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.     

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.     

基于贝叶斯MCMC算法的美式期权定价

鉴于美式期权的定价具有后向迭代搜索特征,本文结合Longstaff和Schwartz提出的美式期权定价的最小二乘模拟方法,研究基于马尔科夫链蒙特卡洛算法对回归方程系数的估计,实现对美式期权的双重模拟定价.通过对无红利美式看跌股票期权定价进行大量实证模拟,从期权价值定价误差等方面同著名的最小二乘蒙特卡洛模拟方法进行对比分析,结果表明基于MCMC回归算法给出的美式期权定价具有更高的精确度.模拟实证结果表明本文提出的对美式期权定价方法具有较好的可行性、有效性与广泛的适用性.该方法的不足之处就是类似于一般的蒙特卡洛方法,会使得求解的计算量有所加大.     

Sampling for Conditional Inference on Contingency Tables

We propose new sequential importance sampling methods for sampling contingency tables with given margins. The proposal for each method is based on asymptotic approximations to the number of tables with fixed margins. These methods generate tables that are very close to the uniform distribution. The tables, along with their importance weights, can be used to approximate the null distribution of test statistics and calculate the total number of tables. We apply the methods to a number of examples and demonstrate an improvement over other methods in a variety of real problems. Supplementary materials are available online.     

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.     

Bayesian Prediction via Partitioning

This article proposes a new Bayesian approach to prediction on continuous covariates. The Bayesian partition model constructs arbitrarily complex regression and classification surfaces by splitting the covariate space into an unknown number of disjoint regions. Within each region the data are assumed to be exchangeable and come from some simple distribution. Using conjugate priors, the marginal likelihoods of the models can be obtained analytically for any proposed partitioning of the space where the number and location of the regions is assumed unknown a priori. Markov chain Monte Carlo simulation techniques are used to obtain predictive distributions at the design points by averaging across posterior samples of partitions.     

Bayesian Filtering for Jump-Diffusions With Application to Stochastic Volatility

In this article, the problem of sequentially learning parameters governing discretely observed jump-diffusions is explored. The estimation framework involves the introduction of latent points between every pair of observations to allow a sufficiently accurate Euler–Maruyama approximation of the underlying (but unavailable) transition densities. Particle filtering algorithms are then implemented to sample the posterior distribution of the latent data and the model parameters online. The methodology is applied to the estimation of parameters governing a stochastic volatility (SV) model with jumps. As well as using S&P 500 Index data, a simulation study is provided. Supplemental materials for this article are available online.     

Conditional Inference on Tables With Structural Zeros

We develop a set of sequential importance sampling (SIS) strategies for sampling nearly uniformly from two-way zero-one or contingency tables with fixed marginal sums and a given set of structural zeros. The SIS procedure samples tables column by column or cell by cell by using appropriate proposal distributions, and enables us to approximate closely the null distributions of a number of test statistics involved in such tables. When structural zeros are on the diagonal or follow certain patterns, more efficient SIS algorithms are developed which guarantee that every generated table is valid. Examples show that our methods can be applied to make conditional inference on zero-one and contingency tables, and are more efficient than other existing Monte Carlo algorithms.     

A Graphical Approach to Diagnosing the Validity of the Conditional Independence Assumptions of a Bayesian Network Given Data

Bayesian networks (BNs) have attained widespread use in data analysis and decision making. Well-studied topics include efficient inference, evidence propagation, parameter learning from data for complete and incomplete data scenarios, expert elicitation for calibrating BN probabilities, and structure learning. It is common for the researcher to assume the structure of the BN or to glean the structure from expert elicitation or domain knowledge. In this scenario, the model may be calibrated through learning the parameters from relevant data. There is a lack of work on model diagnostics for fitted BNs; this is the contribution of this article. We key on the definition of (conditional) independence to develop a graphical diagnostic that indicates whether the conditional independence assumptions imposed, when one assumes the structure of the BN, are supported by the data. We develop the approach theoretically and describe a Monte Carlo method to generate uncertainty measures for the consistency of the data with conditional independence assumptions under the model structure. We describe how this theoretical information and the data are presented in a graphical diagnostic tool. We demonstrate the approach through data simulated from BNs under different conditional independence assumptions. We also apply the diagnostic to a real-world dataset. The results presented in this article show that this approach is most feasible for smaller BNs—this is not peculiar to the proposed diagnostic graphic, but rather is related to the general difficulty of combining large BNs with data in any manner (such as through parameter estimation). It is the authors’ hope that this article helps highlight the need for more research into BN model diagnostics. This article has supplementary materials online.     

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.     

