Prof. Galin Jones, School of Statistics, University of Minnesota (14:00-15:00)

Dr. Flavio Goncalves, Universidade Federal de Minas Gerais, Brazil (15:00-16:00).

Professor Paul Fearnhead, Lancaster University - 14:00-1500

Dr. Sandipan Roy, Department of Mathematical Science, University of Bath (15:00-16:00)

Network Heterogeneity and Strength of Connections

Abstract: Detecting strength of connection in a network is a fundamental problem in understanding the relationship among individuals. Often it is more important to understand how strongly the two individuals are connected rather than the mere presence/absence of the edge. This paper introduces a new concept of strength of connection in a network through a nonparameteric object called “Grafield”. “Grafield” is a piece-wise constant bi-variate kernel function that compactly represents the affinity or strength of ties (or interactions) between every pair of vertices in the graph. We estimate the “Grafield” function through a spectral analysis of the Laplacian matrix followed by a hard thresholding (Gavish & Donoho, 2014) of the singular values. Our estimation methodology is valid for asymmetric directed network also. As a by product we get an efficient procedure for edge probability matrix estimation as well. We validate our proposed approach with several synthetic experiments and compare with existing algorithms for edge probability matrix estimation. We also apply our proposed approach to three real datasets- understanding the strength of connection in (a) a social messaging network, (b) a network of political parties in US senate and (c) a neural network of neurons and synapses in C. elegans, a type of worm.

Speaker: Professor Ingo Scholtez, Department of Informatics, University of Zurich, Switzerland

Speaker: Dr. Ben Calderhead, Department of Mathematics, Imperial College London
Title: Quasi Markov Chain Monte Carlo Methods

Abstract: Quasi-Monte Carlo (QMC) methods for estimating integrals are attractive since the resulting estimators typically converge at a faster rate than pseudo-random Monte Carlo. However, they can be difficult to set up on arbitrary posterior densities within the Bayesian framework, in particular for inverse problems. We introduce a general parallel Markov chain Monte Carlo(MCMC) framework, for which we prove a law of large numbers and a central limit theorem. In that context, non-reversible transitions are investigated. We then extend this approach to the use of adaptive kernels and state conditions, under which ergodicity holds. As a further extension, an importance sampling estimator is derived, for which asymptotic unbiasedness is proven. We consider the use of completely uniformly distributed (CUD) numbers within the above mentioned algorithms, which leads to a general parallel quasi-MCMC (QMCMC) methodology. We prove consistency of the resulting estimators and demonstrate numerically that this approach scales close to n^{-2} as we increase parallelisation, instead of the usual n^{-1} that is typical of standard MCMC algorithms. In practical statistical models we observe multiple orders of magnitude improvement compared with pseudo-random methods.