Browsing by Subject "Markov random fields"
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Item High-dimensional statistics : model specification and elementary estimators(2014-12) Yang, Eunho; Ravikumar, PradeepModern statistics typically deals with complex data, in particular where the ambient dimension of the problem p may be of the same order as, or even substantially larger than, the sample size n. It has now become well understood that even in this type of high-dimensional scaling, statistically consistent estimators can be achieved provided one imposes structural constraints on the statistical models. In spite of great success over the last few decades, we are still experiencing bottlenecks of two distinct kinds: (I) in multivariate modeling, data modeling assumption is typically limited to instances such as Gaussian or Ising models, and hence handling varied types of random variables is still restricted, and (II) in terms of computation, learning or estimation process is not efficient especially when p is extremely large, since in the current paradigm for high-dimensional statistics, regularization terms induce non-differentiable optimization problems, which do not have closed-form solutions in general. The thesis addresses these two distinct but highly complementary problems: (I) statistical model specification beyond the standard Gaussian or Ising models for data of varied types, and (II) computationally efficient elementary estimators for high-dimensional statistical models.Item Spatial applications of Markov random fields and neural networks for spatio-temporal denoising, causal inference and reinforcement learning(2022-08-16) García Tec, Mauricio Benjamín; Scott, James (Statistician); Zigler, Corwin Matthew, 1983-; Zhou, Mingyuan; Walker, Stephen G; Stone, Peter HDiscrete spatial structures are ubiquitous in statistical analysis. They can take the form of images, grids, and more generally, graphs. This work develops novel methodology leading to broadly applicable algorithms of graph smoothing and neural newtorks to improve statistical learning in a variety of tasks and spatially-structured domains, including temporal and sequential decision-making processes. Thus, each chapter corresponds to a case study with applications in spatio-temporal denoising, causal inference, and reinforcement learning. Graph smoothing methods are used in all of them and their effectiveness is evaluated. In addition, some chapters develop more specialized methods that further exploit the spatial and statistical structure of the data. One of the objectives sustained throughout the work will be developing scalable algorithms to handle high-resolution spatial data or other computationally demanding scenarios.