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. This setting is ubiquitous across science and engineering applications, for example in the context of Bayesian inference where a physics-based model governed by an intractable partial differential equation (PDE) appears in the likelihood. A multi-index Sequential Monte Carlo (MISMC) method is used to construct ratio estimators which provably enjoy the complexity improvements of multi-index Monte Carlo (MIMC) as well as the efficiency of Sequential Monte Carlo (SMC) for inference. In particular, the proposed method provably achieves the canonical complexity of MSE$^{-1}$, while single level methods require MSE$^{-\xi}$ for $\xi>1$. This is illustrated on examples of Bayesian inverse problems with an elliptic PDE forward model in $1$ and $2$ spatial dimensions, where $\xi=5/4$ and $\xi=3/2$, respectively. It is also illustrated on a more challenging log Gaussian process models, where single level complexity is approximately $\xi=9/4$ and multilevel Monte Carlo (or MIMC with an inappropriate index set) gives $\xi = 5/4 + \omega$, for any $\omega > 0$, whereas our method is again canonical.

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近似时，对其期望进行估计的问题。这一设定在科学与工程应用中广泛存在，例如在贝叶斯推断中，似然函数会涉及由难以求解的偏微分方程控制的物理模型。本文采用多指标序列蒙特卡罗方法构建比率估计器，其理论性能兼具多指标蒙特卡罗的计算复杂度改进优势与序列蒙特卡罗的推断效率。具体而言，所提方法可证明实现MSE⁻¹的标准复杂度，而单层方法需达到MSE⁻ξ（ξ>1）。我们通过一维和二维空间中以椭圆型偏微分方程为正向模型的贝叶斯逆问题示例进行验证，其中ξ分别为5/4和3/2。此外，在更具挑战性的对数高斯过程模型上，单层方法复杂度约为ξ=9/4，而多层级蒙特卡罗（或使用不恰当指标集的多指标蒙特卡罗）方法可实现ξ=5/4+ω（ω>0），但我们的方法仍保持标准复杂度。