Markov chain Monte Carlo Using an Approximation

This article presents a method for generating samples from an unnormalized posterior distribution f(·) using Markov chain Monte Carlo (MCMC) in which the evaluation of f(·) is very difficult or computationally demanding. Commonly, a less computationally demanding, perhaps local, approximation to f(·) is available, say f^**_x(·). An algorithm is proposed to generate an MCMC that uses such an approximation to calculate acceptance probabilities at each step of a modified Metropolis–Hastings algorithm. Once a proposal is accepted using the approximation, f(·) is calculated with full precision ensuring convergence to the desired distribution. We give sufficient conditions for the algorithm to converge to f(·) and give both theoretical and practical justifications for its usage. Typical applications are in inverse problems using physical data models where computing time is dominated by complex model simulation. We outline Bayesian inference and computing for inverse problems. A stylized example is given of recovering resistor values in a network from electrical measurements made at the boundary. Although this inverse problem has appeared in studies of underground reservoirs, it has primarily been chosen for pedagogical value because model simulation has precisely the same computational structure as a finite element method solution of the complete electrode model used in conductivity imaging, or “electrical impedance tomography.” This example shows a dramatic decrease in CPU time, compared to a standard Metropolis–Hastings algorithm.     

An Efficient Algorithm for Bayesian Nearest Neighbours

K-Nearest Neighbours (k-NN) is a popular classification and regression algorithm, yet one of its main limitations is the difficulty in choosing the number of neighbours. We present a Bayesian algorithm to compute the posterior probability distribution for k given a target point within a data-set, efficiently and without the use of Markov Chain Monte Carlo (MCMC) methods or simulation—alongside an exact solution for distributions within the exponential family. The central idea is that data points around our target are generated by the same probability distribution, extending outwards over the appropriate, though unknown, number of neighbours. Once the data is projected onto a distance metric of choice, we can transform the choice of k into a change-point detection problem, for which there is an efficient solution: we recursively compute the probability of the last change-point as we move towards our target, and thus de facto compute the posterior probability distribution over k. Applying this approach to both a classification and a regression UCI data-sets, we compare favourably and, most importantly, by removing the need for simulation, we are able to compute the posterior probability of k exactly and rapidly. As an example, the computational time for the Ripley data-set is a few milliseconds compared to a few hours when using a MCMC approach.

     

Bayesian variable selection in generalized linear models using a combination of stochastic optimization methods

In this paper the usage of a stochastic optimization algorithm as a model search tool is proposed for the Bayesian variable selection problem in generalized linear models. Combining aspects of three well known stochastic optimization algorithms, namely, simulated annealing, genetic algorithm and tabu search, a powerful model search algorithm is produced. After choosing suitable priors, the posterior model probability is used as a criterion function for the algorithm; in cases when it is not analytically tractable Laplace approximation is used. The proposed algorithm is illustrated on normal linear and logistic regression models, for simulated and real-life examples, and it is shown that, with a very low computational cost, it achieves improved performance when compared with popular MCMC algorithms, such as the MCMC model composition, as well as with “vanilla” versions of simulated annealing, genetic algorithm and tabu search.     

Bayesian Calibration and Uncertainty Analysis for Computationally Expensive Models Using Optimization and Radial Basis Function Approximation

We presenta Bayesian approach to model calibration when evaluation of the model is computationally expensive. Here, calibration is a nonlinear regression problem: given a data vector Y corresponding to the regression model f(β), find plausible values of β. As an intermediate step, Y and f are embedded into a statistical model allowing transformation and dependence. Typically, this problem is solved by sampling from the posterior distribution of β given Y using MCMC. To reduce computational cost, we limit evaluation of f to a small number of points chosen on a high posterior density region found by optimization.Then,we approximate the logarithm of the posterior density using radial basis functions and use the resulting cheap-to-evaluate surface in MCMC.We illustrate our approach on simulated data for a pollutant diffusion problem and study the frequentist coverage properties of credible intervals. Our experiments indicate that our method can produce results similar to those when the true “expensive” posterior density is sampled by MCMC while reducing computational costs by well over an order of magnitude.     

