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 uncertainty 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.