Contributed discussion on “Robust probabilistic inference via a constrained transport metric” by A. Chakraborty, A. Bhattacharya, and D. Pati

We congratulate the authors for an original and stimulating contribution. The proposed D-BETEL framework offers an elegant and effective compromise between robustness and interpretability, where inference remains focused on a finite-dimensional parameter θ, indexing an interpretable parametric model Fθ, while still avoiding a full specification of the data-generating distribution. The procedure is particularly appealing in that inference results from a prior, specified only on the parameter of interest θ, and a modified likelihood contribution, obtained by finding the maximum-entropy reweighting of the empirical distribution of the observations that lies within a neighborhood of Fθ. Hence, values of θ are favored when this distance-based compatibility with Fθ can be achieved with little departure from the uniform empirical weights. In this sense, the modified likelihood rewards models Fθ that explain the bulk of the data while still allowing atypical observations to be downweighted. We view the nonparametric Bayesian interpretation in Section 5, which is shown to arise in an appropriate asymptotic regime, as a distinctive strength of this approach. Unlike many alternative pseudo-likelihood or generalized Bayes approaches, D-BETEL is shown to arise as a limiting marginal posterior under a suitable nonparametric hierarchical construction, thereby providing a genuine probabilistic motivation for its use.
A feature we find especially interesting is that the proposed approach allows the prior distribution to be defined directly on the finite-dimensional parameter of interest θ, while still encompassing data-generating processes more general than the …