# Kernel for MCMC moves in sequential monte carlo

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?

At any "time" point $$t$$, you have $$\theta$$ samples targeting $$p(\theta \mid y_{1:t})$$, and you use those samples to target $$p(\theta \mid y_{1:t+1})$$, and so on and so forth. This sort of thing also goes by the name of data annealing.
It appears you’re correct about the move step being the slow part. He says “[n]ote that, when the rejuvenation is performed by such a kernel, a browse through the whole past subsample is needed in the computation of $$\pi_{n+p}(\theta_j^p)$$, which appears in the acceptance probability.”