Conditioning on robust summaries of the data in a Bayesian model is one way to achieve robustness to model miss-specification. I have called this the “restricted likelihood” here and here since the full data likelihood is replaced with the likelihood conditioned on only the robust summary (i.e. a restricted likelihood).
One of the easiest examples to conceptualize is outliers in a univariate setting. Suppose the true data generating mechanism is a contaminated normal: