# Robust approximate Bayesian inference

## TL;DR

This paper introduces a robust Bayesian inference method using $M$-estimating functions within ABC algorithms, with theoretical analysis, simulations, and a clinical application demonstrating its effectiveness.

## Contribution

It presents a novel approach combining $M$-estimating functions with ABC for robust posterior inference, including theoretical properties and practical implementation.

## Key findings

- The method produces robust posterior distributions in linear mixed models.
- Simulation studies validate the approach's effectiveness.
- Application to clinical data demonstrates practical utility.

## Abstract

We discuss an approach for deriving robust posterior distributions from $M$-estimating functions using Approximate Bayesian Computation (ABC) methods. In particular, we use $M$-estimating functions to construct suitable summary statistics in ABC algorithms. The theoretical properties of the robust posterior distributions are discussed. Special attention is given to the application of the method to linear mixed models. Simulation results and an application to a clinical study demonstrate the usefulness of the method. An R implementation is also provided in the robustBLME package.