Discussion to: Bayesian graphical models for modern biological applications by Y. Ni, V. Baladandayuthapani, M. Vannucci and F.C. Stingo

We contribute to the discussion of the paper by Ni et al. (Stat Methods Appl, 2021. https://doi.org/10.1007/s10260-021-00572-8) by focusing on two aspects: (i) ordering of the variables for directed acyclic graphical models, and (ii) heterogeneity of the data in the presence of covariates. With regard to (i) we claim that an ordering should be assumed only when strongly reliable prior information is available; otherwise one should proceed with an unspecified ordering to guard against order misspecification. Alternatively, one can carry out Bayesian inference on the space of Markov equivalence classes or use a blend of observational and interventional data to alleviate the lack of identification. With regard to (ii) we complement the Authors’ analysis by enlarging the scope to mixed graphs as well as nonparametric Bayesian models.