**A**t the end of [last] August, Jeremy Heng, Adrian Bishop†, George Deligiannidis and Arnaud Doucet arXived a paper on controlled sequential Monte Carlo (SMC). That we read today at the BiPs reading group in Paris-Saclay, when I took these notes. The setting is classical SMC, but with a twist in that the proposals at each time iteration are modified by an importance function. (I was quite surprised to discover that this was completely new in that I was under the false impression that it had been tried ages ago!) This importance sampling setting can be interpreted as a change of measures on both the hidden Markov chain and on its observed version. So that the overall normalising constant remains the same. And then being in an importance sampling setting there exists an optimal choice for the importance functions. That results in a zero variance estimated normalising constant, unsurprisingly. And the optimal solution is actually the backward filter familiar to SMC users.

A large part of the paper actually concentrates on figuring out an implementable version of this optimal solution. Using dynamic programming. And projection of each local generator over a simple linear space with Gaussian kernels (aka Gaussian mixtures). Which becomes feasible through the particle systems generated at earlier iterations of said dynamic programming.

The paper is massive, both in terms of theoretical results and of the range of simulations, and we could not get through it within the 90 minutes Sylvain LeCorff spent on presenting it. I can only wonder at this stage how much Rao-Blackwellisation or AMIS could improve the performances of the algorithm. (A point I find quite amazing in Proposition 1 is that the normalising constant Z of the filtering distribution does not change along observations when using the optimal importance function, which translates into the estimates being nearly constant after a few iterations.)