Studying conditional independence structure among many variables with few
observations is a challenging task. Gaussian Graphical Models (GGMs) tackle
this problem by encouraging sparsity in the precision matrix through an $l_p$
regularization with $p\leq1$. However, since the objective is highly non-convex
for sub-$l_1$ pseudo-norms, most approaches rely on the $l_1$ norm. In this
case frequentist approaches allow to elegantly compute the solution path as a
function of the shrinkage parameter $\lambda$. Instead of optimizing the
penalized likelihood, the Bayesian formulation introduces a Laplace prior on
the precision matrix. However, posterior inference for different $\lambda$
values requires repeated runs of expensive Gibbs samplers. We propose a very
general framework for variational inference in GGMs that unifies the benefits
of frequentist and Bayesian frameworks. Specifically, we propose to approximate
the posterior with a matrix-variate Normalizing Flow defined on the space of
symmetric positive definite matrices. As a key improvement on previous work, we
train a continuum of sparse regression models jointly for all regularization
parameters $\lambda$ and all $l_p$ norms, including non-convex sub-$l_1$
pseudo-norms. This is achieved by conditioning the flow on $p>0$ and on the
shrinkage parameter $\lambda$. We have then access with one model to (i) the
evolution of the posterior for any $\lambda$ and for any $l_p$ (pseudo-) norms,
(ii) the marginal log-likelihood for model selection, and (iii) we can recover
the frequentist solution paths as the MAP, which is obtained through simulated
annealing.

该研究提出了一种变分推断方法，通过定义于对称正定矩阵空间上的矩阵变分流来逼近高斯图模型的后验分布，对任意正则化参数 λ 和任意 l_p (伪 -) 范数，包括非凸的 l1 伪范数进行联合训练，从而获得一个模型，并可以恢复频率主义的解路径。

高斯图模型的条件矩阵流

Conditional Matrix Flows for Gaussian Graphical Models

Both linear mixed models (LMMs) and sparse regression models are widely used
in genetics applications, including, recently, polygenic modeling in
genome-wide association studies. These two approaches make very different
assumptions, so are expected to perform well in different situations. However,
in practice, for a given data set one typically does not know which assumptions
will be more accurate. Motivated by this, we consider a hybrid of the two,
which we refer to as a "Bayesian sparse linear mixed model" (BSLMM) that
includes both these models as special cases. We address several key
computational and statistical issues that arise when applying BSLMM, including
appropriate prior specification for the hyper-parameters, and a novel Markov
chain Monte Carlo algorithm for posterior inference. We apply BSLMM and compare
it with other methods for two polygenic modeling applications: estimating the
proportion of variance in phenotypes explained (PVE) by available genotypes,
and phenotype (or breeding value) prediction. For PVE estimation, we
demonstrate that BSLMM combines the advantages of both standard LMMs and sparse
regression modeling. For phenotype prediction it considerably outperforms
either of the other two methods, as well as several other large-scale
regression methods previously suggested for this problem. Software implementing
our method is freely available from
this http URL

提出了一种 “贝叶斯稀疏线性混合模型”(BSLMM)，将线性混合模型 (LMMs) 和稀疏回归模型两种方法相结合，以实现多基因建模。通过适当的超参数先验规范和新颖的马尔可夫蒙特卡罗算法进行后验推理，BSLMM 的方法在两个多基因建模应用中的估计表现的优势比前文提出的大规模回归方法更好。

贝叶斯稀疏线性混合模型多基因建模

Polygenic Modeling with Bayesian Sparse Linear Mixed Models

Functional neuroimaging can measure the brain?s response to an external
stimulus. It is used to perform brain mapping: identifying from these
observations the brain regions involved. This problem can be cast into a linear
supervised learning task where the neuroimaging data are used as predictors for
the stimulus. Brain mapping is then seen as a support recovery problem. On
functional MRI (fMRI) data, this problem is particularly challenging as i) the
number of samples is small due to limited acquisition time and ii) the
variables are strongly correlated. We propose to overcome these difficulties
using sparse regression models over new variables obtained by clustering of the
original variables. The use of randomization techniques, e.g. bootstrap
samples, and clustering of the variables improves the recovery properties of
sparse methods. We demonstrate the benefit of our approach on an extensive
simulation study as well as two fMRI datasets.

使用稀疏回归模型和变量聚类来解决功能性磁共振成像数据的支持恢复问题。