  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.
