Methods for Reconstructing Networks with Incomplete Information.

Network representations of complex systems are widespread and reconstructing unknown networks from data has been intensively researched in statistical and scientific communities more broadly. Two challenges in network reconstruction problems include having insufficient data to illuminate the full structure of the network and needing to combine information from different data sources. Addressing these challenges, this thesis contributes methodology for network reconstruction in three respects. First, we consider sequentially choosing interventions to discover structure in directed networks focusing on learning a partial order over the nodes. This focus leads to a new model for intervention data under which nodal variables depend on the lengths of paths separating them from intervention targets rather than on parent sets. Taking a Bayesian approach, we present partial-order based priors and develop a novel Markov-Chain Monte Carlo (MCMC) method for computing posterior expectations over directed acyclic graphs. The utility of the MCMC approach comes from designing new proposals for the Metropolis algorithm that move locally among partial orders while independently sampling graphs from each partial order. The resulting Markov Chains mix rapidly and are ergodic. We also adapt an existing strategy for active structure learning, develop an efficient Monte Carlo procedure for estimating the resulting decision function, and evaluate the proposed methods numerically using simulations and benchmark datasets. We next study penalized likelihood methods using incomplete order information as arising from intervention data. To make the notion of incomplete information precise, we introduce and formally define incomplete partial orders which subsumes the important special case of a known total ordering of the nodes. This special case lies along an information lattice and we study the reconstruction performance of penalized likelihood methods at different points along this lattice. Finally, we present a method for ranking a network's potential edges using time-course data. The novelty is our development of a nonparametric gradient-matching procedure and a related summary statistic for measuring the strength of relationships among components in dynamic systems. Simulation studies demonstrate that given sufficient signal moving using this procedure to move from linear to additive approximations leads to improved rankings of potential edges.PhDStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113316/1/jbhender_1.pd

Fast MCMC sampling for Markov jump processes and extensions

Markov jump processes (or continuous-time Markov chains) are a simple and important class of continuous-time dynamical systems. In this paper, we tackle the problem of simulating from the posterior distribution over paths in these models, given partial and noisy observations. Our approach is an auxiliary variable Gibbs sampler, and is based on the idea of uniformization. This sets up a Markov chain over paths by alternately sampling a finite set of virtual jump times given the current path and then sampling a new path given the set of extant and virtual jump times using a standard hidden Markov model forward filtering-backward sampling algorithm. Our method is exact and does not involve approximations like time-discretization. We demonstrate how our sampler extends naturally to MJP-based models like Markov-modulated Poisson processes and continuous-time Bayesian networks and show significant computational benefits over state-of-the-art MCMC samplers for these models.Comment: Accepted at the Journal of Machine Learning Research (JMLR