High-dimensional approximate Bayesian structure learning of undirected graphical models (co-supervised by Camilla Lingjærde)
The aim of this project will be to develop a method for learning the undirected graph structure (or the dependence structure) over a large collection of variables. More specifically, the idea will be to extend the method developed in Pensar et al (2017) and Leppä-aho et al (2017) through Bayesian model averaging, which will enable computation of (approximate) edge inclusion probabilities, i.e. the probability of there being an edge between two nodes in the graph. The project will have more of an "algorithmic" nature and a particular focus will be put on scalability, in the sense that the method should be applicable on systems containing at least hundreds of variables.
Generative probabilistic classification with graphical models
Probabilistic classification models for predicting the class of some categorical variable \(Y\) based on some features \(\boldsymbol{X}=(X_1 , \ldots , X_n)\) can be either discriminative or generative. While discriminative model is trained directly on the objective of interest, \(P(Y \mid \boldsymbol{X})\), a generative model focus on the modelling the class-specific distributions of the features \(P(\boldsymbol{X} \mid Y)\) and produce predictions using Bayes' formula, \(P(Y \mid \boldsymbol{X}) \propto P(\boldsymbol{X} \mid Y) P(Y)\). Discriminative models are most often superior in terms predictive accuracy, but not always (See Ng & Jordan, 2001). On the other hand, generative models can be expected to possess some advantageous properties, for example, when it comes to uncertainty-awareness and imbalanced class distributions. The aim of this project will be to compare the performance of discriminative models against graphical-model-based generative models in different settings.