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61	Addressing Challenges in Graphical Models: MAP estimation, Evidence, Non-Normality, and Subject-Specific Inference Sagar K N Ksheera (15295831) 17 April 2023 (has links) <p>Graphs are a natural choice for understanding the associations between variables, and assuming a probabilistic embedding for the graph structure leads to a variety of graphical models that enable us to understand these associations even further. In the realm of high-dimensional data, where the number of associations between interacting variables is far greater than the available number of data points, the goal is to infer a sparse graph. In this thesis, we make contributions in the domain of Bayesian graphical models, where our prior belief on the graph structure, encoded via uncertainty on the model parameters, enables the estimation of sparse graphs.</p> <p><br></p> <p>We begin with the Gaussian Graphical Model (GGM) in Chapter 2, one of the simplest and most famous graphical models, where the joint distribution of interacting variables is assumed to be Gaussian. In GGMs, the conditional independence among variables is encoded in the inverse of the covariance matrix, also known as the precision matrix. Under a Bayesian framework, we propose a novel prior--penalty dual called the `graphical horseshoe-like' prior and penalty, to estimate precision matrix. We also establish the posterior convergence of the precision matrix estimate and the frequentist consistency of the maximum a posteriori (MAP) estimator.</p> <p><br></p> <p>In Chapter 3, we develop a general framework based on local linear approximation for MAP estimation of the precision matrix in GGMs. This general framework holds true for any graphical prior, where the element-wise priors can be written as a Laplace scale mixture. As an application of the framework, we perform MAP estimation of the precision matrix under the graphical horseshoe penalty.</p> <p><br></p> <p>In Chapter 4, we focus on graphical models where the joint distribution of interacting variables cannot be assumed Gaussian. Motivated by the quantile graphical models, where the Gaussian likelihood assumption is relaxed, we draw inspiration from the domain of precision medicine, where personalized inference is crucial to tailor individual-specific treatment plans. With an aim to infer Directed Acyclic Graphs (DAGs), we propose a novel quantile DAG learning framework, where the DAGs depend on individual-specific covariates, making personalized inference possible. We demonstrate the potential of this framework in the regime of precision medicine by applying it to infer protein-protein interaction networks in Lung adenocarcinoma and Lung squamous cell carcinoma.</p> <p><br></p> <p>Finally, we conclude this thesis in Chapter 5, by developing a novel framework to compute the marginal likelihood in a GGM, addressing a longstanding open problem. Under this framework, we can compute the marginal likelihood for a broad class of priors on the precision matrix, where the element-wise priors on the diagonal entries can be written as gamma or scale mixtures of gamma random variables and those on the off-diagonal terms can be represented as normal or scale mixtures of normal. This result paves new roads for model selection using Bayes factors and tuning of prior hyper-parameters.</p> Applied statistics Biostatistics Computational statistics Statistical data science Statistical theory graphical models non-convex optimization posterior concentration posterior consistency sparsity complete monotonicity graph structure learning graphical horseshoe prior precision matrix estimation Global-local shrinkage priors Precision medicine Quantile regression Varying sparsity model Bayes factor Chib's method Marginal likelihood
62	Development of High-order CENO Finite-volume Schemes with Block-based Adaptive Mesh Refinement (AMR) Ivan, Lucian 31 August 2011 (has links) A high-order central essentially non-oscillatory (CENO) finite-volume scheme in combination with a block-based adaptive mesh refinement (AMR) algorithm is proposed for solution of hyperbolic and elliptic systems of conservation laws on body- fitted multi-block mesh. The spatial discretization of the hyperbolic (inviscid) terms is based on a hybrid solution reconstruction procedure that combines an unlimited high-order k-exact least-squares reconstruction technique following from a fixed central stencil with a monotonicity preserving limited piecewise linear reconstruction algorithm. The limited reconstruction is applied to computational cells with under-resolved solution content and the unlimited k-exact reconstruction procedure is used for cells in which the solution is fully resolved. Switching in the hybrid procedure is determined by a solution smoothness indicator. The hybrid approach avoids the complexity associated with other ENO schemes that require reconstruction on multiple stencils and therefore, would seem very well suited for extension to unstructured meshes. The high-order elliptic (viscous) fluxes are computed based on a k-order accurate average gradient derived from a (k+1)-order accurate reconstruction. A novel h-refinement criterion based on the solution smoothness indicator is used to direct the steady and unsteady refinement of the AMR mesh. The predictive capabilities of the proposed high-order AMR scheme are demonstrated for the Euler and Navier-Stokes equations governing