# Local dimension reduction of summary statistics for likelihood-free inference

## TL;DR

This paper introduces a localized approach to dimension reduction in likelihood-free inference methods like ABC, improving the accuracy of summary statistics by validating transformations in the neighborhood of observed data.

## Contribution

It proposes a localization strategy for projection-based dimension reduction methods, ensuring reliable performance near observed data, which enhances inference accuracy.

## Key findings

- Improved estimation accuracy with localized linear regression.
- Enhanced performance of partial least squares in complex models.
- Validation-based optimization ensures transformation reliability.

## Abstract

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.