  In this paper we consider sequential joint state and static parameter
estimation given discrete time observations associated to a partially observed
stochastic partial differential equation (SPDE). It is assumed that one can
only estimate the hidden state using a discretization of the model. In this
context, it is known that the multi-index Monte Carlo (MIMC) method of [11] can
be used to improve over direct Monte Carlo from the most precise discretizaton.
However, in the context of interest, it cannot be directly applied, but rather
must be used within another advanced method such as sequential Monte Carlo
(SMC). We show how one can use the MIMC method by renormalizing the MI identity
and approximating the resulting identity using the SMC$^2$ method of [5]. We
prove that our approach can reduce the cost to obtain a given mean square error
(MSE), relative to just using SMC$^2$ on the most precise discretization. We
demonstrate this with some numerical examples.
