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.

中文翻译：

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随机点积图的最优贝叶斯估计

我们提出了一种称为后验谱嵌入的贝叶斯方法，用于估计随机点积图中的潜在位置，并证明其最优性。与经典的基于谱的邻接/拉普拉斯谱嵌入不同，后验谱嵌入是一种利用观察到的邻接矩阵的伯努利似然信息的基于完全似然的图估计方法。我们开发了一个用于估计潜在位置的极小极大下界，并表明后验谱嵌入达到了这个下界，因为它既导致极小极大优化后收缩率，又产生了一个点估计器以渐进方式实现极大极小风险。随后将收敛结果应用于随机块模型中的聚类，其结果加强了关于错误聚类顶点数量的现有结果。我们还研究了基于谱的高斯谱嵌入作为邻接谱嵌入的自然贝叶斯类比，但由此产生的后收缩率是次优的，具有额外的对数因子。通过广泛的综合示例和对维基百科图形数据的分析，说明了所提出方法的实际性能。