The predictive approach to Bayesian inference accesses the posterior distribution via a sequence of one-step-ahead predictives, enabling inference via predictive resampling without Markov chain Monte Carlo. In the random-design regression setting, an explicit specification of the predictive design distribution is required, yet the impact of this choice has received little formal attention. We study the role of this predictive design distribution in parametric martingale posteriors for regression, and identify predictive notions of identifiability and design invariance that are essential for valid inference, particularly in the high-dimensional regression setting. Building on these foundations, we introduce a novel class of parametric martingale posteriors for regression that satisfies a weak form of these desiderata, and naturally accommodates the high-dimensional setting through regularization. We then illustrate our method through a simulation.

翻译：预测性贝叶斯推断通过一系列单步向前预测来获取后验分布，使得无需马尔可夫链蒙特卡洛即可通过预测重采样进行推断。在随机设计回归设定中，需明确指定预测设计分布，然而这一选择的影响尚未受到正式关注。我们研究了参数化鞅后验回归中预测设计分布的作用，并识别出对有效推断（尤其在高维回归设定中）至关重要的可识别性与设计不变性的预测性概念。基于这些理论基础，我们引入了一类新颖的参数化鞅后验回归方法，该方法满足这些期望性质的弱形式，并通过正则化自然地适应高维设定。最后，我们通过模拟实验展示了该方法的效果。