论文标题
贝叶斯反问题的多指数顺序蒙特卡洛比率估计器
Multi-index Sequential Monte Carlo ratio estimators for Bayesian Inverse problems
论文作者
论文摘要
我们考虑了估计对目标分布的期望的问题，该目标分布具有未知的归一化常数，即使在有限的分辨率下也需要近似未归一化的目标。在科学和工程应用程序中，这种设置无处不在，例如，在贝叶斯推断的背景下，基于物理的模型由棘手的部分微分方程（PDE）出现在可能性上。多指数顺序的蒙特卡洛（MISMC）方法用于构建比率估计器，该估计量可证明其复杂性改善了多指数蒙特卡洛（MIMC）以及顺序蒙特卡洛（SMC）的效率。特别是，所提出的方法可证明达到了MSE $^{ - 1} $的规范复杂性，而单一级别的方法需要MSE $^{ - ξ} $，对于$ξ> 1 $。这在贝叶斯逆问题的示例中以$ 1 $和$ 2 $的空间尺寸为例，其中$ξ= 5/4 $和$ξ= 3/2 $。在更具挑战性的日志高斯过程模型中也可以说明，其中单级复杂性约为$ξ= 9/4 $，而多级蒙特卡洛（或具有不适当索引集的MIMC）可提供$ξ= 5/4 +ω$，对于任何$ω> 0 $，而我们的方法再次是canonical。
We consider the problem of estimating expectations with respect to a target distribution with an unknown normalizing constant, and where even the unnormalized target needs to be approximated at finite resolution. This setting is ubiquitous across science and engineering applications, for example in the context of Bayesian inference where a physics-based model governed by an intractable partial differential equation (PDE) appears in the likelihood. A multi-index Sequential Monte Carlo (MISMC) method is used to construct ratio estimators which provably enjoy the complexity improvements of multi-index Monte Carlo (MIMC) as well as the efficiency of Sequential Monte Carlo (SMC) for inference. In particular, the proposed method provably achieves the canonical complexity of MSE$^{-1}$, while single level methods require MSE$^{-ξ}$ for $ξ>1$. This is illustrated on examples of Bayesian inverse problems with an elliptic PDE forward model in $1$ and $2$ spatial dimensions, where $ξ=5/4$ and $ξ=3/2$, respectively. It is also illustrated on a more challenging log Gaussian process models, where single level complexity is approximately $ξ=9/4$ and multilevel Monte Carlo (or MIMC with an inappropriate index set) gives $ξ= 5/4 + ω$, for any $ω> 0$, whereas our method is again canonical.
