Riemann manifold Langevin methods on stochastic volatility estimation

In this article, we perform Bayesian estimation of stochastic volatility models with heavy tail distributions using Metropolis adjusted Langevin (MALA) and Riemman manifold Langevin (MMALA) methods. We provide analytical expressions for the application of these methods, assess the performance of these methodologies in simulated data, and illustrate their use on two financial time series datasets.     

Adaptive Step Size Selection for Hessian‐Based Manifold Langevin Samplers

In this paper, I explore the usage of positive definite metric tensors derived from the second derivative information in the context of the simplified manifold Metropolis adjusted Langevin algorithm. I propose a new adaptive step size procedure that resolves the shortcomings of such metric tensors in regions where the log‐target has near zero curvature in some direction. The adaptive step size selection also appears to alleviate the need for different tuning parameters in transient and stationary regimes that is typical of Metropolis adjusted Langevin algorithm. The combination of metric tensors derived from the second derivative information and the adaptive step size selection constitute a large step towards developing reliable manifold Markov chain Monte Carlo methods that can be implemented automatically for models with unknown or intractable Fisher information, and even for target distributions that do not admit factorization into prior and likelihood. Through examples of low to moderate dimension, I show that the proposed methodology performs very well relative to alternative Markov chain Monte Carlo methods.     

Optimal scaling of Metropolis algorithms: Heading toward general target distributions

The authors provide an overview of optimal scaling results for the Metropolis algorithm with Gaussian proposal distribution. They address in more depth the case of high‐dimensional target distributions formed of independent, but not identically distributed components. They attempt to give an intuitive explanation as to why the well‐known optimal acceptance rate of 0.234 is not always suitable. They show how to find the asymptotically optimal acceptance rate when needed, and they explain why it is sometimes necessary to turn to inhomogeneous proposal distributions. Their results are illustrated with a simple example.     

Bayesian Truncated Poisson Regression with Application to Dutch Illegal Immigrant Data

This article presents a Bayesian approach to the regression analysis of truncated data, with a focus on zero-truncated counts from the Poisson distribution. The approach provides inference not only on the regression coefficients but also on the total sample size and the parameters of the covariate distribution. The theory is applied to some illegal immigrant data from The Netherlands. Several models are fitted with the aid of Markov chain Monte Carlo methods and assessed via posterior predictive p-values. Inferences are compared with those obtained elsewhere using other approaches.     

On the Value of derivative evaluations and random walk suppression in Markov Chain Monte Carlo algorithms

Two strategies that can potentially improve Markov Chain Monte Carlo algorithms are to use derivative evaluations of the target density, and to suppress random walk behaviour in the chain. The use of one or both of these strategies has been investigated in a few specific applications, but neither is used routinely. We undertake a broader evaluation of these techniques, with a view to assessing their utility for routine use. In addition to comparing different algorithms, we also compare two different ways in which the algorithms can be applied to a multivariate target distribution. Specifically, the univariate version of an algorithm can be applied repeatedly to one-dimensional conditional distributions, or the multivariate version can be applied directly to the target distribution.     

Scaling limits for the transient phase of local Metropolis–Hastings algorithms

Summary.  The paper considers high dimensional Metropolis and Langevin algorithms in their initial transient phase. In stationarity, these algorithms are well understood and it is now well known how to scale their proposal distribution variances. For the random-walk Metropolis algorithm, convergence during the transient phase is extremely regular—to the extent that the algo-rithm's sample path actually resembles a deterministic trajectory. In contrast, the Langevin algorithm with variance scaled to be optimal for stationarity performs rather erratically. We give weak convergence results which explain both of these types of behaviour and practical guidance on implementation based on our theory.     

Understanding the Hastings Algorithm

The Hastings algorithm is a key tool in computational science. While mathematically justified by detailed balance, it can be conceptually difficult to grasp. Here, we present two complementary and intuitive ways to derive and understand the algorithm. In our framework, it is straightforward to see that the celebrated Metropolis–Hastings algorithm has the highest acceptance probability of all Hastings algorithms.     

Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm

Two new implementations of the EM algorithm are proposed for maximum likelihood fitting of generalized linear mixed models. Both methods use random (independent and identically distributed) sampling to construct Monte Carlo approximations at the E-step. One approach involves generating random samples from the exact conditional distribution of the random effects (given the data) by rejection sampling, using the marginal distribution as a candidate. The second method uses a multivariate t importance sampling approximation. In many applications the two methods are complementary. Rejection sampling is more efficient when sample sizes are small, whereas importance sampling is better with larger sample sizes. Monte Carlo approximation using random samples allows the Monte Carlo error at each iteration to be assessed by using standard central limit theory combined with Taylor series methods. Specifically, we construct a sandwich variance estimate for the maximizer at each approximate E-step. This suggests a rule for automatically increasing the Monte Carlo sample size after iterations in which the true EM step is swamped by Monte Carlo error. In contrast, techniques for assessing Monte Carlo error have not been developed for use with alternative implementations of Monte Carlo EM algorithms utilizing Markov chain Monte Carlo E-step approximations. Three different data sets, including the infamous salamander data of McCullagh and Nelder, are used to illustrate the techniques and to compare them with the alternatives. The results show that the methods proposed can be considerably more efficient than those based on Markov chain Monte Carlo algorithms. However, the methods proposed may break down when the intractable integrals in the likelihood function are of high dimension.     

A joint modeling approach for spatial earthquake risk variations

Modeling spatial patterns and processes to assess the spatial variations of data over a study region is an important issue in many fields. In this paper, we focus on investigating the spatial variations of earthquake risks after a main shock. Although earthquake risks have been extensively studied in the literatures, to our knowledge, there does not exist a suitable spatial model for assessing the problem. Therefore, we propose a joint modeling approach based on spatial hierarchical Bayesian models and spatial conditional autoregressive models to describe the spatial variations in earthquake risks over the study region during two periods. A family of stochastic algorithms based on a Markov chain Monte Carlo technique is then performed for posterior computations. The probabilistic issue for the changes of earthquake risks after a main shock is also discussed. Finally, the proposed method is applied to the earthquake records for Taiwan before and after the Chi-Chi earthquake.     

Differential Evolution Markov Chain with snooker updater and fewer chains

Differential Evolution Markov Chain (DE-MC) is an adaptive MCMC algorithm, in which multiple chains are run in parallel. Standard DE-MC requires at least N=2d chains to be run in parallel, where d is the dimensionality of the posterior. This paper extends DE-MC with a snooker updater and shows by simulation and real examples that DE-MC can work for d up to 50–100 with fewer parallel chains (e.g. N=3) by exploiting information from their past by generating jumps from differences of pairs of past states. This approach extends the practical applicability of DE-MC and is shown to be about 5–26 times more efficient than the optimal Normal random walk Metropolis sampler for the 97.5% point of a variable from a 25–50 dimensional Student t ₃ distribution. In a nonlinear mixed effects model example the approach outperformed a block-updater geared to the specific features of the model.     

An Analysis of Swendsen–Wang and Related Sampling Methods

Convergence rates, statistical efficiency and sampling costs are studied for the original and extended Swendsen–Wang methods of generating a sample path { S _j , j ≥1} with equilibrium distribution π , with r distinct elements, on a finite state space X of size N ₁. Given S _{j -1}, each method uses auxiliary random variables to identify the subset of X from which S _j is to be randomly sampled. Let π_min and π_max denote respectively the smallest and largest elements in π and let N_r denote the number of elements in π with value π_max. For a single auxiliary variable, uniform sampling from the subset and ( N ₁− N_r )π_min+ N_r π_max≈1, our results show rapid convergence and high statistical efficiency for large π_min/π_max or N_r / N ₁ and slow convergence and poor statistical efficiency for small π_min/π_max and N_r / N₁ . Other examples provide additional insight. For extended Swendsen–Wang methods with non-uniform subset sampling, the analysis identifies the properties of a decomposition of π( x ) that favour fast convergence and high statistical efficiency. In the absence of exploitable special structure, subset sampling can be costly regardless of which of these methods is employed.     

Markov Chains and De-initializing Processes

We define a notion of de-initializing Markov chains. We prove that to analyse convergence of Markov chains to stationarity, it suffices to analyse convergence of a de-initializing chain. Applications are given to Markov chain Monte Carlo algorithms and to convergence diagnostics.     

Estimation of origin–destination trip rates in Leicester

The road system in region RA of Leicester has vehicle detectors embedded in many of the network's road links. Vehicle counts from these detectors can provide transportation researchers with a rich source of data. However, for many projects it is necessary for researchers to have an estimate of origin-to-destination vehicle flow rates. Obtaining such estimates from data observed on individual road links is a non-trivial statistical problem, made more difficult in the present context by non-negligible measurement errors in the vehicle counts collected. The paper uses road link traffic count data from April 1994 to estimate the origin–destination flow rates for region RA. A model for the error prone traffic counts is developed, but the resulting likelihood is not available in closed form. Nevertheless, it can be smoothly approximated by using Monte Carlo integration. The approximate likelihood is combined with prior information from a May 1991 survey in a Bayesian framework. The posterior is explored using the Hastings–Metropolis algorithm, since its normalizing constant is not available. Preliminary findings suggest that the data are overdispersed according to the original model. Results for a revised model indicate that a degree of overdispersion exists, but that the estimates of origin–destination flow rates are quite insensitive to the change in model specification.     

