Nonparametric regression using linear combinations of basis functions

This paper discusses a Bayesian approach to nonparametric regression initially proposed by Smith and Kohn (1996. Journal of Econometrics 75: 317–344). In this approach the regression function is represented as a linear combination of basis terms. The basis terms can be univariate or multivariate functions and can include polynomials, natural splines and radial basis functions. A Bayesian hierarchical model is used such that the coefficient of each basis term can be zero with positive prior probability. The presence of basis terms in the model is determined by latent indicator variables. The posterior mean is estimated by Markov chain Monte Carlo simulation because it is computationally intractable to compute the posterior mean analytically unless a small number of basis terms is used. The present article updates the work of Smith and Kohn (1996. Journal of Econometrics 75: 317–344) to take account of work by us and others over the last three years. A careful discussion is given to all aspects of the model specification, function estimation and the use of sampling schemes. In particular, new sampling schemes are introduced to carry out the variable selection methodology.     

Reversible jump Markov chain Monte Carlo algorithms for Bayesian variable selection in logistic mixed models

In this article, to reduce computational load in performing Bayesian variable selection, we used a variant of reversible jump Markov chain Monte Carlo methods, and the Holmes and Held (HH) algorithm, to sample model index variables in logistic mixed models involving a large number of explanatory variables. Furthermore, we proposed a simple proposal distribution for model index variables, and used a simulation study and real example to compare the performance of the HH algorithm with our proposed and existing proposal distributions. The results show that the HH algorithm with our proposed proposal distribution is a computationally efficient and reliable selection method.     

BAYESIAN SUBSET SELECTION AND MODEL AVERAGING USING A CENTRED AND DISPERSED PRIOR FOR THE ERROR VARIANCE

This article proposes a new data‐based prior distribution for the error variance in a Gaussian linear regression model, when the model is used for Bayesian variable selection and model averaging. For a given subset of variables in the model, this prior has a mode that is an unbiased estimator of the error variance but is suitably dispersed to make it uninformative relative to the marginal likelihood. The advantage of this empirical Bayes prior for the error variance is that it is centred and dispersed sensibly and avoids the arbitrary specification of hyperparameters. The performance of the new prior is compared to that of a prior proposed previously in the literature using several simulated examples and two loss functions. For each example our paper also reports results for the model that orthogonalizes the predictor variables before performing subset selection. A real example is also investigated. The empirical results suggest that for both the simulated and real data, the performance of the estimators based on the prior proposed in our article compares favourably with that of a prior used previously in the literature.     

Estimation and variable selection in nonparametric heteroscedastic regression

The article considers a Gaussian model with the mean and the variance modeled flexibly as functions of the independent variables. The estimation is carried out using a Bayesian approach that allows the identification of significant variables in the variance function, as well as averaging over all possible models in both the mean and the variance functions. The computation is carried out by a simulation method that is carefully constructed to ensure that it converges quickly and produces iterates from the posterior distribution that have low correlation. Real and simulated examples demonstrate that the proposed method works well. The method in this paper is important because (a) it produces more realistic prediction intervals than nonparametric regression estimators that assume a constant variance; (b) variable selection identifies the variables in the variance function that are important; (c) variable selection and model averaging produce more efficient prediction intervals than those obtained by regular nonparametric regression.     

Parameter estimation for a deformable template model

In recent years, a number of statistical models have been proposed for the purposes of high-level image analysis tasks such as object recognition. However, in general, these models remain hard to use in practice, partly as a result of their complexity, partly through lack of software. In this paper we concentrate on a particular deformable template model which has proved potentially useful for locating and labelling cells in microscope slides Rue and Hurn (1999). This model requires the specification of a number of rather non-intuitive parameters which control the shape variability of the deformed templates. Our goal is to arrange the estimation of these parameters in such a way that the microscope user's expertise is exploited to provide the necessary training data graphically by identifying a number of cells displayed on a computer screen, but that no additional statistical input is required. In this paper we use maximum likelihood estimation incorporating the error structure in the generation of our training data.     

