Asymptotics for a class of parametric martingale posteriors
研究了参数化鞅后验的渐近性质，提出了两个中心极限定理：预测中心极限定理加速了预测重抽样算法，伯恩斯坦-冯·米塞斯定理为获得良好频率性质提供了方法指导。
Summary The martingale posterior framework replaces the elicitation of the likelihood and prior with that of a sequence of one-step-ahead predictive densities for Bayesian inference. Posterior sampling then involves the imputation of unobserved quantities and can then be carried out in an expedient and parallelizable manner using predictive resampling, without requiring Markov chain Monte Carlo. Recent work has investigated the use of plug-in parametric predictive densities, combined with stochastic gradient descent, to specify a parametric martingale posterior. This paper investigates the asymptotic properties of this class of parametric martingale posteriors. In particular, two central limit theorems based on martingale limit theory are introduced and applied. The first is a predictive central limit theorem, which enables a significant acceleration of the predictive resampling scheme through a hybrid sampling algorithm based on a normal approximation. The second is a Bernstein–von Mises result, which is novel for martingale posteriors, and provides methodological guidance on attaining desirable frequentist properties. We demonstrate the utility of the theoretical results through simulations and a real data example.