Abstract. This chapter provides a survey of the main methods in non-uniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various algorithms, before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods.

We describe and illustrate Bayesian inference in models for density estimation using mixtures of Dirichlet processes. These models provide natural settings for density estimation, and are exemplified by special cases where data are modelled as a sample from mixtures of normal distributions. Efficient simulation methods are used to approximate various prior, posterior and predictive distributions. This allows for direct inference on a variety of practical issues, including problems of local versus global smoothing, uncertainty about density estimates, assessment of modality, and the inference on the numbers of components. Also, convergence results are established for a general class of normal mixture models. Keywords: Kernel estimation; Mixtures of Dirichlet processes; Multimodality; Normal mixtures; Posterior sampling; Smoothing parameter estimation * Michael D. Escobar is Assistant Professor, Department of Statistics and Department of Preventive Medicine and Biostatistics, University ...

One popular method for the recovery of an ideal intensity image from corrupted or indirect measurements is regularization: minimize an objective function which enforces a roughness penalty in addition to coherence with the data. Linear estimates are relatively easy to compute but generally introduce systematic errors; for example, they are incapable of recovering discontinuities and other important image attributes. In contrast, nonlinear estimates are more accurate, but often far less accessible. This is particularly true when the objective function is non-convex and the distribution of each data component depends on many image components through a linear operator with broad support. Our approach is based on an auxiliary array and an extended objective function in which the original variables appear quadratically and the auxiliary variables are decoupled. Minimizing over the auxiliary array alone yields the original function, so the original image estimate can be obtained by joint min...

In order to realize an input-output relation given by noise-contaminated examples, it is effective to use a stochastic model of neural networks. A model network includes hidden units whose activation values are not specified nor observed. It is useful to estimate the hidden variables from the observed or specified input-output data based on the stochastic model. Two algorithms, the EM - and em-algorithms, have so far been proposed for this purpose. The EM-algorithm is an iterative statistical technique of using the conditional expectation, and the em-algorithm is a geometrical one given by information geometry. The em-algorithm minimizes iteratively the Kullback-Leibler divergence in the manifold of neural networks. These two algorithms are equivalent in most cases. The present paper gives a unified information geometrical framework for studying stochastic models of neural networks, by forcussing on the EM and em algorithms, and proves a condition which guarantees their equ...

this paper, a special Metropolis-Hastings type algorithm, Metropolized independent sampling, proposed firstly in Hastings (1970), is studied in full detail. The eigenvalues and eigenvectors of the corresponding Markov chain, as well as a sharp bound for the total variation distance between the n-th updated distribution and the target distribution, are provided. Furthermore, the relationship between this scheme, rejection sampling, and importance sampling are studied with emphasizes on their relative efficiencies. It is shown that Metropolized independent sampling is superior to rejection sampling in two aspects: asymptotic efficiency and ease of computation. Key Words: Coupling, Delta method, Eigen analysis, Importance ratio. 1 1 Introduction

Markov chain sampling has received considerable attention in the recent literature, in particular in the context of Bayesian computation and maximum likelihood estimation. This paper discusses the use of Markov chain splitting, originally developed as a tool for the theoretical analysis of general state space Markov chains, to introduce regeneration times into Markov chain samplers. This allows the use of regenerative methods for analyzing the output of these samplers, and can also provide a useful diagnostic of the performance of the samplers. The general approach is applied to several different samplers and is illustrated in a number of examples. 1 Introduction In Markov chain Monte Carlo, a distribution ß is examined by obtaining sample paths from a Markov chain constructed to have equilibrium distribution ß. This approach was introduced by Metropolis et al. (1953) and has recently received considerable attention as a method for examining posterior distributions in Bayesian infer...