Data reconciliation of nonnormal observations with nonlinear constraints

This paper presents a new method for the reconciliation of data described by arbitrary continuous probability distributions, with the focus on nonlinear constraints. The main idea, already applied to linear constraints in a previous paper, is to restrict the joint prior probability distribution of the observed variables with model constraints to get a joint posterior probability distribution. Because in general the posterior probability density function cannot be calculated analytically, it is shown that it has decisive advantages to sample from the posterior distribution by a Markov chain Monte Carlo (MCMC) method. From the resulting sample of observed and unobserved variables various characteristics of the posterior distribution can be estimated, such as the mean, the full covariance matrix, marginal posterior densities, as well as marginal moments, quantiles, and HPD intervals. The procedure is illustrated by examples from material flow analysis and chemical engineering.     

Estimation of Dynamic Discrete Choice Models Using Artificial Neural Network Approximations

I propose a method for inference in dynamic discrete choice models (DDCM) that utilizes Markov chain Monte Carlo (MCMC) and artificial neural networks (ANNs). MCMC is intended to handle high-dimensional integration in the likelihood function of richly specified DDCMs. ANNs approximate the dynamic-program (DP) solution as a function of the parameters and state variables prior to estimation to avoid having to solve the DP on each iteration. Potential applications of the proposed methodology include inference in DDCMs with random coefficients, serially correlated unobservables, and dependence across individual observations. The article discusses MCMC estimation of DDCMs, provides relevant background on ANNs, and derives a theoretical justification for the method. Experiments suggest this to be a promising approach.     

Bayesian Inference and Forecasting in Dynamic Neural Networks with Fully Markov Switching ARCH Noises

We deal with one-layer feed-forward neural network for the Bayesian analysis of nonlinear time series. Noises are modeled nonlinearly and nonnormally, by means of ARCH models whose parameters are all dependent on a hidden Markov chain. Parameter estimation is performed by sampling from the posterior distribution via Evolutionary Monte Carlo algorithm, in which two new crossover operators have been introduced. Unknown parameters of the model also include the missing values which can occur within the observed series, so, considering future values as missing, it is also possible to compute point and interval multi-step-ahead predictions.     

Bayesian inference in hidden Markov models through the reversible jump Markov chain Monte Carlo method

Hidden Markov models form an extension of mixture models which provides a flexible class of models exhibiting dependence and a possibly large degree of variability. We show how reversible jump Markov chain Monte Carlo techniques can be used to estimate the parameters as well as the number of components of a hidden Markov model in a Bayesian framework. We employ a mixture of zero-mean normal distributions as our main example and apply this model to three sets of data from finance, meteorology and geomagnetism.     

基于吉伯斯样本生成器的向量自回归模型选择

内容提要：向量自回归模型是多元时间序列分析中最常用的方法之一。在建立模型的过程中模型选择是非常重要的一个环节,如果候选模型不是很多时,可以通过比较每个模型的准则值如AIC、AICc、BIC或HQ进行模型选择。可是,当存在大量候选模型时,我们无法一一比较每个模型的准则值。为了解决这个问题,本文提出一个基于吉伯斯样本生成器的向量自回归模型选择方法,结果表明应用该方法能够从大量候选模型中准确、高效地确认准则值最小的模型。     

A Bayesian HMM with random effects and an unknown number of states for DNA copy number analysis

Hidden Markov models (HMMs) have been shown to be a flexible tool for modelling complex biological processes. However, choosing the number of hidden states remains an open question and the inclusion of random effects also deserves more research, as it is a recent addition to the fixed-effect HMM in many application fields. We present a Bayesian mixed HMM with an unknown number of hidden states and fixed covariates. The model is fitted using reversible-jump Markov chain Monte Carlo, avoiding the need to select the number of hidden states. We show through simulations that the estimations produced are more precise than those from a fixed-effect HMM and illustrate its practical application to the analysis of DNA copy number data, a field where HMMs are widely used.     

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.     

Flexible Bayesian Modelling for Survival Data

The analysis of failure time data often involves two strong assumptions. The proportional hazards assumption postulates that hazard rates corresponding to different levels of explanatory variables are proportional. The additive effects assumption specifies that the effect associated with a particular explanatory variable does not depend on the levels of other explanatory variables. A hierarchical Bayes model is presented, under which both assumptions are relaxed. In particular, time-dependent covariate effects are explicitly modelled, and the additivity of effects is relaxed through the use of a modified neural network structure. The hierarchical nature of the model is useful in that it parsimoniously penalizes violations of the two assumptions, with the strength of the penalty being determined by the data.     

