Optimal Bayesian estimation in stochastic block models

Abstract

With the advent of structured data in the form of social networks, genetic circuits and protein interaction networks, statistical analysis of networks has gained popularity over recent years. Stochastic block model constitutes a classical cluster-exhibiting random graph model for networks. There is a substantial amount of literature devoted to proposing strategies for estimating and inferring parameters of the model, both from classical and Bayesian viewpoints. Unlike the classical counterpart, there is however a dearth of theoretical results on the accuracy of estimation in the Bayesian setting. In this article, we undertake a theoretical investigation of the posterior distribution of the parameters in a stochastic block model. In particular, we show that one obtains optimal rates of posterior convergence with routinely used multinomial-Dirichlet priors on cluster indicators and uniform priors on the probabilities of the random edge indicators. En route, we develop geometric embedding techniques to exploit the lower dimensional structure of the parameter space which may be of independent interest.

23 pages

References in corpus (8)

• Stochastic blockmodels and community structure in networks
• Matrix estimation by Universal Singular Value Thresholding
• Needles and Straw in a Haystack: Posterior concentration for possibly sparse sequences
• Asymptotic normality of maximum likelihood and its variational approximation for stochastic blockmodels
• Rate-optimal graphon estimation
• Community extraction for social networks
• Posterior convergence rates for estimating large precision matrices using graphical models
• Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors

Cited by in corpus (3)

• A General Framework for Bayes Structured Linear Models
• Optimal Bayesian Estimation for Random Dot Product Graphs
• Monte Carlo goodness-of-fit tests for degree corrected and related stochastic blockmodels