基于贝叶斯LOO交叉验证的扰动自适应重要性采样
Perturbative adaptive importance sampling for Bayesian LOO cross-validation
AI总结：
本文提出扰动自适应重要性采样方法，通过变换实现高方差问题的解决，结合多种变换策略降低LOO交叉验证的重拟合需求。
摘要：
重要性采样（IS）是贝叶斯模型进行LOO交叉验证时高效替代模型重拟合的方法。IS通过重新加权后验样本来反转单个观测的贝叶斯更新。重要性权重具有高方差，我们通过适应性变换解决此问题。我们观察到移除单个观测会使后验扰动$\mathcal{O}(1/n)$，从而引入生物变换形式$T(θ)=θ+ h Q(θ)$ for $0<h\ll 1$。我们引入了多种此类变换：部分矩匹配，扩展了先前的仿射矩匹配工作，具有可调步长；对数似然下降，部分反转观测的贝叶斯更新；以及梯度流步骤，最小化KL散度或IS方差。梯度流和似然下降变换需要雅可比行列式，可通过自动微分获得；我们还为逻辑回归和浅层ReLU网络推导了闭式表达式。我们在分类（$n\ll p$）、计数回归（泊松和零膨胀负二项式）和生存分析问题中测试了该方法，发现没有单一变换占主导地位，但其组合几乎消除了重拟合的需要。
英文摘要：
Importance sampling (IS) is an efficient stand-in for model refitting in performing (LOO) cross-validation (CV) on a Bayesian model. IS inverts the Bayesian update for a single observation by reweighting posterior samples. The so-called importance weights have high variance -- we resolve this issue through adaptation by transformation. We observe that removing a single observation perturbs the posterior by $\mathcal{O}(1/n)$, motivating bijective transformations of the form $T(θ)=θ+ h Q(θ)$ for $0<h\ll 1.$ We introduce several such transformations: partial moment matching, which generalizes prior work on affine moment-matching with a tunable step size; log-likelihood descent, which partially invert the Bayesian update for an observation; and gradient flow steps that minimize the KL divergence or IS variance. The gradient flow and likelihood descent transformations require Jacobian determinants, which are available via auto-differentiation; we additionally derive closed-form expressions for logistic regression and shallow ReLU networks. We tested the methodology on classification ($n\ll p$), count regression (Poisson and zero-inflated negative binomial), and survival analysis problems, finding that no single transformation dominates but their combination nearly eliminates the need to refit.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
