Finite-horizon quantile martingale posteriors: raw-urn laws and matrix-gain regression

Abstract

Martingale posteriors quantify uncertainty by forward-imputing observations from one-step-ahead predictive distributions, but implementations stop after finitely many imputations. For the empirical P\'olya-urn posterior of a quantile the law of the stopped state is derived. The quantile of the stopped urn measure keeps the familiar martingale tail-sum variance fraction; the deployed stochastic-approximation tracker with frozen gain $c$ does not. Its variance carries an explicit factor $G_a$ with $a=cf_0(q_\tau)$, which may fall below or exceed the tail fraction, and a density-adapted gain restores calibration through a density-free inflation. Shared urn innovations yield the joint law of finitely many quantile levels. For conditional quantile regression, a smoothed martingale posterior started at the ordinary quantile-regression estimator with a full inverse-Jacobian matrix gain satisfies a process Bernstein--von Mises theorem with calibrated finite-horizon bands; scalar or diagonal gains cannot match the sandwich covariance process.