have lower variances than the estimates $\{\bar{\pi}^{N,K}_t(\varphi_i)\}_{i=1}^I$. Lastly, when Algorithm \[algo:pi\_tilde\] is neither a particle filter (i.e. $h_t=0$ for some $t>0$) nor the IBIS algorithm (i.e.  $h_t>0$ for some $t>0$) it does not belong to the class of sequential Monte Carlo (SMC) samplers introduced by @DelDouJas:SMC. This point will be explained more precisely in Section \[sub:theory\] where a convergence result for Algorithm \[algo:pi\_tilde\] is provided.

### Ingredients: Resampling and mutation steps\[sub:resampling\] 

Sequential Monte Carlo algorithms require to specify selection steps, where the particles with large weights are replicated while those with low weights are discarded, and mutation steps, where new particles are generated. Selection steps are performed using a resampling algorithm, that can be informally defined as follow [see e.g. @gerber2019negative for a formal definition].

\[def:resampling\] $\mathcal{R}(\cdot,\cdot)$ is a resampling algorithm if $\mathcal{R}(\{x^m,p^m\}_{m=1}^M, L)$ is a probability distribution on $\{x^1,\dots,x^M\}^L$, where $(p^{1},\dots, p^M)\in[0,1]^M$ and $\sum_{m=1}^M p^m=1$. We say that $\mathcal{R}(\cdot,\cdot)$ is unbiased if the random vector $(X^1,\dots,X^L)\sim \mathcal{R}(\{x^m,p^m\}_{m=1}^M, L)$ satisfies ${\mathbb{P}}(X^l=x^m)=p^m$ for all $l$ and $m$. If in addition the random variables $X^1,\dots, X^L$ are independent then $\mathcal{R}(\cdot,\cdot)$ is the multinomial resampling scheme and is denoted by $\mathcal{R}_{\mathrm{multi}}(\cdot,\cdot)$.

On Line \[exact1\] of Algorithm \[algo:pi\_tilde\] various resampling algorithms $\mathcal{R}(\cdot,\cdot)$ can be 