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We introduce the normal-inverse-gamma summation operator, which combines Bayesian regression results from different data sources and leads to a simple split-and-merge algorithm for big data regressions. The summation operator is also useful for computing the marginal likelihood and facilitates Bayesian model selection methods, including Bayesian LASSO, stochastic search variable selection, Markov chain Monte Carlo model composition, etc. Observations are scanned in one pass and then the sampler iteratively combines normal-inverse-gamma distributions without reloading the data. Simulation studies demonstrate that our algorithms can efficiently handle highly correlated big data. A real-world data set on employment and wage is also analyzed.
We discuss a few principles to guide the design of efficient Metropolis–Hastings proposals for well-behaved target distributions without deeply divided modes. We illustrate them by developing and evaluating novel proposal kernels using a variety of target distributions. Here, efficiency is measured by the variance ratio relative to the independent sampler. The first principle is to introduce negative correlation in the MCMC sample or to reduce positive correlation: to propose something new, propose something different. This explains why single-moded proposals such as the Gaussian random-walk is poorer than the uniform random walk, which is in turn poorer than the bimodal proposals that avoid values very close to the current value. We evaluate three new bimodal proposals called Box, Airplane and StrawHat, and find that they have similar performance to the earlier Bactrian kernels, suggesting that the general shape of the proposal matters, but not the specific distributional form. We propose the “Mirror” kernel, which generates new values around the mirror image of the current value on the other side of the target distribution (effectively the “opposite” of the current value). This introduces negative correlations, leading in many cases to efficiency of . The second principle, applicable to multidimensional targets, is that a sequence of well-designed one-dimensional proposals can be more efficient than a single -dimensional proposal. Thirdly, we suggest that variable transformation be explored as a general strategy for designing efficient MCMC kernels. We apply these principles to a high-dimensional Gaussian target with strong correlations, a logistic regression problem and a molecular clock dating problem to illustrate their practical utility.
A common approach to analyze a covariate-sample count matrix, an element of which represents how many times a covariate appears in a sample, is to factorize it under the Poisson likelihood. We show its limitation in capturing the tendency for a covariate present in a sample to both repeat itself and excite related ones. To address this limitation, we construct negative binomial factor analysis (NBFA) to factorize the matrix under the negative binomial likelihood, and relate it to a Dirichlet-multinomial distribution based mixed-membership model. To support countably infinite factors, we propose the hierarchical gamma-negative binomial process. By exploiting newly proved connections between discrete distributions, we construct two blocked and a collapsed Gibbs sampler that all adaptively truncate their number of factors, and demonstrate that the blocked Gibbs sampler developed under a compound Poisson representation converges fast and has low computational complexity. Example results show that NBFA has a distinct mechanism in adjusting its number of inferred factors according to the sample lengths, and provides clear advantages in parsimonious representation, predictive power, and computational complexity over previously proposed discrete latent variable models, which either completely ignore burstiness, or model only the burstiness of the covariates but not that of the factors.
Constructing gene regulatory networks is a fundamental task in systems biology. We introduce a Gaussian reciprocal graphical model for inference about gene regulatory relationships by integrating messenger ribonucleic acid (mRNA) gene expression and deoxyribonucleic acid (DNA) level information including copy number and methylation. Data integration allows for inference on the directionality of certain regulatory relationships, which would be otherwise indistinguishable due to Markov equivalence. Efficient inference is developed based on simultaneous equation models. Bayesian model selection techniques are adopted to estimate the graph structure. We illustrate our approach by simulations and application in colon adenocarcinoma pathway analysis.
Regular vine copulas are a flexible class of dependence models, but Bayesian methodology for model selection and inference is not yet fully developed. We propose sparsity-inducing but otherwise non-informative priors, and present novel proposals to enable reversible jump Markov chain Monte Carlo posterior simulation for Bayesian model selection and inference. Our method is the first to jointly estimate the posterior distribution of all trees of a regular vine copula. This represents a substantial improvement over existing frequentist and Bayesian strategies, which can only select one tree at a time and are known to induce bias. A simulation study demonstrates the feasibility of our strategy and shows that it combines superior selection and reduced computation time compared to Bayesian tree-by-tree selection. In a real data example, we forecast the daily expected tail loss of a portfolio of nine exchange-traded funds using a fully Bayesian multivariate dynamic model built around Bayesian regular vine copulas to illustrate our model’s viability for financial analysis and risk estimation.
