Some critical aspects of the use of exchangeability in statistics

Summary This expository paper provides a framework for analysing de Finetti's representation theorem for exchangeable finitely additive probabilities. Such an analysis is justified by reasoning of statistical nature, since it is shown that the abandonment of the axiom of σ-additivity has some noteworthy consequences on the common interpretation of the Bayesian paradigm. The usual (strong) fromulation of de Finetti's theorem is deduced from the finitely additive (weak) formulation, and it is used to solve the problem of stating the existence of a stochastic process, with given finite-dimensional probability distributions, whose sample paths are probability distributions. It is of importance, in particular, to specify prior distributions for nonparametric inferential problems in a Bayesian setting. Research partially supported by MPI (40% 1990, Gruppo Nazionale ?Modelli Probabilistici e Statistica Matematica?).     

Book Reviews

This article uses a Bayesian unit-root test in stochastic volatility models. The time series of interest is the volatility that is unobservable. The unit-root testing is based on the posterior odds ratio, which is approximated by Markov-chain Monte Carlo methods. Simulations show that the testing procedure is efficient for moderate sample size. The unit-root hypothesis is rejected in seven market indexes, and some evidence of nonstationarity is observed in the TWSI of Taiwan.     

Efficient Bayesian inference in generalized inverse gamma processes for stochastic volatility

This paper develops a novel and efficient algorithm for Bayesian inference in inverse Gamma stochastic volatility models. It is shown that by conditioning on auxiliary variables, it is possible to sample all the volatilities jointly directly from their posterior conditional density, using simple and easy to draw from distributions. Furthermore, this paper develops a generalized inverse gamma process with more flexible tails in the distribution of volatilities, which still allows for simple and efficient calculations. Using several macroeconomic and financial datasets, it is shown that the inverse gamma and generalized inverse gamma processes can greatly outperform the commonly used log normal volatility processes with Student’s t errors or jumps in the mean equation.     

Rate estimation in partially observed Markov jump processes with measurement errors

We present a simulation methodology for Bayesian estimation of rate parameters in Markov jump processes arising for example in stochastic kinetic models. To handle the problem of missing components and measurement errors in observed data, we embed the Markov jump process into the framework of a general state space model. We do not use diffusion approximations. Markov chain Monte Carlo and particle filter type algorithms are introduced which allow sampling from the posterior distribution of the rate parameters and the Markov jump process also in data-poor scenarios. The algorithms are illustrated by applying them to rate estimation in a model for prokaryotic auto-regulation and the stochastic Oregonator, respectively.     

基于辅助粒子滤波的DSGE模型识别方法及应用

动态随机一般均衡模型中涵盖无法直接观测的变量,同时跨方程约束涉及复杂的非线性关系使方程的解析估计难以实现。在贝叶斯框架下识别动态随机一般均衡模型,基于状态空间方法建立度量方程和状态转移方程,采用辅助粒子滤波预测条件后验分布,建立贝叶斯误差带描述宏观经济变量脉冲响应函数的动态特征。实际数据分析验证了贝叶斯识别方法的有效性。     

On asymptotic properties of Bayesian partially linear models

In this paper, we present large sample properties of a partially linear model from the Bayesian perspective, in which responses are explained by the semiparametric regression model with the additive form of the linear component and the nonparametric component. For this purpose, we investigate asymptotic behaviors of posterior distributions in terms of consistency. Specifically, we deal with a specific Bayesian partially linear regression model with additive noises in which the nonparametric component is modeled using Gaussian process priors. Under the Bayesian partially linear model using Gaussian process priors, we focus on consistency of posterior distribution and consistency of the Bayes factor, and extend these results to generalized additive regression models and study their asymptotic properties. In addition we illustrate the asymptotic properties based on empirical analysis through simulation studies.     

Efficient and flexible model-based clustering of jumps in diffusion processes

Jump–diffusion processes involving diffusion processes with discontinuous movements, called jumps, are widely used to model time-series data that commonly exhibit discontinuity in their sample paths. The existing jump–diffusion models have been recently extended to multivariate time-series data. The models are, however, still limited by a single parametric jump-size distribution that is common across different subjects. Such strong parametric assumptions for the shape and structure of a jump-size distribution may be too restrictive and unrealistic for multiple subjects with different characteristics. This paper thus proposes an efficient Bayesian nonparametric method to flexibly model a jump-size distribution while borrowing information across subjects in a clustering procedure using a nested Dirichlet process. For efficient posterior computation, a partially collapsed Gibbs sampler is devised to fit the proposed model. The proposed methodology is illustrated through a simulation study and an application to daily stock price data for companies in the S&P 100 index from June 2007 to June 2017.     

