Multivariate Gaussian distribution is an often encountered continuous distribution in applied mathematics and statistics due to its well known properties and wide applicability. In the graphical models framework, we make use of graphs to compactly represent the conditional independences between a set of random variables. Combining these two together leads to the class of Gaussian graphical models.
This thesis discusses learning of Gaussian graphical models from multivariate data. Given the data, our goal is to identify the graphical structure that specifies the conditional independence statements between the variables under consideration.
Following the footsteps of Pensar et al [10], we adopt a Bayesian, score-based approach for learning graphical models. Using pseudo-likelihood to approximate the true likelihood allows us to apply results of Consonni et al [4] to compute marginal likelihood integrals in closed form. This results in a method that can be used to make objective comparisons among Gaussian graphical models.
We test the method numerically and show that it can be readily applied in high-dimensional settings. According to our tests, the method presented here outperforms the widely used graphical LASSO method in accuracy.
The structure of this thesis is as follows. The chapters 2-4 discuss graphical models, multivariate Normal distribution and Bayesian model comparison in general. The fifth chapter goes through the results derived by Consonni, which are utilised in the next chapter to develop a scoring function and a learning algorithm for Gaussian graphical model selection. In the sixth chapter, we test the method in practice and present the obtained numerical results. The last appendix chapter is dedicated to the consistency proof, which gives the theoretical justification for the presented method.
