Estimation in tensor Ising models
Estimation in tensor Ising models
复制标题
张量伊辛模型中的估计
DOI：
10.1093/imaiai/iaac007
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Bhattacharya, Bhaswar B
中科院分区：
文献类型：
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作者：
Mukherjee, Somabha;Son, Jaesung;Bhattacharya, Bhaswar B
张量 Ising 模型是一种单参数离散指数族，用于对相关二进制数据进行建模，其中充分统计量是度的多线性形式。这是矩阵伊辛模型的自然推广，它提供了一个方便的数学框架，不仅可以捕获成对的依赖关系，还可以捕获复杂关系数据中的高阶依赖关系。在本文中，我们考虑在给定节点分布中的单个样本的情况下估计张量伊辛模型的自然参数的问题。我们的估计基于最大伪似然（MPL）方法，该方法提供了一种计算有效的算法来估计参数，避免计算棘手的配分函数。我们推导出 MPL 估计一致的一般条件，即它以一定速率收敛到真实参数。我们的条件足够强大，可以处理各种常用的张量伊辛模型，包括具有随机相互作用的自旋玻璃模型和估计速率经历相变的模型。特别是，这包括著名的自旋 Sherrington-Kirkpatrick 模型、一般均匀超图上的自旋系统和超图随机块模型 (HSBM) 上的 Ising 模型中 MPL 估计的结果一致性。事实上，对于 HSBM，我们确定了相变阈值的确切位置，这是由某个平均场变分问题的积极性决定的，使得在该阈值之上，MPL 估计是一致的，而在该阈值之下，没有估计器是一致的。最后，我们推导了张量 Curie-Weiss 模型（完全一致超图上的 Ising 模型）特殊情况下 MPL 估计的精确波动。我们的结果的一个有趣的结果是，Curie-Weiss 模型中的 MPL 估计在高于估计阈值的所有点上都使 Cramer-Rao 下界饱和，也就是说，MPL 估计不会导致可估计性体系中渐近统计效率的损失，即使它是通过仅最小化计算易处理性的真实似然函数的近似值而获得的。
The-tensor Ising model is a one-parameter discrete exponential family for modeling dependent binary data, where the sufficient statistic is a multi-linear form of degree. This is a natural generalization of the matrix Ising model that provides a convenient mathematical framework for capturing, not just pairwise, but higher-order dependencies in complex relational data. In this paper, we consider the problem of estimating the natural parameter of the-tensor Ising model given a single sample from the distribution onnodes. Our estimate is based on the maximum pseudolikelihood (MPL) method, which provides a computationally efficient algorithm for estimating the parameter that avoids computing the intractable partition function. We derive general conditions under which the MPL estimate is-consistent, that is, it converges to the true parameter at rate. Our conditions are robust enough to handle a variety of commonly used tensor Ising models, including spin glass models with random interactions and models where the rate of estimation undergoes a phase transition. In particular, this includes results on-consistency of the MPL estimate in the well-known-spin Sherrington–Kirkpatrick model, spin systems on general-uniform hypergraphs and Ising models on the hypergraph stochastic block model (HSBM). In fact, for the HSBM we pin down the exact location of the phase transition threshold, which is determined by the positivity of a certain mean-field variational problem, such that above this threshold the MPL estimate is-consistent, whereas below the threshold no estimator is consistent. Finally, we derive the precise fluctuations of the MPL estimate in the special case of the-tensor Curie–Weiss model, which is the Ising model on the complete-uniform hypergraph. An interesting consequence of our results is that the MPL estimate in the Curie–Weiss model saturates the Cramer–Rao lower bound at all points above the estimation threshold, that is, the MPL estimate incurs no loss in asymptotic statistical efficiency in the estimability regime, even though it is obtained by minimizing only an approximation of the true likelihood function for computational tractability.