Joint Monitoring of Several Types of Shocks in Normal Dynamic Linear Models: A Bayesian Decision Theoretic Approach

In this article we consider the sequential monitoring process in normal dynamic linear models as a Bayesian sequential decision problem. We use this approach to build a general procedure that jointly analyzes the existence of outliers, level changes, variance changes, and the development of local correlations. In addition, we study the frequentist performance of this procedure and compare it with the monitoring algorithm proposed in an earlier article.     

On solving integral equations using Markov chain Monte Carlo methods

In this paper, we propose an original approach to the solution of Fredholm equations of the second kind. We interpret the standard Von Neumann expansion of the solution as an expectation with respect to a probability distribution defined on a union of subspaces of variable dimension. Based on this representation, it is possible to use trans-dimensional Markov chain Monte Carlo (MCMC) methods such as Reversible Jump MCMC to approximate the solution numerically. This can be an attractive alternative to standard Sequential Importance Sampling (SIS) methods routinely used in this context. To motivate our approach, we sketch an application to value function estimation for a Markov decision process. Two computational examples are also provided.     

Monte Carlo Methods for Bayesian Analysis of Survival Data Using Mixtures of Dirichlet Process Priors

Consider the model in which the data consist of possibly censored lifetimes, and one puts a mixture of Dirichlet process priors on the common survival distribution. The exact computation of the posterior distribution of the survival function is in general impossible to obtain. This article develops and compares the performance of several simulation techniques, based on Markov chain Monte Carlo and sequential importance sampling, for approximating this posterior distribution. One scheme, whose derivation is based on sequential importance sampling, gives an exactly iid sample from the posterior for the case of right censored data. A second contribution of this article is a battery of programs that implement the various schemes discussed here. The programs and methods are illustrated on a dataset of interval-censored times arising from two treatments for breast cancer.     

服从Dirichlet分布的成分数据的贝叶斯分析

本文研究了Dirichlet分布总体的参数和其他感光趣的量的贝叶斯估计。在参数的有实际意义的函数上设置均匀的先验分布，对适当变换后的参数用Metropolis算法得到马尔可夫链蒙特卡罗后验样本，由此即得参数和其他感兴趣的量的贝叶斯估计。     

Efficient Bayesian Inference for Multivariate Factor Stochastic Volatility Models

We discuss efficient Bayesian estimation of dynamic covariance matrices in multivariate time series through a factor stochastic volatility model. In particular, we propose two interweaving strategies to substantially accelerate convergence and mixing of standard MCMC approaches. Similar to marginal data augmentation techniques, the proposed acceleration procedures exploit nonidentifiability issues which frequently arise in factor models. Our new interweaving strategies are easy to implement and come at almost no extra computational cost; nevertheless, they can boost estimation efficiency by several orders of magnitude as is shown in extensive simulation studies. To conclude, the application of our algorithm to a 26-dimensional exchange rate dataset illustrates the superior performance of the new approach for real-world data. Supplementary materials for this article are available online.     

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Most regression modeling is based on traditional mean regression which results in non-robust estimation results for non-normal errors. Compared to conventional mean regression, composite quantile regression (CQR) may produce more robust parameters estimation. Based on a composite asymmetric Laplace distribution (CALD), we build a Bayesian hierarchical model for the weighted CQR (WCQR). The Gibbs sampler algorithm of Bayesian WCQR is developed to implement posterior inference. Finally, the proposed method are illustrated by some simulation studies and a real data analysis.     

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.     

Sequential Monte Carlo Methods for Option Pricing

In this article, we provide a review and development of sequential Monte Carlo (SMC) methods for option pricing. SMC are a class of Monte Carlo-based algorithms, that are designed to approximate expectations w.r.t a sequence of related probability measures. These approaches have been used successfully for a wide class of applications in engineering, statistics, physics, and operations research. SMC methods are highly suited to many option pricing problems and sensitivity/Greek calculations due to the nature of the sequential simulation. However, it is seldom the case that such ideas are explicitly used in the option pricing literature. This article provides an up-to-date review of SMC methods, which are appropriate for option pricing. In addition, it is illustrated how a number of existing approaches for option pricing can be enhanced via SMC. Specifically, when pricing the arithmetic Asian option w.r.t a complex stochastic volatility model, it is shown that SMC methods provide additional strategies to improve estimation.     

Sequential Bayesian learning for stochastic volatility with variance‐gamma jumps in returns

In this work, we investigate sequential Bayesian estimation for inference of stochastic volatility with variance‐gamma (SVVG) jumps in returns. We develop an estimation algorithm that combines the sequential learning auxiliary particle filter with the particle learning filter. Simulation evidence and empirical estimation results indicate that this approach is able to filter latent variances, identify latent jumps in returns, and provide sequential learning about the static parameters of SVVG. We demonstrate comparative performance of the sequential algorithm and off‐line Markov Chain Monte Carlo in synthetic and real data applications.