使用分层矩阵实现可扩展的高斯过程计算

协方差矩阵的层次贝叶斯估计

Hierarchical Bayesian Estimation of Covariance Matrices

提出层次贝叶斯框架估计协方差矩阵，利用O(p)-等变性，推导出Oracle贝叶斯规则，并通过有限Pólya树先验和吉布斯采样逼近，显著优于经典估计量。

基于筛选beta混合先验的稀疏协方差矩阵可扩展最优贝叶斯推断

Scalable and optimal Bayesian inference for sparse covariance matrices via screened beta-mixture prior

本文提出一种结合连续收缩先验与筛选过程的可扩展贝叶斯方法，用于稀疏协方差矩阵估计，解决了“大p，小n”高维问题，并具备确定筛选性质与最优收敛速率。

使用贝叶斯决策理论方法实现可扩展决策

Scalable Decisions using a Bayesian Decision-Theoretic Approach

本文提出一种贝叶斯决策理论框架，通过整合多个目标和权衡，实现更高效和可扩展的决策制定。

Scalable Physics-based Maximum Likelihood Estimation using Hierarchical Matrices

Physics-based covariance models provide a systematic way to construct covariance models that are consistent with the underlying physical laws in Gaussian process analysis. The unknown parameters in the covariance models can be estimated using maximum likelihood estimation, but direct construction of the covariance matrix and classical strategies of computing with it requires $n$ physical model runs, $n^2$ storage complexity, and $n^3$ computational complexity. To address such challenges, we propose to approximate the discretized covariance function using hierarchical matrices. By utilizing ran…

HLIBCov: 并行分层矩阵近似大规模协方差矩阵和似然函数及其在参数识别中的应用

HLIBCov: Parallel Hierarchical Matrix Approximation of Large Covariance Matrices and Likelihoods with Applications in Parameter Identification

本文提出HLIBCov包，利用并行分层矩阵高效近似大规模不均匀协方差矩阵，通过最大化联合高斯对数似然函数估计协方差函数的未知参数，包括方差、平滑度和协方差长度。

Identification of unknown parameters and prediction with hierarchical matrices

Statistical analysis of massive datasets very often implies expensive linear algebra operations with large dense matrices. Typical tasks are an estimation of unknown parameters of the underlying statistical model and prediction of missing values. We developed the H-MLE procedure, which solves these tasks. The unknown parameters can be estimated by maximizing the joint Gaussian log-likelihood function, which depends on a covariance matrix. To decrease high computational cost, we approximate the covariance matrix in the hierarchical (H-) matrix format. The H-matrix technique allows us to work wi…

Scalable Spatio-Temporal Smoothing via Hierarchical Sparse Cholesky Decomposition

We propose an approximation to the forward-filter-backward-sampler (FFBS) algorithm for large-scale spatio-temporal smoothing. FFBS is commonly used in Bayesian statistics when working with linear Gaussian state-space models, but it requires inverting covariance matrices which have the size of the latent state vector. The computational burden associated with this operation effectively prohibits its applications in high-dimensional settings. We propose a scalable spatio-temporal FFBS approach based on the hierarchical Vecchia approximation of Gaussian processes, which has been previously succes…

The R Package BHAM: Fast and Scalable Bayesian Hierarchical Additive Model for High-dimensional Data

BHAM is a freely avaible R pakcage that implments Bayesian hierarchical additive models for high-dimensional clinical and genomic data. The package includes functions that generalized additive model, and Cox additive model with the spike-and-slab LASSO prior. These functions implement scalable and stable algorithms to estimate parameters. BHAM also provides utility functions to construct additive models in high dimensional settings, select optimal models, summarize bi-level variable selection results, and visualize nonlinear effects. The package can facilitate flexible modeling of large-scale …

Likelihood Approximation With Hierarchical Matrices For Large Spatial Datasets

We use available measurements to estimate the unknown parameters (variance, smoothness parameter, and covariance length) of a covariance function by maximizing the joint Gaussian log-likelihood function. To overcome cubic complexity in the linear algebra, we approximate the discretized covariance function in the hierarchical (H-) matrix format. The H-matrix format has a log-linear computational cost and storage $O(kn \log n)$, where the rank $k$ is a small integer and $n$ is the number of locations. The H-matrix technique allows us to work with general covariance matrices in an efficient way, …

Scalable Parallel Factorizations of SDD Matrices and Efficient Sampling for Gaussian Graphical Models

Motivated by a sampling problem basic to computational statistical inference, we develop a nearly optimal algorithm for a fundamental problem in spectral graph theory and numerical analysis. Given an $n\times n$ SDDM matrix ${\bf \mathbf{M}}$, and a constant $-1 \leq p \leq 1$, our algorithm gives efficient access to a sparse $n\times n$ linear operator $\tilde{\mathbf{C}}$ such that $${\mathbf{M}}^{p} \approx \tilde{\mathbf{C}} \tilde{\mathbf{C}}^\top.$$ The solution is based on factoring ${\bf \mathbf{M}}$ into a product of simple and sparse matrices using squaring and spectral sparsificatio…

任意稀疏精度矩阵上的一种新的分层分布

A new hierarchical distribution on arbitrary sparse precision matrices

本文提出基于 Cholesky 分解与 Spike-and-Slab 先验的 S-Bartlett 分层分布，并用定制 MCMC 实现稀疏精度矩阵的贝叶斯估计。