I'm trying to understand how to employ MCMC moves in a sequential Monte Carlo procedure for estimating static parameters as in the setting described by Chopin. He proposes, for example, the usage of a Metropolis-Hastings kernel admitting the posterior $\pi_n$ as invariant distribution.
My problem is that, as far as I understood it, for finding the acceptance probability in such a MH kernel I would have to calculate $p(y_1,\dots,y_n|\Theta')$, $y_i$ being the observations made thus far and $\Theta'$ a new particle drawn from a proposal distribution. When my observations are i.i.d this will boil down to evaluating an $n$-fold product of likelihoods for every particle at every iteration. But, because $\Theta'$ is a completely new particle, I can't calculate this sequentially, and would rather have to evaluate the full product at every time step. Effectively, this would render the algorithm useless for any large set of observations.
Did I get something wrong? what can I do to circumvent the problem?