Statistical methods of inference typically require the likelihood function to be computable\ud in a reasonable amount of time. The class of "likelihood-free" methods termed Approximate\ud Bayesian Computation (ABC) is able to eliminate this requirement, replacing the evaluation\ud of the likelihood with simulation from it. Likelihood-free methods have gained in efficiency\ud and popularity in the past few years, following their integration with Markov Chain Monte\ud Carlo (MCMC) and Sequential Monte Carlo (SMC) in order to better explore the parameter\ud space. They have been applied primarily to the estimation of the parameters of a given\ud model, but can also be used to compare models.\ud Here we present novel likelihood-free approaches to model comparison, based upon the\ud independent estimation of the evidence of each model under study. Key advantages of these\ud approaches over previous techniques are that they allow the exploitation of MCMC or SMC\ud algorithms for exploring the parameter space, and that they do not require a sampler able to\ud mix between models. We validate the proposed methods using a simple exponential family\ud problem before providing a realistic problem from population genetics: the comparison of\ud different growth models based upon observations of human Y chromosome data from the\ud terminal generation

Model comparison for the purposes of selection, averaging, and validation is a problem found throughout statistics. Within the Bayesian paradigm, these problems all require the calculation of the posterior probabilities of models within a particular class. Substantial progress has been made in recent years, but difficulties remain in the implementation of existing schemes. This article presents adaptive sequential Monte Carlo (SMC) sampling strategies to characterize the posterior distribution of a collection of models, as well as the parameters of those models. Both a simple product estimator and a combination of SMC and a path sampling estimator are considered and existing theoretical results are extended to include the path sampling variant. A novel approach to the automatic specification of distributions within SMC algorithms is presented and shown to outperform the state of the art in this area. The performance of the proposed strategies is demonstrated via an extensive empirical study. Comparisons with stateof-the-art algorithms show that the proposed algorithms are always competitive, and often substantially superior to alternative techniques, at equal computational cost and considerably less application-specific implementation effort. Supplementary materials for this article are available online.

The Auxiliary Particle Filter (APF) introduced by Pitt and Shephard (1999) is a very popular alternative to Sequential Importance Sampling and Resampling (SISR) algorithms to perform inference in state-space models. We propose a novel interpretation of the APF as an SISR algorithm. This interpretation allows us to present simple guidelines to ensure good performance of the APF and the first convergence results for this algorithm. Additionally, we show that, contrary to popular belief, the asymptotic variance of APF-based estimators is not always smaller than those of the corresponding SISR estimators -even in the 'perfect adaptation' scenario.

We present an offline, iterated particle filter to facilitate statistical inference in general state space hidden Markov models. Given a model and a sequence of observations, the associated marginal likelihood L is central to likelihood-based inference for unknown statistical parameters. We define a class of "twisted" models: each member is specified by a sequence of positive functions ψ and has an associated ψ-auxiliary particle filter that provides unbiased estimates of L. We identify a sequence ψ * that is optimal in the sense that the ψ * -auxiliary particle filter's estimate of L has zero variance. In practical applications, ψ * is unknown so the ψ * -auxiliary particle filter cannot straightforwardly be implemented. We use an iterative scheme to approximate ψ * , and demonstrate empirically that the resulting iterated auxiliary particle filter significantly outperforms the bootstrap particle filter in challenging settings. Applications include parameter estimation using a particle Markov chain Monte Carlo algorithm.