提出最优矩阵门控得分估计法，提升复杂分布采样精度。
Laplace-Fisher Gate Identities for Optimal Matrix-Gated Blended Score Estimation
- 用矩阵门控融合两种得分估计，动态调节权重以降低方差。
- 推导出理论最优门控公式，适用于奇异或强各向异性目标。
- 适合贝叶斯反问题中需要精确后验评估的研究者使用。
通过反转奥恩斯坦-乌伦贝克扩散采样未归一化目标分布，需估计每层噪声扰动边际分布的得分。现有两种精确恒等式：特威迪恒等式与目标得分恒等式，均可生成无偏有限参考得分估计器。标量加权融合这两种估计器可降方差，但对奇异或强各向异性目标过于僵硬。本文将加权得分估计建模为矩阵值融合系数（即门控）的条件风险最小化问题。核心结果为：最优矩阵门控为 \[ G_/star(y,t) = α_t^2 \left(α_t^2 I_d + γ_t\, \mathbb{E}[H_0(X_0)\mid Y_t=y] \right)^{-1}, \qquad H_0=-\nabla^2\log p_0 \]，其中 $α_t = e^{-t}$，$γ_t = 1-e^{-2t}$ 为 OU 系数，条件期望基于 $X_0$ 在 $Y_t=y$ 下的后验分布。该公式称为拉普拉斯-费舍尔门控恒等式（\ ext{LFGI}）。由于特威迪-目标得分差异的条件均值为零，门控仅改变估计器方差而不影响期望。本文推导出方差最优矩阵门控，给出高斯情形特例，并建立从加权参考样本估计门控的有限参考一致性与稳定性界。随后在贝叶斯反问题中使用有限参考 LFGI 得分估计器进行归一化密度评估。当已有 MCMC 初步样本与导数信息时，LFGI 可利用这些副产品构建后验密度的归一化代理模型。该代理提供单靠 MCMC 样本无法获得的信息：后验能量评估、模型证据估计及下游密度诊断。在约束偏微分方程的反问题基准测试中，相比其他得分估计方法，LFGI 代理显著提升后验密度校准与采样诊断性能。实验验证了在已知模型证据下，无论高斯或非高斯场景，其绝对证据校准效果良好。
原文摘要 · Abstract (English)
Sampling from an unnormalized target density by reversing an Ornstein-Uhlenbeck diffusion requires the score of each noise-perturbed marginal law. Two exact identities are available: Tweedie's identity and a target-score identity, each yielding unbiased finite-reference score estimators for the OU-marginal score. Score estimators induced by scalar blends of Tweedie and TSI score estimators can reduce variance, but they are too rigid for singular or strongly anisotropic targets. We formulate blended score estimation as a conditional risk-minimization problem over matrix valued blending coefficients, referred to as gates. Our central result is to show the optimal matrix valued gate for blended score estimation is given \[ G_\star(y,t) = α_t^2 \left(α_t^2 I_d + γ_t\, \mathbb{E}[H_0(X_0)\mid Y_t=y] \right)^{-1}, \qquad H_0=-\nabla^2\log p_0 .\] Here $α_t = e^{-t}$ and $γ_t = 1-e^{-2t}$ are the OU coefficients, and the conditional expectation is under the OU posterior of $X_0$ given $Y_t=y$. We call this formula the \emph{Laplace-Fisher Gate Identity} (\LFGI{}). Because the Tweedie-TSI disagreement has conditional mean zero, the gate changes the score-estimator variance but not its expected value. We derive the variance-optimal matrix gate, record the Gaussian special case, and establish finite-reference consistency and stability bounds for estimating the gate from weighted reference samples. We then use the finite-reference LFGI score estimator for normalized density evaluation in Bayesian inverse problems. In regimes where MCMC pilot samples and derivative information are already available, LFGI uses those byproducts to construct a normalized surrogate for the posterior density. The resulting surrogate supplies information that the MCMC samples alone do not provide: posterior-energy evaluation, model-evidence estimation, and downstream density-based diagnostics. On a PDE-constrained inverse-problem benchmark, the LFGI surrogate improves posterior-density calibration and sampling diagnostics relative to the other tested score-estimator classes. Experiments using LFGI with known model evidence check absolute evidence calibration in both Gaussian and non-Gaussian settings.
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