Reversed particle filtering for hidden markov models

We present an approach to selecting the distributions in sampling-resampling which improves the efficiency of the weighted bootstrap. To complement the standard scheme of sampling from the prior and reweighting with the likelihood, we introduce a reversed scheme, which samples from the (normalized) likelihood and reweights with the prior. We begin with some motivating examples, before developing the relevant theory. We then apply the approach to the particle filtering of time series, including nonlinear and non-Gaussian Bayesian state-space models, a task that demands efficiency, given the repeated application of the weighted bootstrap. Through simulation studies on a normal dynamic linear model, Poisson hidden Markov model, and stochastic volatility model, we demonstrate the gains in efficiency obtained by the approach, involving the choice of the standard or reversed filter. In addition, for the stochastic volatility model, we provide three real-data examples, including a comparison with importance sampling methods that attempt to incorporate information about the data indirectly into the standard filtering scheme and an extension to multivariate models. We determine that the reversed filtering scheme offers an advantage over such auxiliary methods owing to its ability to incorporate information about the data directly into the sampling, an ability that further facilitates its performance in higher-dimensional settings.