We consider the problem of estimating expectations with respect to a target distribution with an unknown normalizing constant, and where even the unnormalized target needs to be approximated at finite resolution. Under such an assumption, this work builds upon a recently introduced multi-index Sequential Monte Carlo (SMC) ratio estimator, which provably enjoys the complexity improvements of multi-index Monte Carlo (MIMC) and the efficiency of SMC for inference. The present work leverages a randomization strategy to remove bias entirely, which simplifies estimation substantially, particularly in the MIMC context, where the choice of index set is otherwise important. Under reasonable assumptions, the proposed method provably achieves the same canonical complexity of MSE$^{-1}$ as the original method (where MSE is mean squared error), but without discretization bias. It is illustrated on examples of Bayesian inverse and spatial statistics problems.

翻译：我们考虑在目标分布归一化常数未知、且未归一化目标本身需在有限分辨率下逼近的条件下，估计期望值的问题。在此假设下，本文基于近期提出的多指标序贯蒙特卡洛（SMC）比率估计量展开研究，该估计量可证明兼具多指标蒙特卡洛（MIMC）在复杂度上的优势与SMC在推断中的效率。本研究利用随机化策略完全消除偏差，极大简化了估计过程，尤其在MIMC框架中（此时索引集的选择至关重要）。在合理假设下，所提出的方法可证明达到与原方法相同的均方误差（MSE）倒数的标准复杂度，且不含离散化偏差。该方法在贝叶斯反问题与空间统计问题的实例中得到了验证。