  Parametric statistical models that are implicitly defined in terms of a
stochastic data generating process are used in a wide range of scientific
disciplines because they enable accurate modeling. However, learning the
parameters from observed data is generally very difficult because their
likelihood function is typically intractable. Likelihood-free Bayesian
inference methods have been proposed which include the frameworks of
approximate Bayesian computation (ABC), synthetic likelihood, and its recent
generalization that performs likelihood-free inference by ratio estimation
(LFIRE). A major difficulty in all these methods is choosing summary statistics
that reduce the dimensionality of the data to facilitate inference. While
several methods for choosing summary statistics have been proposed for ABC, the
literature for synthetic likelihood and LFIRE is very thin to date. We here
address this gap in the literature, focusing on the important special case of
time-series models. We show that convolutional neural networks trained to
predict the input parameters from the data provide suitable summary statistics
for LFIRE. On a wide range of time-series models, a single neural network
architecture produced equally or more accurate posteriors than alternative
methods.
