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Highly non-linear, chaotic or near chaotic, dynamic models are important infields such as ecology and epidemiology: for example, pest species and diseasesoften display highly non-linear dynamics. However, such models are problematicfrom the point of view of statistical inference. The defining feature ofchaotic and near chaotic systems is extreme sensitivity to small changes insystem states and parameters, and this can interfere with inference. There aretwo main classes of methods for circumventing these difficulties: informationreduction approaches, such as Approximate Bayesian Computation or SyntheticLikelihood and state space methods, such as Particle Markov chain Monte Carlo,Iterated Filtering or Parameter Cascading. The purpose of this article is tocompare the methods, in order to reach conclusions about how to approachinference with such models in practice. We show that neither class of methodsis universally superior to the other. We show that state space methods cansuffer multimodality problems in settings with low process noise or modelmis-specification, leading to bias toward stable dynamics and high processnoise. Information reduction methods avoid this problem but, under the correctmodel and with sufficient process noise, state space methods lead tosubstantially sharper inference than information reduction methods. Morepractically, there are also differences in the tuning requirements of differentmethods. Our overall conclusion is that model development and checking shouldprobably be performed using an information reduction method with low tuningrequirements, while for final inference it is likely to be better to switch toa state space method, checking results against the information reductionapproach.

A comparison of inferential methods for highly non-linear state space models in ecology and epidemiology