Bayesian and Maximum Likelihood Estimation for Gaussian Processes on an Incomplete Lattice

This article proposes a new approach for Bayesian and maximum likelihood parameter estimation for stationary Gaussian processes observed on a large lattice with missing values. We propose a Markov chain Monte Carlo approach for Bayesian inference, and a Monte Carlo expectation-maximization algorithm for maximum likelihood inference. Our approach uses data augmentation and circulant embedding of the covariance matrix, and provides likelihood-based inference for the parameters and the missing data. Using simulated data and an application to satellite sea surface temperatures in the Pacific Ocean, we show that our method provides accurate inference on lattices of sizes up to 512 × 512, and is competitive with two popular methods: composite likelihood and spectral approximations.     

Bayesian estimation for generalized exponential distribution based on progressive type-I interval censoring

In this study, we consider the Bayesian estimation of unknown parameters and reliability function of the generalized exponential distribution based on progressive type-I interval censoring. The Bayesian estimates of parameters and reliability function cannot be obtained as explicit forms by applying squared error loss and Linex loss functions, respectively; thus, we present the Lindley’s approximation to discuss these estimations. Then, the Bayesian estimates are compared with the maximum likelihood estimates by using the Monte Carlo simulations.     

多变量非线性贝叶斯动态模型的参数估计

本文讨论多变量非线性贝叶斯动态模型参数估计 ,将 Monte Carlo最优法用于极大似然函数 ,得到未知参数和状态变量的估计     

Multilevel Monte Carlo in approximate Bayesian computation

In the following article, we consider approximate Bayesian computation (ABC) inference. We introduce a method for numerically approximating ABC posteriors using the multilevel Monte Carlo (MLMC). A sequential Monte Carlo version of the approach is developed and it is shown under some assumptions that for a given level of mean square error, this method for ABC has a lower cost than i.i.d. sampling from the most accurate ABC approximation. Several numerical examples are given.     

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In this paper, we investigate a competing risks model based on exponentiated Weibull distribution under Type-I progressively hybrid censoring scheme. To estimate the unknown parameters and reliability function, the maximum likelihood estimators and asymptotic confidence intervals are derived. Since Bayesian posterior density functions cannot be given in closed forms, we adopt Markov chain Monte Carlo method to calculate approximate Bayes estimators and highest posterior density credible intervals. To illustrate the estimation methods, a simulation study is carried out with numerical results. It is concluded that the maximum likelihood estimation and Bayesian estimation can be used for statistical inference in competing risks model under Type-I progressively hybrid censoring scheme.     

Parameter estimation for a two-parameter bathtub-shaped lifetime distribution

A two-parameter distribution was revisited by Chen (2000) [7]. This distribution can have a bathtub-shaped or increasing failure rate function which enables it to fit real lifetime data sets. Maximum likelihood and Bayes estimates of the two unknown parameters are discussed in this paper. It is assumed in the Bayes case that the unknown parameters have gamma priors. Explicit forms of Bayes estimators cannot be obtained. Different approximations are used to establish point estimates and two sided Bayesian probability intervals for the parameters. Monte Carlo simulations are applied to the comparison between the maximum likelihood estimates and the approximate Bayes estimates obtained under non-informative prior assumptions. Analysis of a real data set is also been presented for illustrative purposes.     

Dynamically Rescaled Hamiltonian Monte Carlo for Bayesian Hierarchical Models

Dynamically rescaled Hamiltonian Monte Carlo is introduced as a computationally fast and easily implemented method for performing full Bayesian analysis in hierarchical statistical models. The method relies on introducing a modified parameterization so that the reparameterized target distribution has close to constant scaling properties, and thus is easily sampled using standard (Euclidian metric) Hamiltonian Monte Carlo. Provided that the parameterizations of the conditional distributions specifying the hierarchical model are “constant information parameterizations” (CIPs), the relation between the modified- and original parameterization is bijective, explicitly computed, and admit exploitation of sparsity in the numerical linear algebra involved. CIPs for a large catalogue of statistical models are presented, and from the catalogue, it is clear that many CIPs are currently routinely used in statistical computing. A relation between the proposed methodology and a class of explicitly integrated Riemann manifold Hamiltonian Monte Carlo methods is discussed. The methodology is illustrated on several example models, including a model for inflation rates with multiple levels of nonlinearly dependent latent variables. Supplementary materials for this article are available online.     

