  We propose a Bayesian approach, called the posterior spectral embedding, for
estimating the latent positions in random dot product graphs, and prove its
optimality. Unlike the classical spectral-based adjacency/Laplacian spectral
embedding, the posterior spectral embedding is a fully-likelihood based graph
estimation method taking advantage of the Bernoulli likelihood information of
the observed adjacency matrix. We develop a minimax-lower bound for estimating
the latent positions, and show that the posterior spectral embedding achieves
this lower bound since it both results in a minimax-optimal posterior
contraction rate, and yields a point estimator achieving the minimax risk
asymptotically. The convergence results are subsequently applied to clustering
in stochastic block models, the result of which strengthens an existing result
concerning the number of mis-clustered vertices. We also study a spectral-based
Gaussian spectral embedding as a natural Bayesian analogy of the adjacency
spectral embedding, but the resulting posterior contraction rate is sub-optimal
with an extra logarithmic factor. The practical performance of the proposed
methodology is illustrated through extensive synthetic examples and the
analysis of a Wikipedia graph data.
