Using information geometry (an area of mathematics that combines statistics and information theory with differential geometry) I'm investigating properties of models from all over science (such as cosmology, statistical physics, and machine learning). By constructing a mathematical object callled a model manifold – the space of all possible predictions for a model – and analyzing its geometry, properties of the model itself can be revealed.
Visualizing Probabilistic Models
Visualizing complex, high-dimensional data in a way that captures important features can reveal emergent phenomena of a model and provide valuable insight. We developped a new way of embedding complex, probabilisitic data that captures intensive properties, meaning it reveals the information density contained in the data.
Bounding Model Predictions
Bounds from approximating model predictions from interpolation theory translate to geometric bounds on the model manifold – the space of all possible predictions for a model.
Random Matrix Theory
Complex, nonlinear models from all over science exibit a hierarchical structure; perturbing certain parameter combinations have no little to no impact on model predictions, whereas others can cause huge variations. Using random matrix theory, we can explain this hierarchy in parameter importance by expading the Fisher Information Matrix and extracting low-rank matrices.
In statistical physics, the Ising Model is used to describe coupled atomic spins on a latice (in a magnetic field and at some temperature). By visualizing the space of all possible Ising models, we reveal important features of the model, namely a (1) singularity near the critical point – which manifests itself as geometrically as a cusp, and (2) how the manifold flows when the system is coarse-grained.
We visualize the space of exponential families using an intensive embedding we developped, which reveals the information density contained in a distribution. We illustrate the connection between this distance and the partition function of exponential families, and consider two special cases:
(1) 1 Spin System, which can be interpreted as an Ising Model with uncoupled spins, and
(2) Gaussians, which are known to create a space of constant negative curvature.
Physics Education Research
I am working on two projects with Prof. Holmes in physics education research.
The first project is exploring how gendered roles manifest in lab spaces, and the impact this has on task division.
By clustering the quantified behaviours of people in labs, we observe a gender-based division of tasks happening in labs which foster decision making and promote collaboration. In particular, we observe men dominating equipment use and women dominating laptop use.
Through an qualitative analysis of individual groups working in physics labs, we explore how gendered roles manifest themselves, how they are assigned and the impact this has on participation.
The second project I'm working on is the development of the Physics Lab Inventory of Critical Thinking (PLIC), a new assessment that measures critical thinking skills in a lab setting.
The PLIC aims to measure critical thinking in labs. In order to test our instrument, I conducted a series of cognitive interviews, and explore the various ways students interpret data and critique experimental methods.
Often, when collecting complex high dimensional data, there are only a handful of underlying factors that determine the distribution. Using factor analysis, I explore thousands of responses to the PLIC to determine what underlying factors influence people's interpretation of data, criticisms of experimental technique, and how they validate or falsify a model.