The focus of this thesis is on graphical models, which provide a structured framework for modeling multivariate data utilizing the graph theory and statistics. These models represent complex dependencies among variables and are widely used in fields such as statistics, machine learning and artificial intelligence. This work explores the Gaussian graphical models. A significant part of the research is devoted to understanding the challenges and approaches for selecting the structure of these graphical models from data, a process known as graphical model selection. This involves estimating the conditional dependencies between variables using penalized likelihood methods, such as the graphical lasso algorithm, which is effective in high-dimensional settings. The thesis explores graphical models based on the multivariate Student’s t-distribution, an extension of the univariate t-distribution to higher dimensions. These models are particularly useful for handling heavy-tailed data and data contaminated by outliers. The t-distributed random vectors, however, lack the pairwise Markov property, making the inference using the MLE more complex. To address this, the thesis introduces the Expectation-Maximization (EM) algorithm for estimating parameters in these models, as well as its penalized variant, the tlasso algorithm. The thesis also develops alternative approaches to handle contamination in high-dimensional data, including a tlasso algorithm based on Monte Carlo methods. These models are then demonstrated through simulations, highlighting their potential to improve model selection in multivariate settings.      «

The focus of this thesis is on graphical models, which provide a structured framework for modeling multivariate data utilizing the graph theory and statistics. These models represent complex dependencies among variables and are widely used in fields such as statistics, machine learning and artificial intelligence. This work explores the Gaussian graphical models. A significant part of the research is devoted to understanding the challenges and approaches for selecting the structure of these grap...     »