Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSparse online variational Bayesian regression
170
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
This work considers variational Bayesian inference as an inexpensive and scalable alternative to a fully Bayesian approach in the context of sparsity-promoting priors. In particular, the priors considered arise from scale mixtures of Normal distributions with a generalized inverse Gaussian mixing distribution. This includes the variational Bayesian LASSO as an inexpensive and scalable alternative to the Bayesian LASSO introduced in [56]. It also includes priors which more strongly promote sparsity. For linear models the method requires only the iterative solution of deterministic least squares problems. Furthermore, for $nrightarrow infty$ data points and p unknown covariates the method can be implemented exactly online with a cost of O(p$^3$) in computation and O(p$^2$) in memory. For large p an approximation is able to achieve promising results for a cost of O(p) in both computation and memory. Strategies for hyper-parameter tuning are also considered. The method is implemented for real and simulated data. It is shown that the performance in terms of variable selection and uncertainty quantification of the variational Bayesian LASSO can be comparable to the Bayesian LASSO for problems which are tractable with that method, and for a fraction of the cost. The present method comfortably handles n = p = 131,073 on a laptop in minutes, and n = 10$^5$, p = 10$^6$ overnight.
rate research
Read More
Deterministic interpolation and quadrature methods are often unsuitable to address Bayesian inverse problems depending on computationally expensive forward mathematical models. While interpolation may give precise posterior approximations, deterministic quadrature is usually unable to efficiently investigate an informative and thus concentrated likelihood. This leads to a large number of required expensive evaluations of the mathematical model. To overcome these challenges, we formulate and test a multilevel adaptive sparse Leja algorithm. At each level, adaptive sparse grid interpolation and quadrature are used to approximate the posterior and perform all quadrature operations, respectively. Specifically, our algorithm uses coarse discretizations of the underlying mathematical model to investigate the parameter space and to identify areas of high posterior probability. Adaptive sparse grid algorithms are then used to place points in these areas, and ignore other areas of small posterior probability. The points are weighted Leja points. As the model discretization is coarse, the construction of the sparse grid is computationally efficient. On this sparse grid, the posterior measure can be approximated accurately with few expensive, fine model discretizations. The efficiency of the algorithm can be enhanced further by exploiting more than two discretization levels. We apply the proposed multilevel adaptive sparse Leja algorithm in numerical experiments involving elliptic inverse problems in 2D and 3D space, in which we compare it with Markov chain Monte Carlo sampling and a standard multilevel approximation.
Due to the demand for tackling the problem of streaming data with high dimensional covariates, we propose an online sparse sliced inverse regression (OSSIR) method for online sufficient dimension reduction. The existing online sufficient dimension reduction methods focus on the case when the dimension $p$ is small. In this article, we show that our method can achieve better statistical accuracy and computation speed when the dimension $p$ is large. There are two important steps in our method, one is to extend the online principal component analysis to iteratively obtain the eigenvalues and eigenvectors of the kernel matrix, the other is to use the truncated gradient to achieve online $L_{1}$ regularization. We also analyze the convergence of the extended Candid covariance-free incremental PCA(CCIPCA) and our method. By comparing several existing methods in the simulations and real data applications, we demonstrate the effectiveness and efficiency of our method.
We describe a numerical scheme for evaluating the posterior moments of Bayesian linear regression models with partial pooling of the coefficients. The principal analytical tool of the evaluation is a change of basis from coefficient space to the space of singular vectors of the matrix of predictors. After this change of basis and an analytical integration, we reduce the problem of finding moments of a density over k + m dimensions, to finding moments of an m-dimensional density, where k is the number of coefficients and k + m is the dimension of the posterior. Moments can then be computed using, for example, MCMC, the trapezoid rule, or adaptive Gaussian quadrature. An evaluation of the SVD of the matrix of predictors is the dominant computational cost and is performed once during the precomputation stage. We demonstrate numerical results of the algorithm. The scheme described in this paper generalizes naturally to multilevel and multi-group hierarchical regression models where normal-normal parameters appear.
Bayesian Additive Regression Trees (BART) is a Bayesian approach to flexible non-linear 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 paper 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.
This paper investigates the high-dimensional linear regression with highly correlated covariates. In this setup, the traditional sparsity assumption on the regression coefficients often fails to hold, and consequently many model selection procedures do not work. To address this challenge, we model the variations of covariates by a factor structure. Specifically, strong correlations among covariates are explained by common factors and the remaining variations are interpreted as idiosyncratic components of each covariate. This leads to a factor-adjusted regression model with both common factors and idiosyncratic components as covariates. We generalize the traditional sparsity assumption accordingly and assume that all common factors but only a small number of idiosyncratic components contribute to the response. A Bayesian procedure with a spike-and-slab prior is then proposed for parameter estimation and model selection. Simulation studies show that our Bayesian method outperforms its lasso analogue, manifests insensitivity to the overestimates of the number of common factors, pays a negligible price in the no correlation case, and scales up well with increasing sample size, dimensionality and sparsity. Numerical results on a real dataset of U.S. bond risk premia and macroeconomic indicators lend strong support to our methodology.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
