Isadora Antoniano Villalobos's contribution to the Discussion of ‘Martingale Posterior Distributions’ by Fong, Holmes and Walker

原文摘要

I thank the authors for their refreshing ideas, providing many insights and intuitions on the process of statistical learning. For years, Bayesians have considered two levels of uncertainty, the likelihood and the prior. By shifting the focus to the predictive distribution as a single expression of uncertainty about a future observation (given data), the sequential nature of statistical learning is made explicit, together with some of its limitations. If each observation carries a piece of information, it follows that any question requiring more than (at most) a countable number of pieces cannot be answered by this process and must, rather, be assumed. This encapsulates in a single idea the need for smooth link functions and kernels, sparsity priors, and other tools used in different contexts to eliminate by design what cannot be learned from data. At the same time, assumptions such as exchangeability may lose relevance. For example, if one wished to learn the upper bound of a certain quantity, in a standard setup, repeated measurements would be assumed exchangeable. However, if measurements 1, 1.1⁠, 2.3 were observed, how much we learn from the third observation (the change in predictive uncertainty) would arguably depend on the order in which the three values were observed. The symmetry relation between measurements is not preserved by the learning process. This indicates that model definition in terms of the predictive sequence, by asking a different question, may allow inference in the presence of more complex forms of data dependence. The definition of a coherent predictive model appears challenging and the empirical distribution is the simplest possible choice. However, if only x1 was available, setting X2=x1 almost surely is hardly a reasonable way to model uncertainty. Mixture models with data-dependent weights and/or kernel parameters might be an alternative worth exploring. Rather than simply achieving continuity or greater flexibility, the choice of the kernel could be driven by the need to better extract (or filter) the information that each new observation contains about possible future samples. This idea is wonderfully illustrated by the bivariate copula construction proposed in Section 4. The copula can be interpreted as a measure of the ‘speed’ at which the predictive is updated (moves) towards its limit (ideally the distribution of interest) as new observations arrive. Studying the relationship between mixtures and copulas could provide a tool to elicit new (perhaps approximate but computationally efficient) predictive models.

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