Likelihood-Based Inference for Partially Observed Multi-Type Markov Branching Processes

Abstract

Markov branching processes are a class of continuous-time Markov chains (CTMCs) frequently used in stochastic modeling with ubiquitous applications. Bivariate or multi-type processes are necessary to model phenomena such as competition, predation, or infection, but often feature large or uncountable state spaces, rendering many general CTMC techniques impractical. We present spectral techniques to compute the transition probabilities and related quantities discretely and unevenly observed multi-type branching processes, enabling likelihood-based inference. Our technique reduces these calculations to low dimensional integration, and analogously enables calculation of related terms such as expected sufficient statistics within an expectation maximization (EM) algorithm. We rigorously assess our EM algorithm in several simulation studies applied to a birth-death-shift (BDS) model, and apply it to estimate intrapatient time evolution of IS6110 transposable element, a genetic marker frequently used during epidemiological studies of Mycobacterium tuberculosis. Further, we incorporate our methods for computing transition probabilities within a compressed sensing framework, demonstrating scalability in the presence of sparsity. Finally, we extend these ideas to loss function estimation of in vivo hematopoietic rates from single-cell lineage tracking data, and develop efficient Bayesian methods for fitting general stochastic epidemic models to discretely observed time series data and partially observed incidence data.