Confidence sets in a sparse stochastic block model with two communities of unknown sizes

In a sparse stochastic block model with two communities of unequal sizes we derive two posterior concentration inequalities, that imply (1) posterior (almost-)exact recovery of the community structure under sparsity bounds comparable to well-known sharp bounds in the planted bi-section model; (2) a construction of confidence sets for the community assignment from credible sets, with finite graph sizes. The latter enables exact frequentist uncertain quantification with Bayesian credible sets at non-asymptotic graph sizes, where posteriors can be simulated well. There turns out to be no proportionality between credible and confidence levels: for given edge probabilities and a desired confidence level, there exists a critical graph size where the required credible level drops sharply from close to one to close to zero. At such graph sizes the frequentist decides to include not most of the posterior support for the construction of his confidence set, but only a small subset of community assignments containing the highest amounts of posterior probability (like the maximum-a-posteriori estimator). It is argued that for the proposed construction of confidence sets, a form of early stopping applies to MCMC sampling of the posterior, which would enable the computation of confidence sets at larger graph sizes.