The alive particle filter

In the following article we develop a particle filter for approximating Feynman-Kac models with indicator potentials. Examples of such models include approximate Bayesian computation (ABC) posteriors associated with hidden Markov models (HMMs) or rare-event problems. Such models require the use of advanced particle filter or Markov chain Monte Carlo (MCMC) algorithms e.g. Jasra et al. (2012), to perform estimation. One of the drawbacks of existing particle filters, is that they may ‘collapse’, in that the algorithm may terminate early, due to the indicator potentials. In this article, using a special case of the locally adaptive particle filter in Lee et al. (2013), which is closely related to Le Gland & Oudjane (2004), we use an algorithm which can deal with this latter problem, whilst introducing a random cost per-time step. This algorithm is investigated from a theoretical perspective and several results are given which help to validate the algorithms and to provide guidelines for their implementation. In addition, we show how this algorithm can be used within MCMC, using particle MCMC (Andrieu et al. 2010). Numerical examples are presented for ABC approximations of HMMs