11 problems

Consider the state-space model and particle approximations $p_{t,\Theta}^N(\mathrm{d} \theta_t|Y_{1:t})$ and $p_{t,\mathsf{X}}^N(\mathrm{d} x_t|Y_{1:t})$ of Corollary 2, with const…

Stochastic rounding schemes assign offspring counts with the stated floor-and-ceiling probabilities; systematic resampling is one such scheme and closely resembles stratified resam…

Weight-sensitive complexity conjecture. The true complexity of sequential Monte Carlo should depend not on the minimum weight, but either on a bottleneck coefficient or on the tota…

Let $X^i$ be the particles in the residual branching particle filter, let $N_n$ be its population size at observation time $t_n$, and let $A^i_{t_n}$ be the particle weights. For a…

Let an SMC algorithm use a stochastic rounding resampling scheme satisfying the standing assumption, and suppose that the incremental transition densities $g_t(x,x')$ satisfy … for…

In sequential Monte Carlo, particles are ordered using a space-filling curve before stratified resampling; the paper considers the Hilbert curve as the ordering mechanism. Gerber e…

Let $\zeta_{\theta,n}^{N}$ be the filter derivative estimator produced by the particle method described in Algorithm 1, and let $\varphi$ be a bounded integrand. Consider the asymp…

Let $\zeta_{\theta,n}^{\mathrm{p},N}$ be the path-space particle approximation of the filter derivative at time $n$, and let $\varphi$ be a bounded integrand. Write the asymptotic…

Let $\{\eta_{\theta,n}\}_{n\geq0}$ be a sequence of filtering measures, let $\zeta_{\theta,n}=\nabla\eta_{\theta,n}$ denote their derivative with respect to $\theta\in\mathbb{R}^d$…

Let $(E,\mathcal{E})$ be the state space, let $\mu$ be the initial distribution, and let $\pi$ be the target invariant distribution. For each $n$ and $0\leq k\leq n$, let…

Conjecture. There is no general way to reduce the variability of the weights in particle sampling further than the method presented here.