Inference for Lévy‐Driven Stochastic Volatility Models via Adaptive Sequential Monte Carlo

Abstract. We investigate simulation methodology for Bayesian inference in Lévy‐driven stochastic volatility (SV) models. Typically, Bayesian inference from such models is performed using Markov chain Monte Carlo (MCMC); this is often a challenging task. Sequential Monte Carlo (SMC) samplers are methods that can improve over MCMC; however, there are many user‐set parameters to specify. We develop a fully automated SMC algorithm, which substantially improves over the standard MCMC methods in the literature. To illustrate our methodology, we look at a model comprised of a Heston model with an independent, additive, variance gamma process in the returns equation. The driving gamma process can capture the stylized behaviour of many financial time series and a discretized version, fit in a Bayesian manner, has been found to be very useful for modelling equity data. We demonstrate that it is possible to draw exact inference, in the sense of no time‐discretization error, from the Bayesian SV model.     

Perfect simulation for Reed-Frost epidemic models

The Reed-Frost epidemic model is a simple stochastic process with parameter q that describes the spread of an infectious disease among a closed population. Given data on the final outcome of an epidemic, it is possible to perform Bayesian inference for q using a simple Gibbs sampler algorithm. In this paper it is illustrated that by choosing latent variables appropriately, certain monotonicity properties hold which facilitate the use of a perfect simulation algorithm. The methods are applied to real data.     

On cross‐validation of Bayesian models

The authors examine several aspects of cross‐validation for Bayesian models. In particular, they propose a computational scheme which does not require a separate posterior sample for each training sample.     

BAYESIAN ANALYSIS OF THE ADDITIVE MIXED MODEL FOR RANDOMIZED BLOCK DESIGNS

This paper deals with the Bayesian analysis of the additive mixed model experiments. Consider b randomly chosen subjects who respond once to each of t treatments. The subjects are treated as random effects and the treatment effects are fixed. Suppose that some prior information is available, thus motivating a Bayesian analysis. The Bayesian computation, however, can be difficult in this situation, especially when a large number of treatments is involved. Three computational methods are suggested to perform the analysis. The exact posterior density of any parameter of interest can be simulated based on random realizations taken from a restricted multivariate t distribution. The density can also be simulated using Markov chain Monte Carlo methods. The simulated density is accurate when a large number of random realizations is taken. However, it may take substantial amount of computer time when many treatments are involved. An alternative Laplacian approximation is discussed. The Laplacian method produces smooth and very accurate approximates to posterior densities, and takes only seconds of computer time. An example of a pipeline cracks experiment is used to illustrate the Bayesian approaches and the computational methods.     

A dynamic power prior for borrowing historical data in noninferiority trials with binary endpoint

Traditionally, noninferiority hypotheses have been tested using a frequentist method with a fixed margin. Given that information for the control group is often available from previous studies, it is interesting to consider a Bayesian approach in which information is “borrowed” for the control group to improve efficiency. However, construction of an appropriate informative prior can be challenging. In this paper, we consider a hybrid Bayesian approach for testing noninferiority hypotheses in studies with a binary endpoint. To account for heterogeneity between the historical information and the current trial for the control group, a dynamic P value–based power prior parameter is proposed to adjust the amount of information borrowed from the historical data. This approach extends the simple test‐then‐pool method to allow a continuous discounting power parameter. An adjusted α level is also proposed to better control the type I error. Simulations are conducted to investigate the performance of the proposed method and to make comparisons with other methods including test‐then‐pool and hierarchical modeling. The methods are illustrated with data from vaccine clinical trials.     

Sampling the Future: A Bayesian Approach to Forecasting From Univariate Time Series Models

The Box–Jenkins methodology for modeling and forecasting from univariate time series models has long been considered a standard to which other forecasting techniques have been compared. To a Bayesian statistician, however, the method lacks an important facet—a provision for modeling uncertainty about parameter estimates. We present a technique called sampling the future for including this feature in both the estimation and forecasting stages. Although it is relatively easy to use Bayesian methods to estimate the parameters in an autoregressive integrated moving average (ARIMA) model, there are severe difficulties in producing forecasts from such a model. The multiperiod predictive density does not have a convenient closed form, so approximations are needed. In this article, exact Bayesian forecasting is approximated by simulating the joint predictive distribution. First, parameter sets are randomly generated from the joint posterior distribution. These are then used to simulate future paths of the time series. This bundle of many possible realizations is used to project the future in several ways. Highest probability forecast regions are formed and portrayed with computer graphics. The predictive density's shape is explored. Finally, we discuss a method that allows the analyst to subjectively modify the posterior distribution on the parameters and produce alternate forecasts.     