Consistency of Bayesian linear model selection with a growing number of parameters

Linear models with a growing number of parameters have been widely used in modern statistics. One important problem about this kind of model is the variable selection issue. Bayesian approaches, which provide a stochastic search of informative variables, have gained popularity. In this paper, we will study the asymptotic properties related to Bayesian model selection when the model dimension p is growing with the sample size n. We consider p≤n and provide sufficient conditions under which: (1) with large probability, the posterior probability of the true model (from which samples are drawn) uniformly dominates the posterior probability of any incorrect models; and (2) the posterior probability of the true model converges to one in probability. Both (1) and (2) guarantee that the true model will be selected under a Bayesian framework. We also demonstrate several situations when (1) holds but (2) fails, which illustrates the difference between these two properties. Finally, we generalize our results to include g-priors, and provide simulation examples to illustrate the main results.     

Sample size problems in ANOVA Bayesian point of view

In this paper we discuss the sample size problem for balanced one-way ANOVA under a posterior Bayesian formulation of the problem. Using the distribution theory of appropriate quadratic forms we derive explicit sample sizes for prespecified posterior precisions. Comparisons with classical sample sizes are made. Instead of extensive tables, a mathematica program for sample size calculation is given. The formulations given in this article form a foundational step towards Bayesian calculation of sample size, in general.     

Bayesian incidence analysis of animal tumorigenicity data

Statistical inference about tumorigenesis should focus on the tumour incidence rate. Unfortunately, in most animal carcinogenicity experiments, tumours are not observable in live animals and censoring of the tumour onset times is informative. In this paper, we propose a Bayesian method for analysing data from such studies. Our approach focuses on the incidence of tumours and accommodates occult tumours and censored onset times without restricting tumour lethality, relying on cause-of-death data, or requiring interim sacrifices. We represent the underlying state of nature by a multistate stochastic process and assume general probit models for the time-specific transition rates. These models allow the incorporation of covariates, historical control data and subjective prior information. The inherent flexibility of this approach facilitates the interpretation of results, particularly when the sample size is small or the data are sparse. We use a Gibbs sampler to estimate the relevant posterior distributions. The methods proposed are applied to data from a US National Toxicology Program carcinogenicity study.     

真实固定效应空间随机前沿模型的贝叶斯估计

忽略个体效应和空间效应会严重干扰效率测算,其中忽略个体效应使得技术无效率项发生偏移,忽略空间相关性导致估计量有偏且不一致。本文基于真实固定效应随机前沿模型（引入了个体效应）,引入因变量和双边误差项的空间滞后项,构建了适用性更佳的真实固定效应空间随机前沿模型。对模型进行组内变化以消除额外参数,使用贝叶斯方法（需推导未知参数的后验分布并执行MCMC抽样）估计参数和技术效率。该方法真正克服了额外参数问题,比同类方法直观、简便。数值模拟结果表明,本文方法对参数、个体截距项及技术无效率项的估计精度均较高,且增加样本容量,估计精度变优。     

Stochastic models of failure in random environments

Stochastic models of failure modes of frequent occurrence in the engineering sciences are considered. The failure-producing stress environment is modelled as a stationary stochastic process. Using theoretical properties of the sample paths of these processes, failure-time distributions which belong to the Birnbaum-Saunders family are obtained. Several examples of particular engineering relevance are treated.     

A bayesian analysis of trend determination in economic time series

In this paper we provide a comprehensive Bayesian posterior analysis of trend determination in general autoregressive models. Multiple lag autoregressive models with fitted drifts and time trends as well as models that allow for certain types of structural change in the deterministic components are considered. We utilize a modified information matrix-based prior that accommodates stochastic nonstationarity, takes into account the interactions between long-run and short-run dynamics and controls the degree of stochastic nonstationarity permitted. We derive analytic posterior densities for all of the trend determining parameters via the Laplace approximation to multivariate integrals. We also address the sampling properties of our posteriors under alternative data generating processes by simulation methods. We apply our Bayesian techniques to the Nelson-Plosser macroeconomic data and various stock price and dividend data. Contrary to DeJong and Whiteman (1989a,b,c), we do not find that the data overwhelmingly favor the existence of deterministic trends over stochastic trends. In addition, we find evidence supporting Perron's (1989) view that some of the Nelson and Plosser data are best construed as trend stationary with a change in the trend function occurring at 1929.     