Best subset selection of autoregressive models with exogenous variables and generalized autoregressive conditional heteroscedasticity errors

Summary.  We develop an efficient way to select the best subset autoregressive model with exogenous variables and generalized autoregressive conditional heteroscedasticity errors. One main feature of our method is to select important autoregressive and exogenous variables, and at the same time to estimate the unknown parameters. The method proposed uses the stochastic search idea. By adopting Markov chain Monte Carlo techniques, we can identify the best subset model from a large of number of possible choices. A simulation experiment shows that the method is very effective. Misspecification in the mean equation can also be detected by our model selection method. In the application to the stock-market data of seven countries, the lagged 1 US return is found to have a strong influence on the other stock-market returns.     

Estimation and inference for exponential smooth transition nonlinear volatility models

A family of threshold nonlinear generalised autoregressive conditionally heteroscedastic models is considered, that allows smooth transitions between regimes, capturing size asymmetry via an exponential smooth transition function. A Bayesian approach is taken and an efficient adaptive sampling scheme is employed for inference, including a novel extension to a recently proposed prior for the smoothing parameter that solves a likelihood identification problem. A simulation study illustrates that the sampling scheme performs well, with the chosen prior kept close to uninformative, while successfully ensuring identification of model parameters and accurate inference for the smoothing parameter. An empirical study confirms the potential suitability of the model, highlighting the presence of both mean and volatility (size) asymmetry; while the model is favoured over modern, popular model competitors, including those with sign asymmetry, via the deviance information criterion.     

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 generalized fused lasso modeling via NEG distribution

The fused lasso penalizes a loss function by the L₁ norm for both the regression coefficients and their successive differences to encourage sparsity of both. In this paper, we propose a Bayesian generalized fused lasso modeling based on a normal-exponential-gamma (NEG) prior distribution. The NEG prior is assumed into the difference of successive regression coefficients. The proposed method enables us to construct a more versatile sparse model than the ordinary fused lasso using a flexible regularization term. Simulation studies and real data analyses show that the proposed method has superior performance to the ordinary fused lasso.     

Bayesian Tobit quantile regression with single-index models

Based on the Bayesian framework of utilizing a Gaussian prior for the univariate nonparametric link function and an asymmetric Laplace distribution (ALD) for the residuals, we develop a Bayesian treatment for the Tobit quantile single-index regression model (TQSIM). With the location-scale mixture representation of the ALD, the posterior inferences of the latent variables and other parameters are achieved via the Markov Chain Monte Carlo computation method. TQSIM broadens the scope of applicability of the Tobit models by accommodating nonlinearity in the data. The proposed method is illustrated by two simulation examples and a labour supply dataset.     

Multivariate Bayesian variable selection and prediction

The multivariate regression model is considered with p regressors. A latent vector with p binary entries serves to identify one of two types of regression coefficients: those close to 0 and those not. Specializing our general distributional setting to the linear model with Gaussian errors and using natural conjugate prior distributions, we derive the marginal posterior distribution of the binary latent vector. Fast algorithms aid its direct computation, and in high dimensions these are supplemented by a Markov chain Monte Carlo approach to sampling from the known posterior distribution. Problems with hundreds of regressor variables become quite feasible. We give a simple method of assigning the hyperparameters of the prior distribution. The posterior predictive distribution is derived and the approach illustrated on compositional analysis of data involving three sugars with 160 near infrared absorbances as regressors.     

Detecting homogeneous segments in DNA sequences by using hidden Markov models

In recent years there has been a rapid growth in the amount of DNA being sequenced and in its availability through genetic databases. Statistical techniques which identify structure within these sequences can be of considerable assistance to molecular biologists particularly when they incorporate the discrete nature of changes caused by evolutionary processes. This paper focuses on the detection of homogeneous segments within heterogeneous DNA sequences. In particular, we study an intron from the chimpanzee α-fetoprotein gene; this protein plays an important role in the embryonic development of mammals. We present a Bayesian solution to this segmentation problem using a hidden Markov model implemented by Markov chain Monte Carlo methods. We consider the important practical problem of specifying informative prior knowledge about sequences of this type. Two Gibbs sampling algorithms are contrasted and the sensitivity of the analysis to the prior specification is investigated. Model selection and possible ways to overcome the label switching problem are also addressed. Our analysis of intron 7 identifies three distinct homogeneous segment types, two of which occur in more than one region, and one of which is reversible.