Algebraic properties of linear structural equation models

Thesis

In this thesis, we explore the problems of identifiability and finding constraints in linear structural equation models (SEMs) and explore the applications of such results.

We start by analysing the effects of interventions on linear SEMs and provide sufficient conditions for when the resulting distribution is identifiable. We then demonstrate how we could use a series of interventions to find constraints in linear SEMs, as long as each resulting distribution is identifiable.

Next, we study algebraic properties of a subset of linear SEMs that can be represented by an \emph{HTC-identifiable graph}. HTC-identifiable graphs are a large family of graphs known to be generically identifiable. We prove that a mixed graphs is HTC-identifiable if and only if all the parameters in the model can be recovered from the covariance matrix using straightforward linear algebra operations. We can then exploit this algebraic property to construct an algorithm that recovers the parameters of the model. We also find a set of polynomials, whose indeterminates are entries of the covariance matrix, that generates the zeroes of the model.

Finally, we prove that the maximum likelihood function is strictly convex in a local neighbourhood around the MLE for HTC-identifiable graphs. In particular, for large sample sizes and assuming we specify the correct model, we find a starting point that is in this convex neighbourhood. We then use hill-climbing to find the MLE of the covariance matrix associated with any HTC-identifiable graphical model G. Using this, we can test whether a particular graphical model G is a good fit for a particular data set.

Authors: Yao, B

• DOI: 10.5287/ora-prxzkqv6r
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: graphical models; causal inference; structural equation models
• Subjects: Statistics; Algebra