  In this article we introduce two new estimates of the normalizing constant
(or marginal likelihood) for partially observed diffusion (POD) processes, with
discrete observations. One estimate is biased but non-negative and the other is
unbiased but not almost surely non-negative. Our method uses the multilevel
particle filter of Jasra et al (2015). We show that, under assumptions, for
Euler discretized PODs and a given $\varepsilon>0$. in order to obtain a mean
square error (MSE) of $\mathcal{O}(\varepsilon^2)$ one requires a work of
$\mathcal{O}(\varepsilon^{-2.5})$ for our new estimates versus a standard
particle filter that requires a work of $\mathcal{O}(\varepsilon^{-3})$. Our
theoretical results are supported by numerical simulations.
