In 1976, Lai constructed a nontrivial confidence sequence for the mean $\mu$ of a Gaussian distribution with unknown variance $\sigma$. Curiously, he employed both an improper (right Haar) mixture over $\sigma$ and an improper (flat) mixture over $\mu$. Here, we elaborate carefully on the details of his construction, which use generalized nonintegrable martingales and an extended Ville's inequality. While this does yield a sequential t-test, it does not yield an ``e-process'' (due to the nonintegrability of his martingale). In this paper, we develop two new e-processes and confidence sequences for the same setting: one is a test martingale in a reduced filtration, while the other is an e-process in the canonical data filtration. These are respectively obtained by swapping Lai's flat mixture for a Gaussian mixture, and swapping the right Haar mixture over $\sigma$ with the maximum likelihood estimate under the null, as done in universal inference. We also analyze the width of resulting confidence sequences, which have a curious dependence on the error probability $\alpha$. Numerical experiments are provided along the way to compare and contrast the various approaches.

翻译：1976年，Lai针对方差未知的高斯分布均值$\mu$构造了非平凡的置信序列。值得注意的是，他同时采用了$\sigma$上的非正常（右哈尔）混合先验与$\mu$上的非正常（平坦）混合先验。本文详细阐述其构造细节，该构造使用了广义不可积鞅与扩展的Ville不等式。虽然该方法确实产生了序贯t检验，但由于鞅的不可积性未能生成"e过程"。针对相同场景，我们开发了两种新的e过程与置信序列：一种是在缩减滤子下的检验鞅，另一种是标准数据滤子下的e过程。分别通过将Lai的平坦混合先验替换为高斯混合先验，以及将$\sigma$上的右哈尔混合先验替换为原假设下的极大似然估计（如通用推断所述）获得。我们还分析了所得置信序列的宽度，该宽度对误差概率$\alpha$存在特殊依赖性。文中穿插数值实验以比较不同方法的性能。