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function, known prior weights and a scale factor. The random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components. For problems involving time series, spatial aggregation and smoothing, the dispersion may be specified in terms of a rank deficient inverse covariance matrix. Approximation of the marginal quasi-likelihood using Laplace's method leads eventually to estimating equations based on penalized quasilikelihood or PQL for the mean parameters and pseudo-likelihood for the variances. Im...

https://www.jstor.org/stable/pdfplus/2290687.pdf

Approximate inference in generalized linear mixed models

A new algorithm for the prediction, filtering, and smoothing of non-Gaussian nonlinear state space models is shown. The algorithm is based on a Monte Carlo method in which successive prediction, filtering (and subsequently smoothing), conditional probability density functions are approximated by many of their realizations. The particular contribution of this algorithm is that it can be applied to a broad class of nonlinear non-Gaussian higher dimensional state space models on the provision that the dimensions of the system noise and the observation noise are relatively low. Several numerical examples are shown.

https://www.jstor.org/stable/pdfplus/1390750.pdf

Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models

Dynamic Bayesian models are developed for application in nonlinear, non-normal time series and regression problems, providing dynamic extensions of standard generalized linear models. A key feature of the analysis is the use of conjugate prior and posterior distributions for the exponential family parameters. This leads to the calculation of closed, standard-form predictive distributions for forecasting and model criticism. The structure of the models depends on the time evolution of underlying state variables, and the feedback of observational information to these variables is achieved using linear Bayesian prediction methods. Data analytic aspects of the models concerning scale parameters and outliers are discussed, and some applications are provided. Dynamic Bayesian models are developed for application in nonlinear, non-normal time series and regression problems, providing dynamic extensions of standard generalized linear models. A key feature of the analysis is the use of conjugate prior and...

https://stat.duke.edu/~mw/MWextrapubs/West1985a.pdf

Dynamic Generalized Linear Models and Bayesian Forecasting

Introduction Information The concept of probability Assessing subjective probabilities An example Linear algebra and probability Notation Outline of the book Elements of Inference Common statistical models Likelihood-based functions Bayes theorem Exchangeability Sufficiency and exponential family Parameter elimination Prior Distribution Entirely subjective specification Specification through functional forms Conjugacy with the exponential family Non-informative priors Hierarchical priors Estimation Introduction to decision theory Bayesian point estimation Classical point estimation Empirical Bayes estimation Comparison of estimators Interval estimation Estimation in the Normal model Approximating Methods The general problem of inference Optimization techniques Asymptotic theory Other analytical approximations Numerical integration methods Simulation methods Hypothesis Testing Introduction Classical hypothesis testing Bayesian hypothesis testing Hypothesis testing and confidence intervals Asymptotic tests Prediction Bayesian prediction Classical prediction Prediction in the Normal model Linear prediction Introduction to Linear Models The linear model Classical estimation of linear models Bayesian estimation of linear models Hierarchical linear models Dynamic linear models Linear models with constraints Sketched Solutions to Selected Exercises List of Distributions References Index Exercises appear at the end of each chapter.

Statistical inference : an integrated approach

We develop a Bayesian analysis based on two different Jeffreys priors for the Student-t regression model with unknown degrees of freedom. It is typically difficult to estimate the number of degrees of freedom: improper prior distributions may lead to improper posterior distributions, whereas proper prior distributions may dominate the analysis. We show that Bayesian analysis with either of the two considered Jeffreys priors provides a proper posterior distribution. Finally, we show that Bayesian estimators based on Jeffreys analysis compare favourably to other Bayesian estimators based on priors previously proposed in the literature.

Objective Bayesian analysis for the Student-t regression model

An analysis of a time series of cross-sectional data is considered under a Bayesian perspective. Information is modelled in terms of prior distributions and stratified parametric linear models developed by Lindley and Smith and dynamic linear models developed by Harrison and Stevens are merged into a general framework. This is shown to include many models proposed in econometrics and experimental design. Properties of the model are derived and shrinkage estimators reassessed. Evolution, smoothing and passage of data information through the levels of the hierarchy are discussed. Inference with an unknown scalar observation variance is drawn and an extension to the non-linear case is proposed

Dynamic Hierarchical Models

http://www.dme.im.ufrj.br/arquivos/publicacoes/arquivo228.pdf

Helio S. Migon

Papers

Dynamic Generalized Linear Models and Bayesian Forecasting

Statistical inference : an integrated approach

Objective Bayesian analysis for the Student-t regression model

Dynamic Hierarchical Models

Dynamic Bayesian beta models