  Highly non-linear, chaotic or near chaotic, dynamic models are important in
fields such as ecology and epidemiology: for example, pest species and diseases
often display highly non-linear dynamics. However, such models are problematic
from the point of view of statistical inference. The defining feature of
chaotic and near chaotic systems is extreme sensitivity to small changes in
system states and parameters, and this can interfere with inference. There are
two main classes of methods for circumventing these difficulties: information
reduction approaches, such as Approximate Bayesian Computation or Synthetic
Likelihood and state space methods, such as Particle Markov chain Monte Carlo,
Iterated Filtering or Parameter Cascading. The purpose of this article is to
compare the methods, in order to reach conclusions about how to approach
inference with such models in practice. We show that neither class of methods
is universally superior to the other. We show that state space methods can
suffer multimodality problems in settings with low process noise or model
mis-specification, leading to bias toward stable dynamics and high process
noise. Information reduction methods avoid this problem but, under the correct
model and with sufficient process noise, state space methods lead to
substantially sharper inference than information reduction methods. More
practically, there are also differences in the tuning requirements of different
methods. Our overall conclusion is that model development and checking should
probably be performed using an information reduction method with low tuning
requirements, while for final inference it is likely to be better to switch to
a state space method, checking results against the information reduction
approach.
