Conditional particle filters with bridge backward sampling

The performance of the conditional particle filter (CPF) with backward sampling is often impressive even with long data records. Two known exceptions are when the observations are weakly informative and the dynamic model is slowly mixing. These are both present when sampling finely time-discretised continuous-time path integral models, but can occur with hidden Markov models too. Multinomial resampling, which is commonly employed in the (backward sampling) CPF, resamples excessively for weakly informative observations and thereby introduces extra variance. A slowly mixing dynamic model renders the backward sampling step ineffective. We detail two conditional resampling strategies suitable for the weakly informative regime: the so-called `killing' resampling and the systematic resampling with mean partial order. To avoid the degeneracy issue of backward sampling, we introduce a generalisation that involves backward sampling with an auxiliary `bridging' CPF step, which is parameterised by a blocking sequence. We present practical tuning strategies for choosing an appropriate blocking. Our experiments demonstrate that the CPF with a suitable resampling and the developed `bridge backward sampling' can lead to substantial efficiency gains in the weakly informative regime.