非高斯状态空间模型的精确拒绝采样
Exact Rejection Sampling for Non-Gaussian State Space Models
AI总结：
本文针对非高斯状态空间模型，构造了满足要求的建议分布，提出基于转换扭转与切线扭转的精确拒绝采样方法，在 T=2000 时实现 75% 的接受概率，大幅提升了采样效率。
AI中文摘要：
拒绝采样需要一个能以已知常数支配目标分布的建议分布，而这通常无法用于非高斯状态空间模型。我们为潜在状态路径构造了这样一个建议分布，从而得到独立的精确平滑 draws（抽样）和一个无偏似然估计量，其相对方差在接受概率为 p 时，每次抽样至多为 1/p - 1。该方法适用于具有仿射高斯动态和对数凹观测密度的标量状态，包括多元观测。Transition twisting（转换扭转）使目标与建议的对数比可分，而切线扭转使每一项非正，从而得到一个可达到的、精确的支配常数。采用 companding node placement（压扩节点放置），对于长度为 T 的样本，若每个时间点有 G 个节点，则累积包络误差为 O(T/G²)，因此 G ∝ √T 可使接受概率远离零；对于随机波动率，所需条件几乎必然成立。一个更简单的以模式为中心的网格在经验上显示出相同的缩放关系。当 T=2000 时，接受概率为 75%，而高斯包络的接受概率约为 10⁻¹⁶。
英文摘要：
Rejection sampling requires a proposal that dominates the target by a known constant, generally unavailable for non-Gaussian state space models. We construct such a proposal for the latent state path, yielding independent exact smoothing draws and an unbiased likelihood estimator whose relative variance is at most $1/p-1$ per draw at acceptance probability $p$. The method covers scalar states with affine Gaussian dynamics and log-concave observation densities, including multivariate observations. Transition twisting makes the log target-to-proposal ratio separable, and tangent-line twists make each term nonpositive, producing an attained, sharp dominating constant. With a companding node placement, the accumulated envelope error is $O(T/G^2)$ for a sample of length $T$ with $G$ nodes per date, so $G\propto\sqrt{T}$ keeps acceptance bounded away from zero; for stochastic volatility, the required conditions hold almost surely. A simpler mode-centered grid shows the same scaling empirically. At $T=2{,}000$, acceptance is $75\%$, versus roughly $10^{-16}$ for the Gaussian envelope.