two-dimensional compressible gaseous flows as well as for advection-diffusion problems characterized by the full range of Peclet numbers, Pe. The ability of the scheme to accurately represent solutions with smooth extrema and yet robustly handle under-resolved and/or non-smooth solution content (i.e., shocks and other discontinuities) is shown for a range of problems. Moreover, the ability to perform mesh refinement in regions of smooth but under-resolved and/or non-smooth solution content to achieve the desired resolution is also demonstrated. computational fluid dynamics finite-volume method high-order scheme adaptive mesh refinement (AMR) ENO scheme essentially non-oscillatory CENO body-fitted multi-block mesh high-order spatial discretization numerical method hyperbolic and elliptic PDE fixed stencil reconstruction piecewise linear reconstruction smoothness indicator Euler and Navier-Stokes equations advection-diffusion equation smooth and non-smooth solution content k-exact least-squares reconstruction hybrid numerical scheme compressible gaseous flows refinement criteria high-performance parallel computing solution monotonicity enforcement large-eddy simulation (LES) interpolation technique structured grid inviscid and viscous flow TVD scheme h-p-adaptation high-order boundary conditions complex curved geometries 0538
63	Development of High-order CENO Finite-volume Schemes with Block-based Adaptive Mesh Refinement (AMR) Ivan, Lucian 31 August 2011 (has links) A high-order central essentially non-oscillatory (CENO) finite-volume scheme in combination with a block-based adaptive mesh refinement (AMR) algorithm is proposed for solution of hyperbolic and elliptic systems of conservation laws on body- fitted multi-block mesh. The spatial discretization of the hyperbolic (inviscid) terms is based on a hybrid solution reconstruction procedure that combines an unlimited high-order k-exact least-squares reconstruction technique following from a fixed central stencil with a monotonicity preserving limited piecewise linear reconstruction algorithm. The limited reconstruction is applied to computational cells with under-resolved solution content and the unlimited k-exact reconstruction procedure is used for cells in which the solution is fully resolved. Switching in the hybrid procedure is determined by a solution smoothness indicator. The hybrid approach avoids the complexity associated with other ENO schemes that require reconstruction on multiple stencils and therefore, would seem very well suited for extension to unstructured meshes. The high-order elliptic (viscous) fluxes are computed based on a k-order accurate average gradient derived from a (k+1)-order accurate reconstruction. A novel h-refinement criterion based on the solution smoothness indicator is used to direct the steady and unsteady refinement of the AMR mesh. The predictive capabilities of the proposed high-order AMR scheme are demonstrated for the Euler and Navier-Stokes equations governing two-dimensional compressible gaseous flows as well as for advection-diffusion problems characterized by the full range of Peclet numbers, Pe. The ability of the scheme to accurately represent solutions with smooth extrema and yet robustly handle under-resolved and/or non-smooth solution content (i.e., shocks and other discontinuities) is shown for a range of problems. Moreover, the ability to perform mesh refinement in regions of smooth but under-resolved and/or non-smooth solution content to achieve the desired resolution is also demonstrated. computational fluid dynamics finite-volume method high-order scheme adaptive mesh refinement (AMR) ENO scheme essentially non-oscillatory CENO body-fitted multi-block mesh high-order spatial discretization numerical method hyperbolic and elliptic PDE fixed stencil reconstruction piecewise linear reconstruction smoothness indicator Euler and Navier-Stokes equations advection-diffusion equation smooth and non-smooth solution content k-exact least-squares reconstruction hybrid numerical scheme compressible gaseous flows refinement criteria high-performance parallel computing solution monotonicity enforcement large-eddy simulation (LES) interpolation technique structured grid inviscid and viscous flow TVD scheme h-p-adaptation high-order boundary conditions complex curved geometries 0538
64	Extremes of log-correlated random fields and the Riemann zeta function, and some asymptotic results for various estimators in statistics Ouimet, Frédéric 05 1900 (has links) No description available. extreme value theory Gaussian free field branching random walk inhomogeneous environment Riemann zeta function Gibbs measure Ghirlanda-Guerra identities ultrametricity large deviations asymptotic statistics complete monotonicity multinomial probabilities Bernstein estimators uniform law of large numbers Laplace distribution goodness-of-fit tests théorie des valeurs extrêmes champ libre gaussien marche aléatoire branchante environnements inhomogènes fonction zêta de Riemann mesure de Gibbs identités de Ghirlanda-Guerra ultramétricité grandes déviations statistique asymptotique monotonicité complète probabilités multinomiales estimateurs de Bernstein loi uniforme des grands nombres loi de Laplace tests d'ajustements probability probabilité statistics statistique champs log-corrélés log-correlated fields mathematics mathématiques Gaussian fields champs gaussiens

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