Structured Secant Methods to Select Smoothing Parameters for General Smooth Models
一般平滑模型中选择平滑参数的结构化割线方法AI 解读 ↓
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一般平滑模型中选择平滑参数的结构化割线方法
- 研究问题
- 一般平滑模型通过边际似然选择平滑参数时，如何减少高阶导数与 Hessian 的计算负担，同时维持可用的估计与推断？
- 方法与思路
- 作者提出 qEFS 拟牛顿方法，以结构化有限记忆割线近似 Hessian，并允许结合部分精确列来改进子块近似。
- 材料陈述的贡献
- 摘要称 qEFS 在一定条件下收敛至 EFS，并通过模拟和模型实例展示可用性；区间覆盖与模型选择仍需部分近似。
- 阅读关注点
- 应核对趋近 EFS 的具体条件、精确 Hessian 列的选择，以及点估计、区间覆盖和模型选择对近似误差的不同敏感性。
- 阅读边界
- 材料为 arXiv 第二维度版本所对应的预印本摘要，不能据此声称已同行评审或期刊接收；具体收敛条件与模拟数值未提供。
原文摘要
General smooth models replace parameters of a regular likelihood with additive models. The models can include parametric terms, Gaussian random effects, and smooth functions of covariates. The latter are parameterized via a reduced-rank spline basis and regularized via weighted quadratic penalties placed on the basis coefficients. Estimates for these weights (i.e., smoothing parameters) can be obtained by optimizing the Laplace-approximate Bayesian marginal likelihood. Existing (second-order) methods require the Hessian of the log-likelihood to solve this optimization problem approximately - exact optimization requires up to fourth order derivatives - which can be difficult to derive and expensive to evaluate. To address these problems, we present a quasi-Newton variant of the second-order Extended Fellner-Schall (EFS) optimization method. Our qEFS method relies on structured limited-memory secant approximations to the Hessian of the log-likelihood and is principally first-order. However, the approximation can also be accumulated for a sub-block of the Hessian, with the remaining columns being constrained to match those of the actual Hessian. The exact columns then provide additional structure for the sub-block approximation, which becomes more accurate as a result. We show that the qEFS method converges to the EFS method under certain conditions and continues to provide good estimates beyond these circumstances, which we illustrate in simulation studies. Secondary tasks involving the Hessian (confidence interval coverage & model selection) require partial approximations to achieve close to nominal performance. We provide Hidden Markov and Tweedie model examples, for which the qEFS method is substantially easier to implement than alternative methods.