A bayesian analysis of trend determination in economic time series

In this paper we provide a comprehensive Bayesian posterior analysis of trend determination in general autoregressive models. Multiple lag autoregressive models with fitted drifts and time trends as well as models that allow for certain types of structural change in the deterministic components are considered. We utilize a modified information matrix-based prior that accommodates stochastic nonstationarity, takes into account the interactions between long-run and short-run dynamics and controls the degree of stochastic nonstationarity permitted. We derive analytic posterior densities for all of the trend determining parameters via the Laplace approximation to multivariate integrals. We also address the sampling properties of our posteriors under alternative data generating processes by simulation methods. We apply our Bayesian techniques to the Nelson-Plosser macroeconomic data and various stock price and dividend data. Contrary to DeJong and Whiteman (1989a,b,c), we do not find that the data overwhelmingly favor the existence of deterministic trends over stochastic trends. In addition, we find evidence supporting Perron's (1989) view that some of the Nelson and Plosser data are best construed as trend stationary with a change in the trend function occurring at 1929.     

A note on reflecting boundaries for solutions of stochastic differential equations

In this note, sufficient conditions are given for a function g(t) to be a reflecting barrier for the sample paths of a solution process X(t) of a stochastic differential equation.     

Bayesian estimation and the application of long memory stochastic volatility models

A new sampling-based Bayesian approach to the long memory stochastic volatility (LMSV) process is presented; the method is motivated by the GPH-estimator in fractionally integrated autoregressive moving average (ARFIMA) processes, which was originally proposed by J. Geweke and S. Porter-Hudak [The estimation and application of long memory time series models, Journal of Time Series Analysis, 4 (1983) 221–238]. In this work, we perform an estimation of the memory parameter in the Bayesian framework; an estimator is obtained by maximizing the posterior density of the memory parameter. Finally, we compare the GPH-estimator and the Bayes-estimator by means of a simulation study and our new approach is illustrated using several stock market indices; the new estimator is proved to be relatively stable for the various choices of frequencies used in the regression.     

Nonparametric Bayes Stochastically Ordered Latent Class Models

Latent class models (LCMs) are used increasingly for addressing a broad variety of problems, including sparse modeling of multivariate and longitudinal data, model-based clustering, and flexible inferences on predictor effects. Typical frequentist LCMs require estimation of a single finite number of classes, which does not increase with the sample size, and have a well-known sensitivity to parametric assumptions on the distributions within a class. Bayesian nonparametric methods have been developed to allow an infinite number of classes in the general population, with the number represented in a sample increasing with sample size. In this article, we propose a new nonparametric Bayes model that allows predictors to flexibly impact the allocation to latent classes, while limiting sensitivity to parametric assumptions by allowing class-specific distributions to be unknown subject to a stochastic ordering constraint. An efficient MCMC algorithm is developed for posterior computation. The methods are validated using simulation studies and applied to the problem of ranking medical procedures in terms of the distribution of patient morbidity.     

Bayesian inference for stable Lévy–driven stochastic differential equations with high‐frequency data

In this paper, we consider parametric Bayesian inference for stochastic differential equations driven by a pure‐jump stable Lévy process, which is observed at high frequency. In most cases of practical interest, the likelihood function is not available; hence, we use a quasi‐likelihood and place an associated prior on the unknown parameters. It is shown under regularity conditions that there is a Bernstein–von Mises theorem associated to the posterior. We then develop a Markov chain Monte Carlo algorithm for Bayesian inference, and assisted with theoretical results, we show how to scale Metropolis–Hastings proposals when the frequency of the data grows, in order to prevent the acceptance ratio from going to zero in the large data limit. Our algorithm is presented on numerical examples that help verify our theoretical findings.     

Selection of Multivariate Stochastic Volatility Models via Bayesian Stochastic Search

We propose a Bayesian stochastic search approach to selecting restrictions on multivariate regression models where the errors exhibit deterministic or stochastic conditional volatilities. We develop a Markov chain Monte Carlo (MCMC) algorithm that generates posterior restrictions on the regression coefficients and Cholesky decompositions of the covariance matrix of the errors. Numerical simulations with artificially generated data show that the proposed method is effective in selecting the data-generating model restrictions and improving the forecasting performance of the model. Applying the method to daily foreign exchange rate data, we conduct stochastic search on a VAR model with stochastic conditional volatilities.     

Bayesian sequential inference for nonlinear multivariate diffusions

In this paper, we adapt recently developed simulation-based sequential algorithms to the problem concerning the Bayesian analysis of discretely observed diffusion processes. The estimation framework involves the introduction of m−1 latent data points between every pair of observations. Sequential MCMC methods are then used to sample the posterior distribution of the latent data and the model parameters on-line. The method is applied to the estimation of parameters in a simple stochastic volatility model (SV) of the U.S. short-term interest rate. We also provide a simulation study to validate our method, using synthetic data generated by the SV model with parameters calibrated to match weekly observations of the U.S. short-term interest rate.