This thesis investigates the implication problem for conditional independence (CI) statements and its connection to the phenomenon of faithlessness in undirected Gaussian graphical models. The implication problem concerns determining whether a set of CI statements logically implies another CI statement. We develop a solution of the implication problem where the CI relations correspond to an undirected graphical model combined with an additional CI statement, which is not obtained by the graph. Violations of faithfulness challenge the standard assumptions used in probabilistic graphical models and have significant implications for model selection, inference, and the interpretation of statistical dependencies. By analyzing the interplay between faithlessness and the implication problem, we provide a deeper understanding of the structural and algebraic challenges inherent in undirected Gaussian graphical models. Using algebraic tools, we present a characterization for when the implication problem is solvable within this framework, including cases where the CI statements cannot be faithfully encoded by a single graphical model but require unions of models. Moreover, we provide a graphical solution for this decomposition.      «

This thesis investigates the implication problem for conditional independence (CI) statements and its connection to the phenomenon of faithlessness in undirected Gaussian graphical models. The implication problem concerns determining whether a set of CI statements logically implies another CI statement. We develop a solution of the implication problem where the CI relations correspond to an undirected graphical model combined with an additional CI statement, which is not obtained by the graph. V...     »