Ramses Mena Chavez's contribution to the Discussion of ‘Martingale Posterior Distributions’ by Fong, Holmes and Walker

I congratulate the authors for introducing a novel approach for Bayesian inference, which avoids directly using the posterior distribution, often not available or computationally expensive. In addition to opening several lines of thinking about the foundations of Bayesian statistics, their proposal unveils many interesting directions to be explored and applied within the world of statistics. With an infinite population and the epistemic view of uncertainty (see, e.g. Goldstein, 2013), namely when the posterior represents the uncertainty due to missing observations, exchangeability arises naturally. Here, by applying a martingale convergence result by Doob (1949), the authors show that the predictive view of Bayesian inference allows one to obtain the posterior distribution, provided the statistic, θn⁠, summarising observations, y1:n⁠, is a martingale. This requirement might indeed lead to other types of symmetries, i.e. not necessarily exchangeability. Within the exchangeability setting, the proposed predictive approach resembles much to that frequently used in Bayesian non-parametrics, e.g. via the Blackwell and MacQueen Pólya-Urn and many of its generalisations used for posterior inference. Hence, with the findings in this paper, one wonders whether the exchangeability requirement is the most practical one. Furthermore, there might be other ways to relax the martingale condition, namely throughout other convergence requirements for θn⁠. Clearly, one of the appealing features of the proposed approach is bypassing the prior to posterior computation, by suitably modelling predictive distributions of conditionally identically distributed (c.i.d.) sequences and θn⁠. For some of the examples in the paper, it is clear what the predictive structure should look like, and though the authors propose a fairly general predictive using their copula approach, there are many inference scenarios where one would need to be very creative to achieve the desired inference, for instance Bayesian inference for phenomena typically modelled throughout continuous time Markov processes. For an Autoregressive of order one, AR(1)⁠, model with transition density p(xi|xi−1;ρ)=N(xi|ρxi−1,1)⁠, and ρ as the parameter of interest, one could use the predictive where a:=∑i=2nxixi−1 and b:=1+∑i=2nxi−12⁠, with functional statistic ρ¯N=a/b⁠. This proposal comes from assuming an N(0,1) prior for ρ and computing the predictive. However, without passing throughout such mechanism, a natural question is how could one propose such predictive and the form to be used for ρ¯N⁠. Indeed, the authors proposal works very well, see Figure 1. Martingale posterior and posterior distribution for ρ⁠. The following contributions were received in writing after the meeting.