Hyper Inverse Wishart Distribution for Non‐decomposable Graphs and its Application to Bayesian Inference for Gaussian Graphical Models

Roverato, Alberto

doi:10.1111/1467-9469.00297

Cited by 183 publications

(215 citation statements)

References 32 publications

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“…The framework, which we refer to as Bayesian functional connectivity (BFC) analysis, makes use of a G-Wishart prior (Roverato, 2002). This prior allows the sparseness structure of estimated precision matrices to be determined by a graph G, corresponding to structural connectivity.…”

Section: Introductionmentioning

confidence: 99%

Structurally-informed Bayesian functional connectivity analysis

et al. 2014

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Functional connectivity refers to covarying activity between spatially segregated brain regions and can be studied by measuring correlation between functional magnetic resonance imaging (fMRI) timeseries. These correlations can be caused either by direct communication via active axonal pathways or indirectly via the interaction with other regions. It is not possible to discriminate between these two kinds of functional interaction simply by considering the covariance matrix. However, the non-diagonal elements of its inverse, the precision matrix, can be naturally related to direct communication between brain areas and interpreted in terms of partial correlations. In this paper, we propose a Bayesian model for functional connectivity analysis which allows estimation of a posterior density over precision matrices, and, consequently, allows one to quantify the uncertainty about estimated partial correlations. In order to make model estimation feasible it is assumed that the sparseness structure of the precision matrices is given by an estimate of structural connectivity obtained using diffusion imaging data. The model was tested on simulated data as well as resting-state fMRI data and compared with a graphical lasso analysis. The presented approach provides a theoretically solid foundation for quantifying functional connectivity in the presence of uncertainty.

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Section: Introductionmentioning

confidence: 99%

Structurally-informed Bayesian functional connectivity analysis

et al. 2014

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“…However, this approach is most useful in the context of conjugate Bayesian inference, since a prior concentration matrix would have to be specified. Also, it is rather tedious to sample large sparse non-decomposable models (Roverato, 2002).…”

Section: Numerical Results For Simulated Datamentioning

confidence: 99%

Low-Order Conditional Independence Graphs for Inferring Genetic Networks

Wille¹,

Bühlmann

2006

Statistical Applications in Genetics and Molecular Biology

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As a powerful tool for analyzing full conditional (in-)dependencies between random variables, graphical models have become increasingly popular to infer genetic networks based on gene expression data. However, full (unconstrained) conditional relationships between random variables can be only estimated accurately if the number of observations is relatively large in comparison to the number of variables, which is usually not fulfilled for high-throughput genomic data.Recently, simplified graphical modeling approaches have been proposed to determine dependencies between gene expression profiles. For sparse graphical models such as genetic networks, it is assumed that the zero-and first-order conditional independencies still reflect reasonably well the full conditional independence structure between variables. Moreover, loworder conditional independencies have the advantage that they can be accurately estimated even when having only a small number of observations. Therefore, using only zero-and first-order conditional dependencies to infer the complete graphical model can be very useful. Here, we analyze the statistical and probabilistic properties of these low-order conditional independence graphs (called 0-1 graphs). We find that for faithful graphical models, the 0-1 graph contains at least all edges of the full conditional independence graph (concentration graph). For simple structures such as Markov trees, the 0-1 graph even coincides with the concentration graph. Furthermore, we present some asymptotic results and we demonstrate in a simulation study that despite their simplicity, 0-1 graphs are generally good estimators of sparse graphical models. Finally, the biological relevance of some applications is summarized.

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“…This means that, given S 0 , we can use algorithms for maximum-likelihood estimation or Bayesian conjugate inference (Roverato 2002) as matrix completion algorithms. To this end, we first draw S 0 from a Wishart distribution…”

Section: Simulation Of Parameters Of a Homogeneous Mixed Gmmmentioning

confidence: 99%

Mapping eQTL Networks with Mixed Graphical Markov Models

2014

Self Cite

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Expression quantitative trait loci (eQTL) mapping constitutes a challenging problem due to, among other reasons, the highdimensional multivariate nature of gene-expression traits. Next to the expression heterogeneity produced by confounding factors and other sources of unwanted variation, indirect effects spread throughout genes as a result of genetic, molecular, and environmental perturbations. From a multivariate perspective one would like to adjust for the effect of all of these factors to end up with a network of direct associations connecting the path from genotype to phenotype. In this article we approach this challenge with mixed graphical Markov models, higherorder conditional independences, and q-order correlation graphs. These models show that additive genetic effects propagate through the network as function of gene-gene correlations. Our estimation of the eQTL network underlying a well-studied yeast data set leads to a sparse structure with more direct genetic and regulatory associations that enable a straightforward comparison of the genetic control of gene expression across chromosomes. Interestingly, it also reveals that eQTLs explain most of the expression variability of network hub genes.

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Hyper Inverse Wishart Distribution for Non‐decomposable Graphs and its Application to Bayesian Inference for Gaussian Graphical Models

Cited by 183 publications

References 32 publications

Structurally-informed Bayesian functional connectivity analysis

Structurally-informed Bayesian functional connectivity analysis

Low-Order Conditional Independence Graphs for Inferring Genetic Networks

Mapping eQTL Networks with Mixed Graphical Markov Models

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