Recentered Importance Sampling With Applications to Bayesian Model Validation

Since its introduction in the early 1990s, the idea of using importance sampling (IS) with Markov chain Monte Carlo (MCMC) has found many applications. This article examines problems associated with its application to repeated evaluation of related posterior distributions with a particular focus on Bayesian model validation. We demonstrate that, in certain applications, the curse of dimensionality can be reduced by a simple modification of IS. In addition to providing new theoretical insight into the behavior of the IS approximation in a wide class of models, our result facilitates the implementation of computationally intensive Bayesian model checks. We illustrate the simplicity, computational savings, and potential inferential advantages of the proposed approach through two substantive case studies, notably computation of Bayesian p-values for linear regression models and simulation-based model checking. Supplementary materials including the Appendix and the R code for Section are available online.     

Adaptive approximate Bayesian computation for complex models

We propose a new approximate Bayesian computation (ABC) algorithm that aims at minimizing the number of model runs for reaching a given quality of the posterior approximation. This algorithm automatically determines its sequence of tolerance levels and makes use of an easily interpretable stopping criterion. Moreover, it avoids the problem of particle duplication found when using a MCMC kernel. When applied to a toy example and to a complex social model, our algorithm is 2–8 times faster than the three main sequential ABC algorithms currently available.     

Parallel Bayesian Additive Regression Trees

Bayesian additive regression trees (BART) is a Bayesian approach to flexible nonlinear regression which has been shown to be competitive with the best modern predictive methods such as those based on bagging and boosting. BART offers some advantages. For example, the stochastic search Markov chain Monte Carlo (MCMC) algorithm can provide a more complete search of the model space and variation across MCMC draws can capture the level of uncertainty in the usual Bayesian way. The BART prior is robust in that reasonable results are typically obtained with a default prior specification. However, the publicly available implementation of the BART algorithm in the R package BayesTree is not fast enough to be considered interactive with over a thousand observations, and is unlikely to even run with 50,000 to 100,000 observations. In this article we show how the BART algorithm may be modified and then computed using single program, multiple data (SPMD) parallel computation implemented using the Message Passing Interface (MPI) library. The approach scales nearly linearly in the number of processor cores, enabling the practitioner to perform statistical inference on massive datasets. Our approach can also handle datasets too massive to fit on any single data repository.     

艾拉姆咖分布参数变点的Bayes估计

研究了艾拉姆咖分布变点估计的非迭代抽样算法(IBF)和MCMC算法.在贝叶斯框架下,选取无信息先验分布,得到关于变点位置的后验分布和各参数的满条件分布,并且详细介绍了IBF算法和MCMC方法的实施步骤.最后进行随机模拟试验,结果表明两种算法都能够有效的估计变点位置,并且IBF算法的计算速度优于MCMC方法.     

Adaptive Markov Chain Monte Carlo for Bayesian Variable Selection

We describe adaptive Markov chain Monte Carlo (MCMC) methods for sampling posterior distributions arising from Bayesian variable selection problems. Point-mass mixture priors are commonly used in Bayesian variable selection problems in regression. However, for generalized linear and nonlinear models where the conditional densities cannot be obtained directly, the resulting mixture posterior may be difficult to sample using standard MCMC methods due to multimodality. We introduce an adaptive MCMC scheme that automatically tunes the parameters of a family of mixture proposal distributions during simulation. The resulting chain adapts to sample efficiently from multimodal target distributions. For variable selection problems point-mass components are included in the mixture, and the associated weights adapt to approximate marginal posterior variable inclusion probabilities, while the remaining components approximate the posterior over nonzero values. The resulting sampler transitions efficiently between models, performing parameter estimation and variable selection simultaneously. Ergodicity and convergence are guaranteed by limiting the adaptation based on recent theoretical results. The algorithm is demonstrated on a logistic regression model, a sparse kernel regression, and a random field model from statistical biophysics; in each case the adaptive algorithm dramatically outperforms traditional MH algorithms. Supplementary materials for this article are available online.     

A Marginal Sampler for σ-Stable Poisson–Kingman Mixture Models

We investigate the class of σ-stable Poisson–Kingman random probability measures (RPMs) in the context of Bayesian nonparametric mixture modeling. This is a large class of discrete RPMs, which encompasses most of the popular discrete RPMs used in Bayesian nonparametrics, such as the Dirichlet process, Pitman–Yor process, the normalized inverse Gaussian process, and the normalized generalized Gamma process. We show how certain sampling properties and marginal characterizations of σ-stable Poisson–Kingman RPMs can be usefully exploited for devising a Markov chain Monte Carlo (MCMC) algorithm for performing posterior inference with a Bayesian nonparametric mixture model. Specifically, we introduce a novel and efficient MCMC sampling scheme in an augmented space that has a small number of auxiliary variables per iteration. We apply our sampling scheme to a density estimation and clustering tasks with unidimensional and multidimensional datasets, and compare it against competing MCMC sampling schemes. Supplementary materials for this article are available online.     

