  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.
