标题： 对称伯努利假设下序概率比检验最优性的简单几何证明 显示英文标题

标题： A Simple Geometric Proof of the Optimality of the Sequential Probability Ratio Test for Symmetric Bernoulli Hypotheses

摘要： 本文重新审视了确定加权硬币偏差的经典问题，其中偏差已知为$p = 1/2 + \varepsilon$或$p = 1/2 - \varepsilon$，同时最小化期望的硬币投掷次数和错误概率。 该问题的最优策略由沃尔德的序贯概率比检验（SPRT）给出，该检验通过将对数似然比与固定阈值进行比较来确定停止时间。 此类结果的经典证明通常依赖于分析性、连续性和非构造性论证。 在本文中，我们提出了一个离散且自包含的证明，证明SPRT在此问题中的最优性。 我们将问题建模为二维（正面，反面）整数格点上的有偏随机游走，并将策略建模为该格点上的标记停止时间。 我们的证明采用直接的贪心方法，展示了如何通过一系列改进贝叶斯风险目标的局部扰动，将任意策略转换为对应于SPRT的最优并行线“差分策略”。 显示英文摘要

摘要： This paper revisits the classical problem of determining the bias of a weighted coin, where the bias is known to be either $p = 1/2 + \varepsilon$ or $p = 1/2 - \varepsilon$, while minimizing the expected number of coin tosses and the error probability. The optimal strategy for this problem is given by Wald's Sequential Probability Ratio Test (SPRT), which compares the log-likelihood ratio against fixed thresholds to determine a stopping time. Classical proofs of this result typically rely on analytical, continuous, and non-constructive arguments. In this paper, we present a discrete, self-contained proof of the optimality of the SPRT for this problem. We model the problem as a biased random walk on the two-dimensional (heads, tails) integer lattice, and model strategies as marked stopping times on this lattice. Our proof takes a straightforward greedy approach, showing how any arbitrary strategy may be transformed into the optimal, parallel-line "difference policy" corresponding to the SPRT, via a sequence of local perturbations that improve a Bayes risk objective.