Falling and explosive,dormant, and rising markets via multiple‐regime financial time series models

A multiple‐regime threshold nonlinear financial time series model, with a fat‐tailed error distribution, is discussed and Bayesian estimation and inference are considered. Furthermore, approximate Bayesian posterior model comparison among competing models with different numbers of regimes is considered which is effectively a test for the number of required regimes. An adaptive Markov chain Monte Carlo (MCMC) sampling scheme is designed, while importance sampling is employed to estimate Bayesian residuals for model diagnostic testing. Our modeling framework provides a parsimonious representation of well‐known stylized features of financial time series and facilitates statistical inference in the presence of high or explosive persistence and dynamic conditional volatility. We focus on the three‐regime case where the main feature of the model is to capturing of mean and volatility asymmetries in financial markets, while allowing an explosive volatility regime. A simulation study highlights the properties of our MCMC estimators and the accuracy and favourable performance as a model selection tool, compared with a deviance criterion, of the posterior model probability approximation method. An empirical study of eight international oil and gas markets provides strong support for the three‐regime model over its competitors, in most markets, in terms of model posterior probability and in showing three distinct regime behaviours: falling/explosive, dormant and rising markets. Copyright © 2009 John Wiley & Sons, Ltd.     

Posterior Simulation with Priors Specified on Functionals

Abstract

Many Bayesian analyses use Markov chain Monte Carlo (MCMC) techniques. MCMC techniques work fastest (per iteration) when the prior distribution of the parameters is chosen conveniently, such as a conjugate prior. However, this is sometimes at odds with the prior desired by the investigator. We describe two motivating examples where nonconjugate priors are preferred. One is a Dirichlet process where it is difficult to implement alternative, nonconjugate priors. We develop a method that allows computation to be done with a convenient prior but adjusts the equilibrium distribution of the Markov chain to be the posterior distribution from the desired prior. In addition to allowing more freedom in choosing prior distributions, the method enables the investigator to perform quick sensitivity analyses, even in nonparametric settings.     

Bayesian Conditional Density Filtering

We propose a conditional density filtering (C-DF) algorithm for efficient online Bayesian inference. C-DF adapts MCMC sampling to the online setting, sampling from approximations to conditional posterior distributions obtained by propagating surrogate conditional sufficient statistics (a function of data and parameter estimates) as new data arrive. These quantities eliminate the need to store or process the entire dataset simultaneously and offer a number of desirable features. Often, these include a reduction in memory requirements and runtime and improved mixing, along with state-of-the-art parameter inference and prediction. These improvements are demonstrated through several illustrative examples including an application to high dimensional compressed regression. In the cases where dimension of the model parameter does not grow with time, we also establish sufficient conditions under which C-DF samples converge to the target posterior distribution asymptotically as sampling proceeds and more data arrive. Supplementary materials of C-DF are available online.     

Frequency domain characteristics of linear operator to decompose a time series into the multi-components

Frequency domain properties of the operators to decompose a time series into the multi-components along the Akaike's Bayesian model (Akaike (1980, Bayesian Statistics, 143–165, University Press, Valencia, Spain)) are shown. In that analysis a normal disturbance-linear-stochastic regression prior model is applied to the time series. A prior distribution, characterized by a small number of hyperparameters, is specified for model parameters. The posterior distribution is a linear function (filter) of observations. Here we use frequency domain analysis or filter characteristics of several prior models parametrically as a function of the hyperparameters.     

Sparse Variational Analysis of Linear Mixed Models for Large Data Sets

It is increasingly common to be faced with longitudinal or multi-level data sets that have large numbers of predictors and/or a large sample size. Current methods of fitting and inference for mixed effects models tend to perform poorly in such settings. When there are many variables, it is appealing to allow uncertainty in subset selection and to obtain a sparse characterization of the data. Bayesian methods are available to address these goals using Markov chain Monte Carlo (MCMC), but MCMC is very computationally expensive and can be infeasible in large p and/or large n problems. As a fast approximate Bayes solution, we recommend a novel approximation to the posterior relying on variational methods. Variational methods are used to approximate the posterior of the parameters in a decomposition of the variance components, with priors chosen to obtain a sparse solution that allows selection of random effects. The method is evaluated through a simulation study, and applied to an epidemiological application.     

