Filippo Ascolani, Antonio Lijoi, and Igor Prünster’s contribution to the Discussion of ‘Martingale Posterior Distributions’ by Fong, Holmes and Walker
Filippo Ascolani, Antonio Lijoi, Igor Prünster · Journal of the Royal Statistical Society Series B (Statistical Methodology) · 2023
We would like to congratulate the authors on their fine and insightful contribution, which provides an original perspective on Bayesian inference and opens up new exciting research directions. The key point is a novel interpretation of the role of prediction: in a Bayesian framework all the uncertainty lies in the unobserved data Yn+1,∞ and, once they are imputed through the predictive distributions, inference is straightforward. Therefore, prediction rules, besides allowing forecasting and extrapolation, are crucial also to infer parameters of interest. In Bayesian non-parametrics there, is a large stream of works focussing on the m-step ahead prediction for exchangeable species sampling data (see, e.g. Favaro et al., 2009; Lijoi et al., 2007), with recent contributions also in the partially exchangeable set-up (Camerlenghi et al., 2017). However, the predictive distributions are always determined through an indirect procedure that relies on the specification of a non-parametric prior and derives the prediction rule as a posterior expected value. The authors adopt a different, and more direct, approach by considering conditionally identically distributed sequences (Berti et al., 2023) that are only asymptotically exchangeable: in this case, the predictive distributions are available in closed form, but predictions may depend on the order of the observed data Y1:n. This seems in contrast with the assumption of independent and identically distributed data that should imply invariance of inferential results with respect to permutation of the observations. Therefore, one may wonder whether the analysis could be extended so to come up with novel exchangeable predictives, without explicit reference to an underlying prior. To overcome the lack of invariance with respect to the ordering of the data, the authors suggest to average the predictions over different permutations of Y1:n. While being computationally unfeasible, averaging over all the permutations induces a symmetry condition which is reminiscent of exchangeability. It would be interesting to check whether the ensuing prediction mechanism actually identifies an exchangeable sequence. This would boil down to showing invariance of the two-step ahead predictives (Fortini et al., 2000). If this were actually the case, the natural goal would be to identify the underlying prior. In general, the standard prior-likelihood mechanism may still be a plus when it comes to describing the dependence structure among the observations (i.e. the generative model). For instance, hierarchical models, that distinguish global and group-specific parameters, have proven to be useful in multiple fields. We wonder whether it would be possible to encode these structures directly into the predictive distributions: in particular, it would be interesting to ascertain whether the neat recursive expressions of Section 4, that allow fast computations, can be retained.