Yanan Fan

Sample Quantiles in Statistical Packages

Abstract There are a large number of different definitions used for sample quantiles in statistical computer packages. Often within the same package one definition will be used to compute a quantile explicitly, while other definitions may be used when producing a boxplot, a probability plot, or a QQ plot. We compare the most commonly implemented sample quantile definitions by writing them in a common notation and investigating their motivation and some of their properties. We argue that there is a need to adopt a standard definition for sample quantiles so that the same answers are produced by different packages and within each package. We conclude by recommending that the median-unbiased estimator be used because it has most of the desirable properties of a quantile estimator and can be defined independently of the underlying distribution.

Sequential Monte Carlo without likelihoods.

Recent new methods in Bayesian simulation have provided ways of evaluating posterior distributions in the presence of analytically or computationally intractable likelihood functions. Despite representing a substantial methodological advance, existing methods based on rejection sampling or Markov chain Monte Carlo can be highly inefficient and accordingly require far more iterations than may be practical to implement. Here we propose a sequential Monte Carlo sampler that convincingly overcomes these inefficiencies. We demonstrate its implementation through an epidemiological study of the transmission rate of tuberculosis.

Sexual dimorphism, survival and dispersal in red deer

A detailed and extensive mark-recapture-recovery study of red deer on the island of Rum forms the basis of the modeling of this article. We analyze male and female deer separately, and report results for both in this article, but use the female data to demonstrate our modeling approach. We provide a model-selection procedure that allows us to describe the survival by a combination of age-classes, with common survival within each class, and senility, which is modeled continuously as a parametric function of age. Dispersal out of the study area is modeled separately. Survival and dispersal probabilities are examined for the possible influence of both environmental and individual covariates, including a range of alternative measures of population density. The resulting model is succinct and biologically realistic. We compare and contrast survival rates of male and female deer of different ages and compare the factors that affect their survival. We demonstrate large differences in the rate of senescence between males and females even though their senescence begins at the same age. The differences between the sexes suggest that, in population modeling of sexually size-dimorphic species, it is important to identify sex-specific survival functions.

On sequential Monte Carlo, partial rejection control and approximate Bayesian computation

We present a variant of the sequential Monte Carlo sampler by incorporating the partial rejection control mechanism of Liu (2001). We show that the resulting algorithm can be considered as a sequential Monte Carlo sampler with a modified mutation kernel. We prove that the new sampler can reduce the variance of the incremental importance weights when compared with standard sequential Monte Carlo samplers, and provide a central limit theorem. Finally, the sampler is adapted for application under the challenging approximate Bayesian computation modelling framework.

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.

Approximate Bayesian Computation and Bayes’ Linear Analysis: Toward High-Dimensional ABC

Bayes’ linear analysis and approximate Bayesian computation (ABC) are techniques commonly used in the Bayesian analysis of complex models. In this article, we connect these ideas by demonstrating that regression-adjustment ABC algorithms produce samples for which first- and second-order moment summaries approximate adjusted expectation and variance for a Bayes’ linear analysis. This gives regression-adjustment methods a useful interpretation and role in exploratory analysis in high-dimensional problems. As a result, we propose a new method for combining high-dimensional, regression-adjustment ABC with lower-dimensional approaches (such as using Markov chain Monte Carlo for ABC). This method first obtains a rough estimate of the joint posterior via regression-adjustment ABC, and then estimates each univariate marginal posterior distribution separately in a lower-dimensional analysis. The marginal distributions of the initial estimate are then modified to equal the separately estimated marginals, thereby providing an improved estimate of the joint posterior. We illustrate this method with several examples. Supplementary materials for this article are available online.

Perfect forward simulation via simulated tempering

Several authors discuss how the simulated tempering scheme provides a very simple mechanism for introducing regenerations within a Markov chain. In this article we explain how regenerative simulated tempering schemes provide a very natural mechanism for perfect simulation. We use this to provide a perfect simulation algorithm, which uses a single-sweep forward-simulation without the need for recursively searching through negative times. We demonstrate this algorithm in the context of several examples.

Bayesian modelling of prehistoric corbelled domes

The field of archaeology provides a rich source of complex, non-standard problems ideally suited to Bayesian inference. We discuss the application of Bayesian methodology to prehistoric corbelled tomb data collected from a variety of sites around Europe. We show how the corresponding analyses may be carried out with the aid of reversible jump Markov chain Monte Carlo simulation techniques and, by calculating posterior model probabilities, we show how to distinguish between competing models. In particular, we discuss how earlier analyses of tomb data by Cavanagh and Laxton and by Buck and co-workers, where structural changes are anticipated in the shape of the tomb at different depths, can be extended and improved by considering a wider range of models. We also discuss the extent to which these analyses may be useful in addressing questions concerning the origin of tomb building technologies, particularly in distinguishing between corbelled domes built by different civilizations, as well as the processes involved in their construction.

Output Assessment for Monte Carlo Simulations via the Score Statistic

This article presents several applications of the score statistic in the context of output assessment for Monte Carlo simulations. We begin by observing that the expected value of the score statistic U is zero, and that when the inverse of the information matrix I = E(UUT) exists, the asymptotic distribution of UTI−1U is χ2. Thus, we may monitor the sample mean of this statistic throughout a simulation as a means to determine whether or not the simulation has been run for a sufficiently long time. We also demonstrate a second convergence assessment method based upon the idea of path sampling, but first show how the score statistic can be used to accurately estimate the stationary density using only a small number of simulated values. These methods provide a powerful suite of tools which can be generically applied when alternatives such as the Rao-Blackwell density estimator are not available. Our second convergence assessment method is based upon these density estimates. By running several replications of the chain, the corresponding estimated densities may be compared to assess how “close” the chains are to one another and to the true stationary distribution. We explain how this may be done using both L1 and L2 distance measures. We first illustrate these new methods via the analysis of MCMC output arising from some simulated examples, emphasizing the advantages of our methods over existing diagnostics. We further illustrate the utility of our methods with three examples: analyzing a set of real time series data, a collection of censored survival data, and bivariate normal data using a model with a nonidentified parameter.

Automating and evaluating reversible jump MCMC proposal distributions

The reversible jump Markov chain Monte Carlo (MCMC) sampler (Green in Biometrika 82:711–732, 1995) has become an invaluable device for Bayesian practitioners. However, the primary difficulty with the sampler lies with the efficient construction of transitions between competing models of possibly differing dimensionality and interpretation. We propose the use of a marginal density estimator to construct between-model proposal distributions. This provides both a step towards black-box simulation for reversible jump samplers, and a tool to examine the utility of common between-model mapping strategies. We compare the performance of our approach to well established alternatives in both time series and mixture model examples.