Advancements in structure learning of undirected graphical models: from novel specifications to Bayesian and variational inference

This thesis introduces some advances in the field of structure learning of undirected graphical modeling, with a focus on Bayesian inference and variational approximations. First, in the context of count data, a novel model specification based on the Poisson distribution is proposed that demonstrates that, under certain conditions, the induced joint distribution satisfies the properties of a Markov graph. Second, two Bayesian structure learning algorithms are developed: Chain-LPGM, which leverages the proposed model specification, and Bayes-LPGM, which builds upon local Poisson graphical models as introduced by Allen and Liu (2013). Third, posterior inference is addressed within the framework of a general Bayes approach (Bissiri et al.,2016), employing Markov chain Monte Carlo sampling and mean field variational approximation methods. Lastly, a novel approach for structure learning of differential graphical models for exponential family distributions is proposed. This approach utilizes density ratio estimation results and integrates concepts from probabilistic classifiers to allow easy inference. Collectively, these contributions advance theoretical foundations and practical methodologies for graphical modeling and Bayesian inference in complex data settings.