Local Derivative-Free Approximation of Computationally Expensive Posterior Densities

Bayesian inference using Markov chain Monte Carlo (MCMC) is computationally prohibitive when the posterior density of interest, π, is computationally expensive to evaluate. We develop a derivative-free algorithm GRIMA to accurately approximate π by interpolation over its high-probability density (HPD) region, which is initially unknown. Our local approach reduces the waste of computational budget on approximation of π in the low-probability region, which is inherent in global experimental designs. However, estimation of the HPD region is nontrivial when derivatives of π are not available or are not informative about the shape of the HPD region. Without relying on derivatives, GRIMA iterates (a) sequential knot selection over the estimated HPD region of π to refine the surrogate posterior and (b) re-estimation of the HPD region using an MCMC sample from the updated surrogate density, which is inexpensive to obtain. GRIMA is applicable to approximation of general unnormalized posterior densities. To determine the range of tractable problem dimensions, we conduct simulation experiments on test densities with linear and nonlinear component-wise dependence, skewness, kurtosis and multimodality. Subsequently, we use GRIMA in a case study to calibrate a computationally intensive nonlinear regression model to real data from the Town Brook watershed. Supplemental materials for this article are available online.     

On Monte Carlo methods for Bayesian multivariate regression models with heavy-tailed errors

We consider Bayesian analysis of data from multivariate linear regression models whose errors have a distribution that is a scale mixture of normals. Such models are used to analyze data on financial returns, which are notoriously heavy-tailed. Let π denote the intractable posterior density that results when this regression model is combined with the standard non-informative prior on the unknown regression coefficients and scale matrix of the errors. Roughly speaking, the posterior is proper if and only if n≥d+k, where n is the sample size, d is the dimension of the response, and k is number of covariates. We provide a method of making exact draws from π in the special case where n=d+k, and we study Markov chain Monte Carlo (MCMC) algorithms that can be used to explore π when n>d+k. In particular, we show how the Haar PX-DA technology studied in Hobert and Marchev (2008) [11] can be used to improve upon Liu’s (1996) [7] data augmentation (DA) algorithm. Indeed, the new algorithm that we introduce is theoretically superior to the DA algorithm, yet equivalent to DA in terms of computational complexity. Moreover, we analyze the convergence rates of these MCMC algorithms in the important special case where the regression errors have a Student’s t distribution. We prove that, under conditions on n, d, k, and the degrees of freedom of the t distribution, both algorithms converge at a geometric rate. These convergence rate results are important from a practical standpoint because geometric ergodicity guarantees the existence of central limit theorems which are essential for the calculation of valid asymptotic standard errors for MCMC based estimates.     

Weakly Informative Reparameterizations for Location-Scale Mixtures

While mixtures of Gaussian distributions have been studied for more than a century, the construction of a reference Bayesian analysis of those models remains unsolved, with a general prohibition of improper priors due to the ill-posed nature of such statistical objects. This difficulty is usually bypassed by an empirical Bayes resolution. By creating a new parameterization centered on the mean and possibly the variance of the mixture distribution itself, we manage to develop here a weakly informative prior for a wide class of mixtures with an arbitrary number of components. We demonstrate that some posterior distributions associated with this prior and a minimal sample size are proper. We provide Markov chain Monte Carlo (MCMC) implementations that exhibit the expected exchangeability. We only study here the univariate case, the extension to multivariate location-scale mixtures being currently under study. An R package called Ultimixt is associated with this article. Supplementary material for this article is available online.     

A general approach to Bayesian portfolio optimization

We develop a general approach to portfolio optimization taking account of estimation risk and stylized facts of empirical finance. This is done within a Bayesian framework. The approximation of the posterior distribution of the unknown model parameters is based on a parallel tempering algorithm. The portfolio optimization is done using the first two moments of the predictive discrete asset return distribution. For illustration purposes we apply our method to empirical stock market data where daily asset log-returns are assumed to follow an orthogonal MGARCH process with t-distributed perturbations. Our results are compared with other portfolios suggested by popular optimization strategies.     

Bayesian detection of structural changes

A Bayesian solution is given to the problem of making inferences about an unknown number of structural changes in a sequence of observations. Inferences are based on the posterior distribution of the number of change points and on the posterior probabilities of possible change points. Detailed analyses are given for binomial data and some regression problems, and numerical illustrations are provided. In addition, an approximation procedure to compute the posterior probabilities is presented.