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加速稀疏高斯图模型的贝叶斯结构学习AI 解读

Bayesian structure learning in Gaussian graphical models is often done by search algorithms over the graph space.The conjugate prior for the precision matrix satisfying graphical constraints is the well-known G-Wishart.With this prior, the transition probabilities in the search algorithms necessitate evaluating the ratios of the prior normalizing constants of G-Wishart.In moderate to high-dimensions, this ratio is often approximated by using sampling-based methods as…

具有对称不变性的高斯模型空间中的模型选择AI 解读

We consider multivariate centered Gaussian models for the random variable Z=(Z1,…,Zp), invariant under the action of a subgroup of the group of permutations on {1,…,p}. Using the representation theory of the symmetric group on the field of reals, we derive the distribution of the maximum likelihood estimate of the covariance parameter Σ and also the analytic expression of the normalizing constant of the Diaconis–Ylvisaker conjugate prior for the precision parameter K=Σ−1. We…

贝叶斯因子与离散层次对数线性模型的几何AI 解读

A standard tool for model selection in a Bayesian framework is the Bayes factor which compares the marginal likelihood of the data under two given different models. In this paper, we consider the class of hierarchical loglinear models for discrete data given under the form of a contingency table with multinomial sampling. We assume that the prior distribution on the loglinear parameters is the Diaconis–Ylvisaker conjugate prior, and the uniform is the prior distribution on…

离散层级对数线性模型的共轭先验AI 解读

In Bayesian analysis of multi-way contingency tables, the selection of a prior distribution for either the log-linear parameters or the cell probabilities parameters is a major challenge. In this paper, we define a flexible family of conjugate priors for the wide class of discrete hierarchical log-linear models, which includes the class of graphical models. These priors are defined as the Diaconis–Ylvisaker conjugate priors on the log-linear parameters subject to “baseline…

高斯图模型中的灵活协方差估计AI 解读

In this paper, we propose a class of Bayes estimators for the covariance matrix of graphical Gaussian models Markov with respect to a decomposable graph G. Working with the WPG family defined by Letac and Massam [Ann. Statist. 35 (2007) 1278–1323] we derive closed-form expressions for Bayes estimators under the entropy and squared-error losses. The WPG family includes the classical inverse of the hyper inverse Wishart but has many more shape parameters, thus allowing for…

Wishart 矩阵子式的矩AI 解读

For a random matrix following a Wishart distribution, we derive formulas for the expectation and the covariance matrix of compound matrices. The compound matrix of order m is populated by all m×m-minors of the Wishart matrix. Our results yield first and second moments of the minors of the sample covariance matrix for multivariate normal observations. This work is motivated by the fact that such minors arise in the expression of constraints on the covariance matrix in many…

可分解图上的 Wishart 分布AI 解读

When considering a graphical Gaussian model ${\mathcal{N}}_{G}$ Markov with respect to a decomposable graph G, the parameter space of interest for the precision parameter is the cone PG of positive definite matrices with fixed zeros corresponding to the missing edges of G. The parameter space for the scale parameter of ${\mathcal{N}}_{G}$ is the cone QG, dual to PG, of incomplete matrices with submatrices corresponding to the cliques of G being positive definite. In this…

非可分解高斯图模型边际似然的蒙特卡洛计算AI 解读

A centred Gaussian model that is Markov with respect to an undirected graph G is characterised by the parameter set of its precision matrices which is the cone M+(G) of positive definite matrices with entries corresponding to the missing edges of G constrained to be equal to zero. In a Bayesian framework, the conjugate family for the precision parameter is the distribution with Wishart density with respect to the Lebesgue measure restricted to M+(G). We call this…

欧氏若尔当代数上格条件独立模型的估计与检验AI 解读

In this paper we generalize the major results of Andersson and Perlman on LCI models to the setting of symmetric cones and give an explicit closed form formula for the estimate of the covariance matrix in the generalized LCI models that we define. To this end, we replace the cone $H_I^+(\mathbb{R})$ sitting inside the Jordan algebra of symmetric real $I \times I$-matrices by the symmetric cone $\Omega$ of an Euclidean Jordan algebra $V$. We introduce the Andersson-Perlman…

Wishart分布的二次回归与逆回归刻画AI 解读

If $U$ and $V$ are independent random variables which are gamma distributed with the same scale parameter, then there exist $a$ and $b$ in $\\mathbb{R}$ such that $$\\mathbb{E}(U|U + V) = a(U + V)$$ and $$\\mathbb{E}(U^2|U + V) = b(U + V)^2$$. This, in fact, is characteristic of gamma distributions. Our paper extends this property to the Wishart distributions in a suitable way, by replacing the real number $U^2$ by a pair of quadratic functions of the symmetric matrix $U$.…