Bayesian Additive Machine: classification with a semiparametric discriminant function

In this paper, we propose a new Bayesian inference approach for classification based on the traditional hinge loss used for classical support vector machines, which we call the Bayesian Additive Machine (BAM). Unlike existing approaches, the new model has a semiparametric discriminant function where some feature effects are nonlinear and others are linear. This separation of features is achieved automatically during model fitting without user pre-specification. Following the literature on sparse regression of high-dimensional models, we can also identify the irrelevant features. By introducing spike-and-slab priors using two sets of indicator variables, these multiple goals are achieved simultaneously and automatically, without any parameter tuning such as cross-validation. An efficient partially collapsed Markov chain Monte Carlo algorithm is developed for posterior exploration based on a data augmentation scheme for the hinge loss. Our simulations and three real data examples demonstrate that the new approach is a strong competitor to some approaches that were proposed recently for dealing with challenging classification examples with high dimensionality.     

A Bayesian random effects model for analyzing mixed negative binomial and continuous longitudinal responses

ABSTRACT

A general Bayesian random effects model for analyzing longitudinal mixed correlated continuous and negative binomial responses with and without missing data is presented. This Bayesian model, given some random effects, uses a normal distribution for the continuous response and a negative binomial distribution for the count response. A Markov Chain Monte Carlo sampling algorithm is described for estimating the posterior distribution of the parameters. This Bayesian model is illustrated by a simulation study. For sensitivity analysis to investigate the change of parameter estimates with respect to the perturbation from missing at random to not missing at random assumption, the use of posterior curvature is proposed. The model is applied to a medical data, obtained from an observational study on women, where the correlated responses are the negative binomial response of joint damage and continuous response of body mass index. The simultaneous effects of some covariates on both responses are also investigated.     

A Bayesian Approach to Modelling Reticulation Events with Application to the Ribosomal Protein Gene rps11 of Flowering Plants

Traditional phylogenetic inference assumes that the history of a set of taxa can be explained by a tree. This assumption is often violated as some biological entities can exchange genetic material giving rise to non‐treelike events often called reticulations. Failure to consider these events might result in incorrectly inferred phylogenies. Phylogenetic networks provide a flexible tool which allows researchers to model the evolutionary history of a set of organisms in the presence of reticulation events. In recent years, a number of methods addressing phylogenetic network parameter estimation have been introduced. Some of them are based on the idea that a phylogenetic network can be defined as a directed acyclic graph. Based on this definition, we propose a Bayesian approach to the estimation of phylogenetic network parameters which allows for different phylogenies to be inferred at different parts of a multiple DNA alignment. The algorithm is tested on simulated data and applied to the ribosomal protein gene rps11 data from five flowering plants, where reticulation events are suspected to be present. The proposed approach can be applied to a wide variety of problems which aim at exploring the possibility of reticulation events in the history of a set of taxa.     

Bayesian prediction of waiting times in stochastic models

The authors show how saddlepoint techniques lead to highly accurate approximations for Bayesian predictive densities and cumulative distribution functions in stochastic model settings where the prior is tractable, but not necessarily the likelihood or the predictand distribution. They consider more specifically models involving predictions associated with waiting times for semi‐Markov processes whose distributions are indexed by an unknown parameter θ. Bayesian prediction for such processes when they are not stationary is also addressed and the inverse‐Gaussian based saddlepoint approximation of Wood, Booth & Butler (1993) is shown to accurately deal with the nonstationarity whereas the normal‐based Lugannani & Rice (1980) approximation cannot, Their methods are illustrated by predicting various waiting times associated with M/M/q and M/G/1 queues. They also discuss modifications to the matrix renewal theory needed for computing the moment generating functions that are used in the saddlepoint methods.     

