David Rossell's contribution to the Discussion of ‘Martingale Posterior Distributions’ by Fong, Holmes and Walker

原文摘要

Congratulations on a thought-provoking piece. Building Bayesian inference from a (likelihood, predictive) pair, rather than a (likelihood, prior), enriches the paradigm and provides new ways to think about, formulate, and solve problems. A few respectful remarks. First, although the authors never claim this, it is worth emphasising that the framework is not prior-free. There is a posterior and a likelihood, hence the prior is proportional to their ratio. The key is that said prior is data-dependent, providing an interesting avenue to develop objective Bayes methods, at the cost of losing the coherence property in belief updating. Inspecting the prior can be informative. Figure 1 shows a Bernoulli example where truly θ=0.5 but the implied prior places little mass around that value, and a Gaussian example where the prior is centred around the sample mean.1 This apparently erratic prior behaviour might be problematic for model choice via Bayes factors, e.g. returning a very small integrated likelihood in the Bernoulli example. Likelihood, posterior, and prior densities for Bernoulli(0.5) and Normal(5,1) examples with n=100⁠. Second, while sometimes it is easier to elicit a predictive than a prior, in my experience the reverse is often true. For example, in regression, a prior on parameters defines a prior on the R2 coefficient, an easy-to-interpret quantity, whereas eliciting predictives may be less intuitive for nonstatisticians. Further, note that computational considerations elegantly discussed by the authors severely restrict the range of predictives one may consider in practice, limiting the flexibility of the framework. Third, I am afraid I disagree on the frameworks’ computational convenience. Doing a single optimisation may be faster than sampling, but the framework requires solving many optimisations. This is not cheaper than posterior sampling in a standard (likelihood, prior) construction, also the latter offers fast nonsampling based tools, e.g. Laplace approximations and extensions. It would be interesting to consider analogues for the predictive framework. Finally, a remark on assuming that at n=∞ there is no uncertainty left. In some settings this is not true, e.g. in high-dimensional regression with p≫n (one adds higher-order polynomial terms as n grows, say) and a normal prior on the parameters there remains posterior uncertainty even as n→∞⁠. The proposed framework does not account for such uncertainty, unless suitable adjustments are made.

摘要来源：OpenAlex