  Many application areas rely on models that can be readily simulated but lack
a closed-form likelihood, or an accurate approximation under arbitrary
parameter values. Existing parameter estimation approaches in this setting are
generally approximate. Recent work on using neural network models to
reconstruct the mapping from the data space to the parameters from a set of
synthetic parameter-data pairs suffers from the curse of dimensionality,
resulting in inaccurate estimation as the data size grows. We propose a
dimension-reduced approach to likelihood-free estimation which combines the
ideas of reconstruction map estimation with dimension-reduction approaches
based on subject-specific knowledge. We examine the properties of
reconstruction map estimation with and without dimension reduction and explore
the trade-off between approximation error due to information loss from reducing
the data dimension and approximation error. Numerical examples show that the
proposed approach compares favorably with reconstruction map estimation,
approximate Bayesian computation, and synthetic likelihood estimation.