Addressing potential prior‐data conflict when using informative priors in proof‐of‐concept studies

Bayesian methods are increasingly used in proof‐of‐concept studies. An important benefit of these methods is the potential to use informative priors, thereby reducing sample size. This is particularly relevant for treatment arms where there is a substantial amount of historical information such as placebo and active comparators. One issue with using an informative prior is the possibility of a mismatch between the informative prior and the observed data, referred to as prior‐data conflict. We focus on two methods for dealing with this: a testing approach and a mixture prior approach. The testing approach assesses prior‐data conflict by comparing the observed data to the prior predictive distribution and resorting to a non‐informative prior if prior‐data conflict is declared. The mixture prior approach uses a prior with a precise and diffuse component. We assess these approaches for the normal case via simulation and show they have some attractive features as compared with the standard one‐component informative prior. For example, when the discrepancy between the prior and the data is sufficiently marked, and intuitively, one feels less certain about the results, both the testing and mixture approaches typically yield wider posterior‐credible intervals than when there is no discrepancy. In contrast, when there is no discrepancy, the results of these approaches are typically similar to the standard approach. Whilst for any specific study, the operating characteristics of any selected approach should be assessed and agreed at the design stage; we believe these two approaches are each worthy of consideration. Copyright © 2015 John Wiley & Sons, Ltd.     

Modified spline regression based on randomly right-censored data: A comparative study

ABSTRACT

In this paper, we propose modified spline estimators for nonparametric regression models with right-censored data, especially when the censored response observations are converted to synthetic data. Efficient implementation of these estimators depends on the set of knot points and an appropriate smoothing parameter. We use three algorithms, the default selection method (DSM), myopic algorithm (MA), and full search algorithm (FSA), to select the optimum set of knots in a penalized spline method based on a smoothing parameter, which is chosen based on different criteria, including the improved version of the Akaike information criterion (AICc), generalized cross validation (GCV), restricted maximum likelihood (REML), and Bayesian information criterion (BIC). We also consider the smoothing spline (SS), which uses all the data points as knots. The main goal of this study is to compare the performance of the algorithm and criteria combinations in the suggested penalized spline fits under censored data. A Monte Carlo simulation study is performed and a real data example is presented to illustrate the ideas in the paper. The results confirm that the FSA slightly outperforms the other methods, especially for high censoring levels.     

SPACE–TIME MODELLING OF SYDNEY HARBOUR WINDS

This paper develops a space‐time statistical model for local forecasting of surface‐level wind fields in a coastal region with complex topography. The statistical model makes use of output from deterministic numerical weather prediction models which are able to produce forecasts of surface wind fields on a spatial grid. When predicting surface winds at observing stations , errors can arise due to sub‐grid scale processes not adequately captured by the numerical weather prediction model , and the statistical model attempts to correct for these influences. In particular , it uses information from observing stations within the study region as well as topographic information to account for local bias. Bayesian methods for inference are used in the model , with computations carried out using Markov chain Monte Carlo algorithms. Empirical performance of the model is described , illustrating that a structured Bayesian approach to complicated space‐time models of the type considered in this paper can be readily implemented and can lead to improvements in forecasting over traditional methods.     

Bayesian analysis of outlier problems using the Gibbs sampler

We consider the Bayesian analysis of outlier models. We show that the Gibbs sampler brings considerable conceptual and computational simplicity to the problem of calculating posterior marginals. Although other techniques for finding posterior marginals are available, the Gibbs sampling approach is notable for its ease of implementation. Allowing the probability of an outlier to be unknown introduces an extra parameter into the model but this turns out to involve only minor modification to the algorithm. We illustrate these ideas using a contaminated Gaussian distribution, at-distribution, a contaminated binomial model and logistic regression.     

