Martingale Posterior Predictive Coherence: Hausdorff Moment Hierarchy
Nicholas G. Polson, Daniel Zantedeschi

TL;DR
This paper investigates the conditions under which a martingale posterior fully determines multi-step predictive distributions in exchangeable Bernoulli sequences, highlighting the importance of the posterior law's uniqueness.
Contribution
It establishes that a martingale posterior determines all multi-step predictives if and only if the terminal value's conditional law is uniquely specified, clarifying predictive completeness conditions.
Findings
Posterior mean alone is insufficient for multi-step predictive identification.
The plug-in predictive is dominated by the Bayes predictive under proper scoring rules.
A closure theorem links predictive completeness to the uniqueness of the terminal law.
Abstract
For an exchangeable Bernoulli sequence with de Finetti mixing measure Pi, the k-step predictive probability P(X_{n+1}=...=X_{n+k}=0 | F_n) equals the posterior expectation E[(1-theta)^k | F_n]. By binomial expansion, this depends on all posterior moments up to order k. We show that the first moment alone is not sufficient to uniquely identify these quantities: for k >= 2, the mapping from posterior mean to k-step predictive is set-valued. The martingale posterior framework of Fong, Holmes, and Walker (which constrains only the first conditional moment of the terminal value) does not, in general, uniquely identify multi-step predictive distributions. Under any strictly proper scoring rule, the plug-in predictive is strictly dominated by the Bayes predictive whenever the posterior is non-degenerate. A closure theorem establishes that a martingale posterior determines all k-step…
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