Coupling random inputs for parameter estimation in complex models

Complex stochastic models, such as individual-based models, are becoming increasingly popular. However this complexity can often mean that the likelihood is intractable. Performing parameter estimation on the model can then be difficult. One way of doing this when the complex model is relatively quick to simulate from is approximate Bayesian computation (ABC). Rejection-ABC algorithm is not always efficient so numerous other algorithms have been proposed. One such method is ABC with Markov chain Monte Carlo (ABC–MCMC). Unfortunately for some models this method does not perform well and some alternatives have been proposed including the fsMCMC algorithm (Neal and Huang, in: Scand J Stat 42:378–396, 2015) that explores the random inputs space as well unknown model parameters. In this paper we extend the fsMCMC algorithm and take advantage of the joint parameter and random input space in order to get better mixing of the Markov Chain. We also introduce a Gibbs step that conditions on the current accepted model and allows the parameters to move as well as the random inputs conditional on this accepted model. We show empirically that this improves the efficiency of the ABC–MCMC algorithm on a queuing model and an individual-based model of the group-living bird, the woodhoopoe.