论文标题
随机块模型中的不确定性定量，类数量未知类
Uncertainty quantification in the stochastic block model with an unknown number of classes
论文作者
论文摘要
我们研究了随机块模型的贝叶斯统计推断的频繁特性，其大小不同。我们为顶点标签的空间配备了类上的先验，并有条件地在标签上有先验。根据图的稀疏性，类可能会随着顶点的数量而生长到无穷大。我们得出表格$ p_ {θ_{0，n}}π_n（b_n \ mid x^n）\leε_n$的非反应后后收缩率，其中$ x^n $是观察到的图形，根据$ p_ {θ_{θ_{0，n} $，$ b_n $，or y is o o y is or y is Or n} \} $或在非常稀疏的情况下，已知范围$θ_{0，n} $的球，而$ε_n$是明确的收敛速率。 这些结果使可信集转换为置信度集。在稀疏的情况下，可靠的测试被证明是置信度集。在非常稀疏的情况下，可靠的集合会放大以形成置信度集。对于可靠水平和收敛速度的函数，对于每个$ n $，置信度水平是明确的。 在后赔率的帮助下，考虑了类数量之间的假设检验，并被证明是一致的。对第一类和第二类的错误以及测试功能的明确下限的明确上限。
We study the frequentist properties of Bayesian statistical inference for the stochastic block model, with an unknown number of classes of varying sizes. We equip the space of vertex labellings with a prior on the number of classes and, conditionally, a prior on the labels. The number of classes may grow to infinity as a function of the number of vertices, depending on the sparsity of the graph. We derive non-asymptotic posterior contraction rates of the form $P_{θ_{0,n}}Π_n(B_n\mid X^n)\le ε_n$, where $X^n$ is the observed graph, generated according to $P_{θ_{0,n}}$, $B_n$ is either $\{θ_{0, n}\}$ or, in the very sparse case, a ball around $θ_{0,n}$ of known extent, and $ε_n$ is an explicit rate of convergence. These results enable conversion of credible sets to confidence sets. In the sparse case, credible tests are shown to be confidence sets. In the very sparse case, credible sets are enlarged to form confidence sets. Confidence levels are explicit, for each $n$, as a function of the credible level and the rate of convergence. Hypothesis testing between the number of classes is considered with the help of posterior odds, and is shown to be consistent. Explicit upper bounds on errors of the first and second type and an explicit lower bound on the power of the tests are given.
