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

原文摘要

To see the effect of the paper on statistical practice, consider the following hypothetical dialogue between a Quantitative Scientist (QS) and a Predictive Statistician (PS). QS: I’ve been thinking about the data generating mechanism and what we can safely assume pre-experimentally. PS: Actually, I’m more interested in what you think your n+1 data point would be. If you were going to predict your n+1 measurement, how would you do it? QS: I’d guess what the true model is and take its mean or median. But, I don’t know what the true model is. PS: But you do know what the predictive density m(yn+1|y1:n) is, right? QS: If I assume my likelihood and prior are right. But, I don’t know that. PS: Do you believe there is an overall probability model for Y1:n+1? QS: Sure: M. PS: Then, we can use De Finetti to backform a likelihood and prior if we want. But let’s not. Let’s think about the martingale model m(yn+1|y1:n)⁠, m(yn+2|y1:n+1) and so on. QS: You mean what on average a new random draw would look like given the past data? If I’d seen y1:n+k I’d draw Yn+k+1 from some m(⋅|y¯n+k)⁠. I’d start with a density shaped like this [draws a density] and let it tighten a bit as n increases. PS: So that’s step k in your martingale model. We could use it, but let’s not. Instead, I’m going to form one-step ahead predictive DF’s using y1:n⁠, y1:n+1⁠, and so forth. I’m going to draw one data point from each DF to find the next. For each step up to N, I’ll find a version of the posterior mean. I’ll do this many times and then use the final posterior means to estimate W(⋅|y1:n)⁠. QS: You’re using repeated generation of data I don’t have to find the posterior I do have? PS: Yes, I’m completing your experiment conceptually by filling out your missing data. Over and over. Then, I can quantify the information you don’t have and derive the variability left in data you do have i.e. form the posterior. QS: And you’re ignoring everything we know about the data generator besides the data. Where does Pθ come in? PS: It doesn’t. We’re relying on martingale convergence in the mixture M. By changing the probability defining the mode of convergence we get the limits we want for Θ¯ and the DF—that’s why we can use them to estimate the posterior. QS: So, what role do the prior and likelihood—or even conditional mixtures—have anymore? If you’re right, why would we bother with them? Some sort of robustness analysis? PS: We could. But, we let’s not. Let’s think about the predictive process directly. QS: What’s the magic? PS: No magic. We just changed the mode from Pθ to M since we’re thinking about the whole countably infinite product space Y∞⁠. QS: Is θ still a parameter? PS: It’s a function value. Take Θn=Θn(Y1:n) and think of the posterior mean. Or better take n=∞⁠. Each Θ=θ has a Vθ={y1:∞|θ(y∞)=θ}⁠. QS: Don’t θ and a y1:∞ that I might conceptually get have to be compatible in that if y1:∞∉Vθ then θ can’t be true? PS: Yes. If you’re looking for magic, it’s that at infinity Θ=Θ(Y1:∞)=E(Θ|Y1:∞) pointwise in Y∞ and as random variables. Then the y1:∞ you get gives you your θ. So, in the limit, our computational procedure mimics what you would get if you had y1:∞ —but, we don’t. What you don’t sample gives an assessment of your uncertainty. QS: But I’m really using a sequence of predictive DF’s not posterior means or conditional predictives. PS: Yes. That way you get objectivity in your predictive thinking.

摘要来源：OpenAlex 索引摘要（稿件版本未注明）