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Predictive Coherence and the Moment Hierarchy: Martingale Posteriors for Exchangeable Bernoulli Sequences

Nicholas G. Polson, Daniel Zantedeschi

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 predictives if and only if the conditional law of the terminal value is uniquely specified. Hill's A_(n) rule under the Jeffreys Beta(1/2,1/2) prior is a positive example. The discrepancy is O(Var(theta | F_n)) and vanishes as the posterior concentrates. These results clarify the structural requirements for predictive completeness under exchangeability.

Predictive Coherence and the Moment Hierarchy: Martingale Posteriors for Exchangeable Bernoulli Sequences

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 predictives if and only if the conditional law of the terminal value is uniquely specified. Hill's A_(n) rule under the Jeffreys Beta(1/2,1/2) prior is a positive example. The discrepancy is O(Var(theta | F_n)) and vanishes as the posterior concentrates. These results clarify the structural requirements for predictive completeness under exchangeability.
Paper Structure (34 sections, 30 theorems, 29 equations, 2 figures, 4 tables)

This paper contains 34 sections, 30 theorems, 29 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Sanov, de Finetti, and the sample-path picture
  3. Setup and notation
  4. Exchangeable Bernoulli sequences
  5. Martingale posteriors and the terminal value
  6. Martingale versus exchangeability
  7. The predictive moment hierarchy
  8. Moment representation
  9. Injectivity
  10. Binomial inversion
  11. Explicit expansion for small k
  12. KL geometry and the moment hierarchy
  13. The Sanov Gap
  14. Moderate deviations and the second moment as first nontrivial Sanov term
  15. Insufficiency of first-moment coherence
  16. ...and 19 more sections

Key Result

Theorem 1

The martingale condition identifies the posterior mean; we characterize what additional structure is required for predictive completeness. Specifically: for $k\ge 2$, the mapping $m_n\mapsto\mathbb{E}[(1-\theta)^k\mid\mathcal{F}_n]$ is set-valued, so first-moment information does not uniquely identi

Figures (2)

  • Figure 1: Sanov geometry ($m_n=0.4$, $n=12$). Blue: the Sanov rate $D_{\mathrm{KL}}(m_n\|\theta)$, which governs posterior shape through \ref{['eq:sanov-posterior']}. Dashed red: its quadratic (Fisher information) approximation, retaining only the second-order term. Green: the resulting posterior density. The dotted vertical at $m_n$ marks what first-moment information identifies. One-step prediction requires only this location; two-step additionally requires the curvature $\sigma_n^2$ (equation \ref{['eq:k2-identity']}); $k$-step prediction for $k\ge 3$ requires further moments.
  • Figure 2: The predictive moment hierarchy. Bottom: the martingale / mean-only tier (cf. linear Bayes) constrains only the first moment and determines one-step predictives. Middle: specifying $J$ moments (Goldstein (goldstein1975) prevision tier) determines $k$-step predictives for $k\le J$ but not for $k>J$ (Theorem \ref{['thm:insuff']}). Top: the full conditional law determines all $k$-step predictives (predictive completeness; Theorem \ref{['thm:closure']}).

Theorems & Definitions (78)

  • Theorem : Main results, informal
  • Theorem 2.1: Sanov--Moment Bridge
  • proof
  • Definition 3.1
  • Theorem 3.2: de Finetti
  • proof
  • Remark 3.3: Sufficiency
  • Proposition 3.5: Terminal value
  • proof
  • Remark 3.6
  • ...and 68 more