We propose two new sequential Monte Carlo (SMC) smoothing methods for general state-space models with unknown parameters. The first is a modification of the particle learning and smoothing (PLS) algorithm of Carvalho, Johannes, Lopes, and Polson (2010), with an adjustment in the backward resampling weights. The second, called Refiltering, is a two-stage method that combines sequential parameter learning and particle smoothing algorithms. We illustrate the methods on three benchmark models using simulated data, and apply them to a stochastic volatility model for daily S&P 500 index returns during the financial crisis. We show that both new methods outperform existing SMC approaches, and that Refiltering is competitive with smoothing approaches based on Markov chain Monte Carlo (MCMC) and Particle MCMC.
Robust Bayesian models are appealing alternatives to standard models, providing protection from data that contains outliers or other departures from the model assumptions. Historically, robust models were mostly developed on a case-by-case basis; examples include robust linear regression, robust mixture models, and bursty topic models. In this paper we develop a general approach to robust Bayesian modeling. We show how to turn an existing Bayesian model into a robust model, and then develop a generic computational strategy for it. We use our method to study robust variants of several models, including linear regression, Poisson regression, logistic regression, and probabilistic topic models. We discuss the connections between our methods and existing approaches, especially empirical Bayes and James–Stein estimation.
The matrix- distribution is presented as prior for covariance matrices as an alternative to the conjugate inverted Wishart distribution. A special case of the univariate distribution for a variance parameter is equivalent to a half- distribution for a standard deviation, which is becoming increasingly popular in the Bayesian literature. The matrix- distribution can be conveniently modeled as a Wishart mixture of Wishart or inverse Wishart distributions, which allows straightforward implementation in a Gibbs sampler. By mixing the covariance matrix of a multivariate normal distribution with a matrix- distribution, a multivariate horseshoe type prior is obtained which is useful for modeling sparse signals. Furthermore, it is shown that the intrinsic prior for testing covariance matrices in non-hierarchical models has a matrix- distribution. This intrinsic prior is also useful for testing inequality constrained hypotheses on variances. Finally through simulation it is shown that the matrix-variate distribution has good frequentist properties as prior for the random effects covariance matrix in generalized linear mixed models.
We obtain the optimal Bayesian minimax rate for the unconstrained large covariance matrix of multivariate normal sample with mean zero, when both the sample size, , and the dimension, , of the covariance matrix tend to infinity. Traditionally the posterior convergence rate is used to compare the frequentist asymptotic performance of priors, but defining the optimality with it is elusive. We propose a new decision theoretic framework for prior selection and define Bayesian minimax rate. Under the proposed framework, we obtain the optimal Bayesian minimax rate for the spectral norm for all rates of . We also considered Frobenius norm, Bregman divergence and squared log-determinant loss and obtain the optimal Bayesian minimax rate under certain rate conditions on . A simulation study is conducted to support the theoretical results.
A Markov equivalence class contains all the Directed Acyclic Graphs (DAGs) encoding the same conditional independencies, and is represented by a Completed Partially Directed Acyclic Graph (CPDAG), also named Essential Graph (EG). We approach the problem of model selection among noncausal sparse Gaussian DAGs by directly scoring EGs, using an objective Bayes method. Specifically, we construct objective priors for model selection based on the Fractional Bayes Factor, leading to a closed form expression for the marginal likelihood of an EG. Next we propose a Markov Chain Monte Carlo (MCMC) strategy to explore the space of EGs using sparsity constraints, and illustrate the performance of our method on simulation studies, as well as on a real dataset. Our method provides a coherent quantification of inferential uncertainty, requires minimal prior specification, and shows to be competitive in learning the structure of the data-generating EG when compared to alternative state-of-the-art algorithms.
We propose a spatiotemporal Bayesian variable selection model for detecting activation in functional magnetic resonance imaging (fMRI) settings. Following recent research in this area, we use binary indicator variables for classifying active voxels. We assume that the spatial dependence in the images can be accommodated by applying an areal model to parcels of voxels. The use of parcellation and a spatial hierarchical prior (instead of the popular Ising prior) results in a posterior distribution amenable to exploration with an efficient Markov chain Monte Carlo (MCMC) algorithm. We study the properties of our approach by applying it to simulated data and an fMRI data set.