Computational Aspects of Bayesian Spectral Density Estimation

Gaussian time-series models are often specified through their spectral density. Such models present several computational challenges, in particular because of the nonsparse nature of the covariance matrix. We derive a fast approximation of the likelihood for such models. We propose to sample from the approximate posterior (i.e., the prior times the approximate likelihood), and then to recover the exact posterior through importance sampling. We show that the variance of the importance sampling weights vanishes as the sample size goes to infinity. We explain why the approximate posterior may typically be multimodal, and we derive a Sequential Monte Carlo sampler based on an annealing sequence to sample from that target distribution. Performance of the overall approach is evaluated on simulated and real datasets. In addition, for one real-world dataset, we provide some numerical evidence that a Bayesian approach to semiparametric estimation of spectral density may provide more reasonable results than its frequentist counterparts. The article comes with supplementary materials, available online, that contain an Appendix with a proof of our main Theorem, a Python package that implements the proposed procedure, and the Ethernet dataset.     

MCMC calibration of spot‐prices models in electricity markets

The calibration of some stochastic differential equation used to model spot prices in electricity markets is investigated. As an alternative to relying on standard likelihood maximization, the adoption of a fully Bayesian paradigm is explored, that relies on Markov chain Monte Carlo (MCMC) stochastic simulation and provides the posterior distributions of the model parameters. The proposed method is applied to one‐ and two‐factor stochastic models, using both simulated and real data. The results demonstrate good agreement between the maximum likelihood and MCMC point estimates. The latter approach, however, provides a more complete characterization of the model uncertainty, an information that can be exploited to obtain a more realistic assessment of the forecasting error. In order to further validate the MCMC approach, the posterior distribution of the Italian electricity price volatility is explored for different maturities and compared with the corresponding maximum likelihood estimates.     

Bayesian Registration of Functions With a Gaussian Process Prior

We present a Bayesian framework for registration of real-valued functional data. At the core of our approach is a series of transformations of the data and functional parameters, developed under a differential geometric framework. We aim to avoid discretization of functional objects for as long as possible, thus minimizing the potential pitfalls associated with high-dimensional Bayesian inference. Approximate draws from the posterior distribution are obtained using a novel Markov chain Monte Carlo (MCMC) algorithm, which is well suited for estimation of functions. We illustrate our approach via pairwise and multiple functional data registration, using both simulated and real datasets. Supplementary material for this article is available online.     

Bayesian Free-Knot Monotone Cubic Spline Regression

This article proposes a new Bayesian approach for monotone curve fitting based on the isotonic regression model. The unknown monotone regression function is approximated by a cubic spline and the constraints are represented by the intersection of quadratic cones. We treat the number and locations of knots as free parameters and use reversible jump Markov chain Monte Carlo to obtain posterior samples of knot configurations. Given the number and locations of the knots, second-order cone programming is used to estimate the remaining parameters. Simulation results suggest the method performs well and we illustrate the approach using the ASA car data.     

Bayesian Variable Selection via Particle Stochastic Search

We focus on Bayesian variable selection in regression models. One challenge is to search the huge model space adequately, while identifying high posterior probability regions. In the past decades, the main focus has been on the use of Markov chain Monte Carlo (MCMC) algorithms for these purposes. In this article, we propose a new computational approach based on sequential Monte Carlo (SMC), which we refer to as particle stochastic search (PSS). We illustrate PSS through applications to linear regression and probit models.     

Improving Approximate Bayesian Computation via Quasi-Monte Carlo

ABC (approximate Bayesian computation) is a general approach for dealing with models with an intractable likelihood. In this work, we derive ABC algorithms based on QMC (quasi-Monte Carlo) sequences. We show that the resulting ABC estimates have a lower variance than their Monte Carlo counter-parts. We also develop QMC variants of sequential ABC algorithms, which progressively adapt the proposal distribution and the acceptance threshold. We illustrate our QMC approach through several examples taken from the ABC literature.     

Distribution-free posterior analysis of econometric models

Very often, one needs to perform (classical or Bayesian) inference, when essentially nothing is known about the distribution of the dependent variable given certain covariates. The paper proposes to approximate the unknown distribution by its non-parametric counterpart—a step function—and treat the points of the support and the corresponding density values, as parameters, whose posterior distributions should be determined based on the available data. The paper proposes distributions should be determined based on the available data. The paper proposes Markov chain Monte Carlo methods to perform posterior analysis, and applies the new method to an analysis of stock returns. Copyright © 1999 John Wiley & Sons, Ltd.     

An Adaptive Interacting Wang–Landau Algorithm for Automatic Density Exploration

While statisticians are well-accustomed to performing exploratory analysis in the modeling stage of an analysis, the notion of conducting preliminary general-purpose exploratory analysis in the Monte Carlo stage (or more generally, the model-fitting stage) of an analysis is an area that we feel deserves much further attention. Toward this aim, this article proposes a general-purpose algorithm for automatic density exploration. The proposed exploration algorithm combines and expands upon components from various adaptive Markov chain Monte Carlo methods, with the Wang–Landau algorithm at its heart. Additionally, the algorithm is run on interacting parallel chains—a feature that both decreases computational cost as well as stabilizes the algorithm, improving its ability to explore the density. Performance of this new parallel adaptive Wang–Landau algorithm is studied in several applications. Through a Bayesian variable selection example, we demonstrate the convergence gains obtained with interacting chains. The ability of the algorithm’s adaptive proposal to induce mode-jumping is illustrated through a Bayesian mixture modeling application. Last, through a two-dimensional Ising model, the authors demonstrate the ability of the algorithm to overcome the high correlations encountered in spatial models. Supplemental materials are available online.     

