M.Sc. Janne Leppä-aho defends his doctoral thesis *Methods for Learning Directed and Undirected Graphical Models* on Friday the 24th of January 2020 at 12 o'clock noon in the University of Helsinki Exactum building, Auditorium B123 (Pietari Kalmin katu 5, 1st floor). His opponent is Senior Researcher Brandon Malone (NEC Laboratories Europe, Germany) and custos Professor Teemu Roos (University of Helsinki). The defence will be held in English.

The thesis of Janne Leppä-aho is a part of research done in the Department of Computer Science and in the Information, Complexity and Learning research group at the University of Helsinki. His supervisor has been Professor Teemu Roos (University of Helsinki).

#### Methods for Learning Directed and Undirected Graphical Models

Probabilistic graphical models provide a general framework for modeling relationships between multiple random variables. The main tool in this framework is a mathematical object called graph which visualizes the assertions of conditional independence between the variables. This thesis investigates methods for learning these graphs from observational data.

Regarding undirected graphical models, we propose a new scoring criterion for learning a dependence structure of a Gaussian graphical model. The scoring criterion is derived as an approximation to often intractable Bayesian marginal likelihood. We prove that the scoring criterion is consistent and demonstrate its applicability to high-dimensional problems when combined with an efficient search algorithm.

Secondly, we present a non-parametric method for learning undirected graphs from continuous data. The method combines a conditional mutual information estimator with a permutation test in order to perform conditional independence testing without assuming any specific parametric distributions for the involved random variables. Accompanying this test with a constraint-based structure learning algorithm creates a method which performs well in numerical experiments when the data generating mechanisms involve non-linearities.

For directed graphical models, we propose a new scoring criterion for learning Bayesian network structures from discrete data. The criterion approximates a hard-to-compute quantity called the normalized maximum likelihood. We study the theoretical properties of the score and compare it experimentally to popular alternatives. Experiments show that the proposed criterion provides a robust and safe choice for structure learning and prediction over a wide variety of different settings.

Finally, as an application of directed graphical models, we derive a closed form expression for Bayesian network Fisher kernel. This provides us with a similarity measure over discrete data vectors, capable of taking into account the dependence structure between the components. We illustrate the similarity measured by this kernel with an example where we use it to seek sets of observations that are important and representative of the underlying Bayesian network model.

#### Availability of the dissertation

An electronic version of the doctoral dissertation is available on the e-thesis site of the University of Helsinki at http://urn.fi/URN:ISBN:978-951-51-5772-0.

Printed copies will be available on request from Janne Leppä-aho: janne.leppa-aho@helsinki.fi.