On the (non-)convergence of particle filters with Gaussian importance distributions

We consider convergence properties of particle filters with Gaussian importance distributions for certain time-varying Poisson regression models. We analyze both the classical bounded-importance-weight condition and a more recent moment condition. We show that Gaussian importance distributions based on Laplace approximations or non-linear Kalman filters lead to particle filters that are not guaranteed to converge. We also suggest avoiding the problem by a certain split-Gaussian modification that naturally arises from ensuring bounded weights. Although in this paper we concentrate on the time-varying Poisson regression model, we argue that our findings have implications in more general particle filtering problems.