潜变量模型估计的计算：统一随机近端框架
Computation for Latent Variable Model Estimation: A Unified Stochastic Proximal Framework
AI总结：
针对含大量潜变量、混合变量类型、参数约束与惩罚的潜变量模型估计问题，本文提出统一优化表述及拟牛顿随机近端算法，并证明其理论性质与计算优势。
摘要：
潜变量模型在心理测量学及相关领域一直发挥着核心作用。在许多现代应用中，基于潜变量模型的推断涉及以下一个或多个特征：（1）存在大量潜变量；（2）观测变量和潜变量为连续型、离散型或两者的组合；（3）参数受到约束；（4）对参数施加惩罚以实现模型简约。估计过程通常涉及最大化基于边际似然/伪似然的目标函数，并可能伴随参数约束和/或惩罚。由于上述特征带来的复杂性，求解这一优化问题极具挑战性。尽管已有若干高效算法被提出，但仍缺乏一个能够兼顾所有这些特征的统一计算框架。本文填补了这一空白。具体而言，我们对该优化问题给出了统一表述，随后提出一种拟牛顿随机近端算法。我们建立了所提算法的理论性质，并通过潜变量模型估计多种设置下的模拟研究，展示了其计算效率与稳健性。
英文摘要：
Latent variable models have been playing a central role in psychometrics and related fields. In many modern applications, the inference based on latent variable models involves one or several of the following features: (1) the presence of many latent variables, (2) the observed and latent variables being continuous, discrete, or a combination of both, (3) constraints on parameters, and (4) penalties on parameters to impose model parsimony. The estimation often involves maximizing an objective function based on a marginal likelihood/pseudo-likelihood, possibly with constraints and/or penalties on parameters. Solving this optimization problem is highly non-trivial, due to the complexities brought by the features mentioned above. Although several efficient algorithms have been proposed, there lacks a unified computational framework that takes all these features into account. In this paper, we fill the gap. Specifically, we provide a unified formulation for the optimization problem and then propose a quasi-Newton stochastic proximal algorithm. Theoretical properties of the proposed algorithms are established. The computational efficiency and robustness are shown by simulation studies under various settings for latent variable model estimation.
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