  We introduce a new notion of stability--which we call stable list
decoding--and demonstrate its applicability in designing differentially private
density estimators. This definition is weaker than global stability [ABLMM22]
and is related to the notions of replicability [ILPS22] and list replicability
[CMY23]. We show that if a class of distributions is stable list decodable,
then it can be learned privately in the agnostic setting. As the main
application of our framework, we prove the first upper bound on the sample
complexity of private density estimation for Gaussian Mixture Models in the
agnostic setting, extending the realizable result of Afzali et al. [AAL24].
