Advisor - Vladimir Minin
Abstract - Markov branching processes are a class of continuous-time Markov chains (CTMCs) with ubiquitous modeling applications. Multi-type processes are necessary to model phenomena such as competition, predation, or infection, but often feature large or uncountable state spaces, rendering general CTMC techniques impractical. We present new methodology motivated by processes arising in molecular epidemiology, cellular differentiation, and infectious disease dynamics. We present a spectral technique enabling likelihood-based inference of discretely and unevenly observed data from a multi-type branching process, and demonstrate scalability of the method in the presence of sparsity via compressed sensing. We extend these ideas to develop a moment-based loss function estimator for large systems in which the branching process is latent, and apply this technique to estimate rates of cell fate decisions from single-cell lineage tracking data of hematopoiesis, the process of blood cell production. Finally, we investigate a two-type branching approximation to the susceptible-infected-removed (SIR) model, which yields closed forms for transition probabilities. We demonstrate the use of this model toward exact and approximate Bayesian methods for fitting general stochastic epidemic models to partially observed data.