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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.
We propose a one-step procedure to estimate the latent positions in random dot product graphs efficiently. Unlike the classical spectral-based methods such as the adjacency and Laplacian spectral embedding, the proposed one-step procedure takes advan
We derive the optimal proposal density for Approximate Bayesian Computation (ABC) using Sequential Monte Carlo (SMC) (or Population Monte Carlo, PMC). The criterion for optimality is that the SMC/PMC-ABC sampler maximise the effective number of sampl
We study a nonparametric Bayesian approach to estimation of the volatility function of a stochastic differential equation driven by a gamma process. The volatility function is modelled a priori as piecewise constant, and we specify a gamma prior on i
We consider high-dimensional measurement errors with high-frequency data. Our focus is on recovering the covariance matrix of the random errors with optimality. In this problem, not all components of the random vector are observed at the same time an
We consider nonparametric inference of finite dimensional, potentially non-pathwise differentiable target parameters. In a nonparametric model, some examples of such parameters that are always non pathwise differentiable target parameters include pro