Jiaqi Gu and Guosheng Yin’s contribution to the Discussion of ‘Martingale Posterior Distributions’ by Fong, Holmes and Walker

We congratulate the authors for their thought-provoking paper, especially in the three aspects as follows. Instead of imposing a prior π(θ) on the parameter θ of a likelihood function fθ(y)⁠, where both π and fθ are elicited subjectively, the authors proposed the framework of martingale posterior to directly model the predictive p(yn+1:∞∣y1:n) for which only the prior predictive p(y) needs to be specified. As a result, Bayesian inference can be conducted on any parameter (or statistical functional) of the true sampling distribution. As the hyperparameter ρ of the copula is chosen in such a data-driven way that p(y1:n) fits the observed data y1:n well, the impact of the initial guess for the prior predictive, p0(y)⁠, on p(yn+1:∞∣y1:n) can be adjusted properly. Unlike existing MCMC methods which generate autocorrelated posterior samples, the proposed predictive resampling algorithm is GPU-friendly and parallelisable. Thus, independent posterior samples can be obtained for inference, leading to improvement of both computational and statistical efficiency. The proposed martingale posterior framework, we believe, has the further potentials in several directions. Because variable selection can generally improve statistical efficiency of regression analysis, it is of interest to investigate how Bayesian variable selection can be conducted with control of the Bayesian false discovery rate under the framework of martingale posterior. Although a default choice is suggested for the initial guess p0(y)⁠, it is possible to leverage some information from the observed data y1:n to initialise p0(y)⁠, similar to empirical Bayes methods. This raises a question on how the martingale posterior connects to nonparametric empirical Bayes, which also estimates the prior from the data.