Speaker: Professor Sayan Mukherjee, Department of Statistical Science, Duke University
Title: Stochastic topology and inference
Abstract: We will discuss three examples where stochastic topology is relevant to statistical inference or probability theory.
Modeling surfaces: Given morphological data in the form of meshes we introduce two related statistical summaries, the Euler Characteristic Transform and the Persistent Homology Transform. We use the PHT and ECT to represent shapes and execute operations such as computing distances between shapes. We prove the transforms are injective maps and satisfy a formal definition of statistical sufficiency. In addition, the ECT provides for a simple exponential family formulation which allows for likelihood based modeling of surfaces without the need for landmarks. We present results on a set heel bones of 106 extinct and extant primates.
Percolation on manifolds: Given n points drawn from a point process on a manifold, consider the random set which consists of the union of balls of radius r around the points. As n goes to infinity, r is sent to zero at varying rates. For this stochastic process we provide scaling limits and phase transitions on the counts of Betti numbers and critical points. This study falls into the category of higher-dimensional notions of percolation.
Random hypergraph models: It may be of interest to model conditional independence structure beyond graphs with the goal of capturing higher-order interactions. We develop a framework for posterior inference and prior specification for random hypergraphs using ideas from computational geometry and spatial point processes. We illustrate the utility of this approach on simulated data.
Joint work with: Katharine Turner and Doug Boyer, Omer Bobrowski, Simon Lunagomez, Robert Wolpert, Edo Airoldi