De Finetti + Sanov = Bayes: Exchangeable Prediction under Moment Constraints

We study exchangeable prediction when an empirical-moment constraint is primitive rather than a one-time completed-sample event.

For each active finite horizon N, the relevant law is the de Finetti mixture conditioned on E_N = {Phat_N in E_{eps_N}}.

Since the underlying law and the constraint are permutation invariant, the prediction target may be any fixed block of m coordinates contained in the active horizon, including coordinates interpreted as future relative to an arbitrary finite cut.

Conditionally on the directing measure mu, the Gibbs-conditioning principle sends the law of such a block to the m-fold product of the I-projection P*_mu = argmin_{Q in E} D(Q || mu).

On a finite alphabet we give an elementary master inequality for general polyhedral moment windows.

After mixing over the constraint posterior Pi_{N,E}, and under weak convergence plus posterior-averaged component control, the finite-dimensional marginals converge to a consistent exchangeable law whose random directing measure is the I-projection P*_mu, with mu drawn from the weak limit Pi_E.

Sequential prediction under this limiting law is therefore Bayesian prediction from a mixture of componentwise I-projections.

What survives is a dichotomy on the subfamily minimizing the constraint rate: a reachable constraint leaves the projection asymptotically inactive, an unreachable one leaves genuine projections but drives the prior onto that subfamily.

In each of our examples one mechanism or the other is gone in the limit, while both act at every finite N.

The master bound also reads as an equivalence of ensembles.

We reserve "maximum entropy" for a uniform or flat baseline and use "minimum relative entropy" or "I-projection" in general.