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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…

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…

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…

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 $U$.…