PhD Defense Cecilie Olesen Recke
Title: Identifiability in Statistical Models via Algebraic Geometry
Abstract:
If we are only given partial information about a probability distribution how much about said distribution or its generating process can we recover? This broad question of identifiability will be posed and answered in different ways throughout this thesis using tools from algebraic geometry. The first chapter serves as an introduction to the different types of identifiability in the thesis and how algebraic geometry can be used to answer such questions.
The second chapter introduces the graphical discrete Lyapunov models which arise as the steady-state distributions of first order vector autoregressive models. It is a model of cross-sectional observations from a dynamical system. A directed graph encodes the sparsity pattern of the parameter matrix. Under the assumption of non-Gaussian error terms the tensor equations for the higher-order cumulants are derived, the so-called discrete Lyapunov equations. In this framework we prove generic identifiability of the parameter matrix for any directed acyclic graph with all self-loops and no isolated nodes and local identifiability for all directed graphs with all self-loops and no isolated nodes.
The third chapter introduces a different model of cross-sectional observations from a dynamical system, the continuous Lyapunov model. They arise as the steady-state distributions of a stochastic differential equation with a linear drift matrix whose sparsity pattern is encoded by a directed graph. We derive the higher-order cumulant tensor equations for the steady-state, a generalization of the continuous covariance Lyapunov equation. We prove generic identifiability of the drift matrix for any connected graph with all self-loops. We propose a new semiparametric estimator of the drift matrix and derive its asymptotic distribution by utilizing the identifiability result.
The fourth chapter is on-going work, where we propose an observation noise on top of the steady-state distributions in the previous chapter. First we discuss how to give the models a causal interpretation and then show how the identifiability result from Chapter 3 translates to this set-up when the observation noise is Poisson.
The fifth chapter considers a different type of identifiability than the parameter identifiability discussed in the previous chapters. Instead we consider the completion problem for log-linear models. So we assume that we only observe some coordinates of a discrete probability distribution and discuss when it can completed to a full probability distribution in a given log-linear model. In the case where there are finitely many completions we show that there is either a one or two completions to the log-linear model.
Supervisor: Professor Niels Richard Hansen, University of Copenhagen
Assessment Committee:
Professor Carsten Wiuf (chair), University of Copenhagen
Associate Professor Liam Solus, KTH Royal Institute of Technology
Associate Professor Kaie Kubjas, Aalto University