Using Bayesian analysis in repeated preclinical in vivo studies for a more effective use of animals

Whilst innovative Bayesian approaches are increasingly used in clinical studies, in the preclinical area Bayesian methods appear to be rarely used in the reporting of pharmacology data. This is particularly surprising in the context of regularly repeated in vivo studies where there is a considerable amount of data from historical control groups, which has potential value. This paper describes our experience with introducing Bayesian analysis for such studies using a Bayesian meta‐analytic predictive approach. This leads naturally either to an informative prior for a control group as part of a full Bayesian analysis of the next study or using a predictive distribution to replace a control group entirely. We use quality control charts to illustrate study‐to‐study variation to the scientists and describe informative priors in terms of their approximate effective numbers of animals. We describe two case studies of animal models: the lipopolysaccharide‐induced cytokine release model used in inflammation and the novel object recognition model used to screen cognitive enhancers, both of which show the advantage of a Bayesian approach over the standard frequentist analysis. We conclude that using Bayesian methods in stable repeated in vivo studies can result in a more effective use of animals, either by reducing the total number of animals used or by increasing the precision of key treatment differences. This will lead to clearer results and supports the “3Rs initiative” to Refine, Reduce and Replace animals in research. Copyright © 2016 John Wiley & Sons, Ltd.     

Bayesian inference in the triangular cointegration model using a jeffreys prior

This paper presents a strategy for conducting Bayesian inference in the triangular cointegration model. A Jeffreys prior is used to circumvent an identification problem in the parameter region in which there is a near lack of cointegration. Sampling experiments are used to compare the repeated sampling performance of the approach with alternative classical cointegration methods. The Bayesian procedure is applied to testing for substitution between private and public consumption for a range of countries, with posterior estimates produced via Markov Chain Monte Carlo simulators.     

An efficient Monte Carlo EM algorithm for Bayesian lasso

The lasso is a popular technique of simultaneous estimation and variable selection in many research areas. The marginal posterior mode of the regression coefficients is equivalent to estimates given by the non-Bayesian lasso when the regression coefficients have independent Laplace priors. Because of its flexibility of statistical inferences, the Bayesian approach is attracting a growing body of research in recent years. Current approaches are primarily to either do a fully Bayesian analysis using Markov chain Monte Carlo (MCMC) algorithm or use Monte Carlo expectation maximization (MCEM) methods with an MCMC algorithm in each E-step. However, MCMC-based Bayesian method has much computational burden and slow convergence. Tan et al. [An efficient MCEM algorithm for fitting generalized linear mixed models for correlated binary data. J Stat Comput Simul. 2007;77:929–943] proposed a non-iterative sampling approach, the inverse Bayes formula (IBF) sampler, for computing posteriors of a hierarchical model in the structure of MCEM. Motivated by their paper, we develop this IBF sampler in the structure of MCEM to give the marginal posterior mode of the regression coefficients for the Bayesian lasso, by adjusting the weights of importance sampling, when the full conditional distribution is not explicit. Simulation experiments show that the computational time is much reduced with our method based on the expectation maximization algorithm and our algorithms and our methods behave comparably with other Bayesian lasso methods not only in prediction accuracy but also in variable selection accuracy and even better especially when the sample size is relatively large.     

A Bayesian approach to zero-inflated data in extremes

Abstract

The generalized extreme value (GEV) distribution is known as the limiting result for the modeling of maxima blocks of size n, which is used in the modeling of extreme events. However, it is possible for the data to present an excessive number of zeros when dealing with extreme data, making it difficult to analyze and estimate these events by using the usual GEV distribution. The Zero-Inflated Distribution (ZID) is widely known in literature for modeling data with inflated zeros, where the inflator parameter w is inserted. The present work aims to create a new approach to analyze zero-inflated extreme values, that will be applied in data of monthly maximum precipitation, that can occur during months where there was no precipitation, being these computed as zero. An inference was made on the Bayesian paradigm, and the parameter estimation was made by numerical approximations of the posterior distribution using Markov Chain Monte Carlo (MCMC) methods. Time series of some cities in the northeastern region of Brazil were analyzed, some of them with predominance of non-rainy months. The results of these applications showed the need to use this approach to obtain more accurate and with better adjustment measures results when compared to the standard distribution of extreme value analysis.     

Model weights for model choice and averaging

A method is suggested to estimate posterior model probabilities and model averaged parameters via MCMC sampling under a Bayesian approach. The estimates use pooled output for J models (J>1) whereby all models are updated at each iteration. Posterior probabilities are based on averages of continuous weights obtained for each model at each iteration, while samples of averaged parameters are obtained from iteration specific averages that are based on these weights. Parallel sampling of models assists in deriving posterior densities for parameter contrasts between models and in assessing hypotheses regarding model averaged parameters. Four worked examples illustrate application of the approach, two involving fixed effect regression, and two involving random effects.