非参数问题中近似贝叶斯计算的极小极大速率最优性
On the minimax-rate optimality of approximate Bayesian computation in nonparametric problems
- La Trobe University（拉筹伯大学）
- Kyushu University（九州大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究证明近似贝叶斯计算（ABC）在非参数场景下可达到极小极大速率最优，提出基于统计量筛和局部先验质量条件的ABC后验收缩定理，并验证其在高斯序列估计、密度估计中的有效性。
AI中文摘要：
近似贝叶斯计算（ABC）通过模拟观测数据与合成数据并进行比较，替代了传统贝叶斯计算中的似然评估。我们证明ABC在非参数场景下可达到极小极大速率最优。主要结果是基于统计量筛和局部先验质量条件的ABC后验的一般收缩定理。将该定理应用于Sobolev椭球上的高斯序列估计，以及有界Sobolev型类上的密度估计，在特定模型条件下，构造出其理想后验和后验均值达到对应极小极大速率的ABC方法。在从指定先验采样的条件下，当模拟提议数量在有效维度上以足够大的指数速率增长时，传统蒙特卡洛拒绝ABC算法可继承相同速率。
英文摘要：
Approximate Bayesian computation (ABC) replaces likelihood evaluation by simulation and comparison of observed and synthetic data. We establish minimax-rate guarantees for nonparametric ABC under random-series priors with simulable finite-dimensional coordinates. The contraction theorem uses local prior mass, bounds on ABC acceptance probabilities, and control of prior mass outside a sieve. In fixed-design orthogonal-series regression with centered $g$-and-$k$ errors, an infinite Gaussian series prior with a compact scale hyperprior yields minimax-rate contraction and a minimax-rate clipped posterior mean. In compound Poisson decompounding, only random sums are observed and the target is the underlying jump density. With an unknown count intensity in a fixed compact subinterval of $(0,π/2)$, we prove stability of the zero-count-augmented trigonometric population summaries and use a square-root Gaussian series prior on the space of probability density functions. Over bounded periodic Sobolev classes of smoothness $α>d/2$, a polynomially enlarged synthetic sample yields ABC contraction at rate $n^{-α/(2α+d)}$ and posterior mean squared risk of order $n^{-2α/(2α+d)}$, matching a lower bound for the aggregate-observation model. Rejection-ABC Monte Carlo approximations inherit these rates under sufficient sampling budgets.
