Algebraic Aspects of Hidden Variable Models

Hidden variables are random variables that we cannot observe in reality but they are important for understanding the phenomenon of our interest because they affect the observable variables. Hidden variable models aim to represent the effect of the presence of hidden variables which are theoretically thought to exist but we have no data on them. In this thesis, we focus on two hidden variable models in phylogenetics and statistics. In phylogenetics, we seek answers to two important questions related to modeling evolution. First, we study the embedding problem in the group-based models and the strand symmetric model and its higher order generalizations. In Publication I, we provide some embeddability criteria in the group-based models equipped with certain labeling. In Publication III, we characterize the embeddability in the strand symmetric model. These results allow us to measure approximately the proportion of the set of embeddable Markov matrices within the space of Markov matrices. These results generalize the previously established embeddability results on the Jukes-Cantor and Kimura models. The second question of our interest concerns with the distinguishability of phylogenetic network models which is related to the notion of generic identifiability. In Publication II, we provide some conditions on the network topology that ensure the distinguishability of their associated phylogenetic network models under some group-based models. The last part of this thesis is dedicated to studying the factor analysis model which is a statistical model that seeks to reduce a large number of observable variables into a fewer number of hidden variables. The factor analysis model assumes that the observed variables can be presented as a linear combination of the hidden variables together with some error terms. Moreover, the observed and the hidden variables together with the error terms are assumed to be Gaussian. We generalize the factor analysis model by dropping the Gaussianity assumption and introduce the higher order factor analysis model. In Publication IV, we provide the dimension of the higher order factor analysis model and present some conditions under which the model has positive codimension.