Markov chain Monte Carlo (MCMC) sampling strategies can be used to simulate hidden Markov model (HMM) parameters from their posterior distribution given observed data. Some MCMC methods (for computing likelihood, conditional probabilities of hidden states, and the most likely sequence of states) used in practice can be improved by incorporating established recursive algorithms. The most important is a set of forward-backward recursions calculating conditional distributions of the hidden states given observed data and model parameters. We show how to use the recursive algorithms in an MCMC context and demonstrate mathematical and empirical results showing a Gibbs sampler using the forward-backward recursions mixes more rapidly than another sampler often used for HMM's. We introduce an augmented variables technique for obtaining unique state labels in HMM's and finite mixture models. We show how recursive computing allows statistically efficient use of MCMC output when estimating the hidden states. We directly calculate the posterior distribution of the hidden chain's state space size by MCMC, circumventing asymptotic arguments underlying the Bayesian information criterion, which is shown to be inappropriate for a frequently analyzed data set in the HMM literature. The use of log-likelihood for assessing MCMC convergence is illustrated, and posterior predictive checks are used to investigate application specific questions of model adequacy.

We consider the problem of inference about a quantity of interest given different sources of information linked by a deterministic population dynamics model. Our approach consists of translating all the available information into a joint pre-model distribution on all the model inputs and outputs, and then restricting this to the submanifold defined by the model to obtain the joint post-model distribution. Marginalizing this yields inference, conditional on the model, about quantities of interest which can be functions of model inputs, model outputs, or both. Samples from the postmodel distribution are obtained by importance sampling and Rubin's SIR algorithm. The framework includes as a special case the situation where the pre-model information about the outputs consists of measurements with error; this reduces to standard Bayesian inference. The results are in the form of a sample from the post-model distribution and so can be examined using the full range of exploratory data analysis...

this paper we develop an original and general framework for automatically optimizing the statistical properties of Markov chain Monte Carlo (MCMC) samples, which are typically used to evaluate complex integrals. The Metropolis-Hastings algorithm is the basic building block of classical MCMC methods and requires the choice of a proposal distribution, which usually belongs to a parametric family. The correlation properties together with the exploratory ability of the Markov chain heavily depend on the choice of the proposal distribution. By monitoring the simulated path, our approach allows us to learn "on the fly" the optimal parameters of the proposal distribution for several statistical criteria. Keywords: Monte Carlo, adaptive MCMC, calibration, stochastic approximation, gradient method, optimal scaling, random walk, Langevin, Gibbs, controlled Markov chain, learning algorithm, reversible jump MCMC. 1. Motivation 1.1. Introduction Markov chain Monte Carlo (MCMC) is a general strategy for generating samples x i (i = 0; 1; : : :) from complex high-dimensional distributions, say defined on the space X ae R nx , from which integrals of the type I (f) = Z X f (x) (x) dx; can be calculated using the estimator b I N (f) = 1 N + 1 N X i=0 f (x i ) ; provided that the Markov chain produced is ergodic. The main building block of this class of algorithms is the Metropolis-Hastings (MH) algorithm. It requires the definition of a proposal distribution q whose role is to generate possible transitions for the Markov chain, say from x to y, which are then accepted or rejected according to the probabilityy ff (x; y) = min ae 1; (y) q (y; x) (x) q (x; y) oe : The simplicity and universality of this algorithm are both its strength and weakness. The choice of ...

A new algorithm is presented for generating the conditional mean estimates of functions of target positions, orientations and type in recognition, and tracking of an unknown nmnber of targets and target types. Taking a Bayesian approach, a posterior measure is defined on the tracking/target parameter space by combining a narrowband sensor array manifold model with a high resolution imaging model, and a prior based on airplane dynanfics. The Newtoninn force equations governing rigid body dynamic s are utilized to form the prior density on airplane motion. The conditional mean estimates are generated using a random sampling algorithm based on jump-diffusion processes [1] i)r empirically generating MMSE estimates of functions of these random target positions, orientations, and type under the posterior measure. Results are presented on target tracking and identification from an implementation of the algorithm on a networked Silicon Graphics workstation and DECmpp/MasPar parallel machine.