Bayesian Model Assessment in Joint Modeling of Longitudinal and Survival Data With Applications to Cancer Clinical Trials

Joint models for longitudinal and survival data are routinely used in clinical trials or other studies to assess a treatment effect while accounting for longitudinal measures such as patient-reported outcomes. In the Bayesian framework, the deviance information criterion (DIC) and the logarithm of the pseudo-marginal likelihood (LPML) are two well-known Bayesian criteria for comparing joint models. However, these criteria do not provide separate assessments of each component of the joint model. In this article, we develop a novel decomposition of DIC and LPML to assess the fit of the longitudinal and survival components of the joint model, separately. Based on this decomposition, we then propose new Bayesian model assessment criteria, namely, ΔDIC and ΔLPML, to determine the importance and contribution of the longitudinal (survival) data to the model fit of the survival (longitudinal) data. Moreover, we develop an efficient Monte Carlo method for computing the conditional predictive ordinate statistics in the joint modeling setting. A simulation study is conducted to examine the empirical performance of the proposed criteria and the proposed methodology is further applied to a case study in mesothelioma. Supplementary materials for this article are available online.     

Prediction and Inference for Truncated Spatial Data

Abstract

Spatial data in mining, hydrology, and pollution monitoring commonly have a substantial proportion of zeros. One way to model such data is to suppose that some pointwise transformation of the observations follows the law of a truncated Gaussian random field. This article considers Monte Carlo methods for prediction and inference problems based on this model. In particular, a method for computing the conditional distribution of the random field at an unobserved location, given the data, is described. These results are compared to those obtained by simple kriging and indicator cokriging. Simple kriging is shown to give highly misleading results about conditional distributions; indicator cokriging does quite a bit better but still can give answers that are substantially different from the conditional distributions. A slight modification of this basic technique is developed for calculating the likelihood function for such models, which provides a method for computing maximum likelihood estimates of unknown parameters and Bayesian predictive distributions for values of the process at unobserved locations.     

Chain ladder method: Bayesian bootstrap versus classical bootstrap

The intention of this paper is to estimate a Bayesian distribution-free chain ladder (DFCL) model using approximate Bayesian computation (ABC) methodology. We demonstrate how to estimate quantities of interest in claims reserving and compare the estimates to those obtained from classical and credibility approaches. In this context, a novel numerical procedure utilizing a Markov chain Monte Carlo (MCMC) technique, ABC and a Bayesian bootstrap procedure was developed in a truly distribution-free setting. The ABC methodology arises because we work in a distribution-free setting in which we make no parametric assumptions, meaning we cannot evaluate the likelihood point-wise or in this case simulate directly from the likelihood model. The use of a bootstrap procedure allows us to generate samples from the intractable likelihood without the requirement of distributional assumptions; this is crucial to the ABC framework. The developed methodology is used to obtain the empirical distribution of the DFCL model parameters and the predictive distribution of the outstanding loss liabilities conditional on the observed claims. We then estimate predictive Bayesian capital estimates, the value at risk (VaR) and the mean square error of prediction (MSEP). The latter is compared with the classical bootstrap and credibility methods.     

The Monte Carlo EM method for estimating multinomial probit latent variable models

We propose a multinomial probit (MNP) model that is defined by a factor analysis model with covariates for analyzing unordered categorical data, and discuss its identification. Some useful MNP models are special cases of the proposed model. To obtain maximum likelihood estimates, we use the EM algorithm with its M-step greatly simplified under Conditional Maximization and its E-step made feasible by Monte Carlo simulation. Standard errors are calculated by inverting a Monte Carlo approximation of the information matrix using Louis’s method. The methodology is illustrated with a simulated data.     

基于DMC方法的城市空气质量SUR模型研究

我国现阶段城市化的日益发展,使城市空气质量的宏观调控面临越来越大的压力。本文建立了关于空气质量的似乎不相关(SUR)模型,采用Jeffreys's的不变先验分析直接蒙特卡罗(DMC)方法,计算各参数的贝叶斯后验密度和未来值的预测密度。对中国厦门市区三项污染指标及四项外部驱动因素的数据进行实证分析,并将其与贝叶斯分层模型得出的结果进行比较。