A geometric approach to transdimensional markov chain monte carlo

The authors present theoretical results that show how one can simulate a mixture distribution whose components live in subspaces of different dimension by reformulating the problem in such a way that observations may be drawn from an auxiliary continuous distribution on the largest subspace and then transformed in an appropriate fashion. Motivated by the importance of enlarging the set of available Markov chain Monte Carlo (MCMC) techniques, the authors show how their results can be fruitfully employed in problems such as model selection (or averaging) of nested models, or regeneration of Markov chains for evaluating standard deviations of estimated expectations derived from MCMC simulations.     

Bayesian multivariate GARCH models with dynamic correlations and asymmetric error distributions

The main goal in this paper is to develop and apply stochastic simulation techniques for GARCH models with multivariate skewed distributions using the Bayesian approach. Both parameter estimation and model comparison are not trivial tasks and several approximate and computationally intensive methods (Markov chain Monte Carlo) will be used to this end. We consider a flexible class of multivariate distributions which can model both skewness and heavy tails. Also, we do not fix tail behaviour when dealing with fat tail distributions but leave it subject to inference.     

Generalized multiple-point Metropolis algorithms for approximate Bayesian computation

It is well known that the approximate Bayesian computation algorithm based on Markov chain Monte Carlo methods suffers from the sensitivity to the choice of starting values, inefficiency and a low acceptance rate. To overcome these problems, this study proposes a generalization of the multiple-point Metropolis algorithm, which proceeds by generating multiple-dependent proposals and then by selecting a candidate among the set of proposals on the basis of weights that can be chosen arbitrarily. The performance of the proposed algorithm is illustrated by using both simulated and real data.     

Adaptive optimal scaling of Metropolis–Hastings algorithms using the Robbins–Monro process

ABSTRACT

We present an adaptive method for the automatic scaling of random-walk Metropolis–Hastings algorithms, which quickly and robustly identifies the scaling factor that yields a specified overall sampler acceptance probability. Our method relies on the use of the Robbins–Monro search process, whose performance is determined by an unknown steplength constant. Based on theoretical considerations we give a simple estimator of this constant for Gaussian proposal distributions. The effectiveness of our method is demonstrated with both simulated and real data examples.     

Bayesian image analysis with Markov chain Monte Carlo and coloured continuum triangulation models

It is now possible to carry out Bayesian image segmentation from a continuum parametric model with an unknown number of regions. However, few suitable parametric models exist. We set out to model processes which have realizations that are naturally described by coloured planar triangulations. Triangulations are already used, to represent image structure in machine vision, and in finite element analysis, for domain decomposition. However, no normalizable parametric model, with realizations that are coloured triangulations, has been specified to date. We show how this must be done, and in particular we prove that a normalizable measure on the space of triangulations in the interior of a fixed simple polygon derives from a Poisson point process of vertices. We show how such models may be analysed by using Markov chain Monte Carlo methods and we present two case-studies, including convergence analysis.     

Sampling based approach for one-hit and multi-hit models in quantal bioassay

Bayesian methods for estimating the dose response curves with the one-hit model, the gamma multi-hit model, and their modified versions with Abbott's correction are studied. The Gibbs sampling approach with data augmentation and with the Metropolis algorithm is employed to compute the Bayes estimates of the potency curves. In addition, estimation of the relative additional risk and the virtually safe dose is studied. Model selection based on conditional predictive ordinates from cross-validated data is developed.     

Improving Convergence of the HastingsMetropolis Algorithm with an Adaptive Proposal

The HastingsMetropolis algorithm is a general MCMC method for sampling from a density known up to a constant. Geometric convergence of this algorithm has been proved under conditions relative to the instrumental (or proposal) distribution. We present an inhomogeneous HastingsMetropolis algorithm for which the proposal density approximates the target density, as the number of iterations increases. The proposal density at the n th step is a non-parametric estimate of the density of the algorithm, and uses an increasing number of i.i.d. copies of the Markov chain. The resulting algorithm converges (in n ) geometrically faster than a HastingsMetropolis algorithm with any fixed proposal distribution. The case of a strictly positive density with compact support is presented first, then an extension to more general densities is given. We conclude by proposing a practical way of implementation for the algorithm, and illustrate it over simulated examples.