The Alive Particle Filter and its use in Particle Markov chain Monte

In the following article we investigate a particle filter for approximating Feynman-Kac models with indicator potentials and we use this algorithm within Markov chain Monte Carlo (MCMC) to learn static parameters of the model. 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 MCMC algorithms e.g. [13], 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 newly developed special case of the locally adaptive particle filter in [14], which is closely related to [16], we use an algorithm which can deal with this latter problem, whilst introducing a random cost per-time step. In particular, we show how this algorithm can be used within MCMC, using particle MCMC [2]. It is established that, when not taking into account computational time, when the new MCMC algorithm is applied to a simplified model it has a lower asymptotic variance in comparison to a standard particle MCMC algorithm. Numerical examples are presented for ABC approximations of HMMs.