Graphical Models: Selecting causal and statistical models

The topic of this dissertation is graphical models with the major theme being methods for selecting graphical models from observations. I describe how graphical models can be given a statistical and a causal interpretation and investigate what one can and cannot learn about the structure of statistical and causal models from various types of observations under various assumptions. The two key concepts in this investigation are the concepts of distinguishability and representation. I investigate the distinguishability relationships between models for a various types of observations. I also investigate different types of representations for capturing what can and cannot be distinguished regarding the structure of a graphical model within a class of observationally equivalent graphical models. The remainder of the dissertation investigates how one can identify the correct equivalence class of graphical models from observations. I contrast the two basic approaches, the independence approach and the scoring approach, and discuss their application to the problem of model selection. I also discuss the asymptotic properties of several search algorithms for model selection and the asymptotic properties of scores. The search algorithms used for the independence and scoring approaches are asymptotically correct under an assumption called faithfulness. I demonstrate several results regarding faithfulness and argue for the reasonableness of this assumption.