  Approximate Bayesian computation (ABC) and other likelihood-free inference
methods have gained popularity in the last decade, as they allow rigorous
statistical inference for complex models without analytically tractable
likelihood functions. A key component for accurate inference with ABC is the
choice of summary statistics, which summarize the information in the data, but
at the same time should be low-dimensional for efficiency. Several dimension
reduction techniques have been introduced to automatically construct
informative and low-dimensional summaries from a possibly large pool of
candidate summaries. Projection-based methods, which are based on learning
simple functional relationships from the summaries to parameters, are widely
used and usually perform well, but might fail when the assumptions behind the
transformation are not satisfied. We introduce a localization strategy for any
projection-based dimension reduction method, in which the transformation is
estimated in the neighborhood of the observed data instead of the whole space.
Localization strategies have been suggested before, but the performance of the
transformed summaries outside the local neighborhood has not been guaranteed.
In our localization approach the transformation is validated and optimized over
validation datasets, ensuring reliable performance. We demonstrate the
improvement in the estimation accuracy for localized versions of linear
regression and partial least squares, for three different models of varying
complexity.
