We examine three landmark clinical trials -- ECMO, CALGB~49907, and I-SPY~2 -- through a unified Bayesian framework connecting prior specification, sequential adaptation, and decision-theoretic optimisation. For ECMO, the posterior probability of treatment superiority is robust across the range of priors examined. For CALGB, predictive probability monitoring stopped enrolment at 633 instead of 1800 patients. For I-SPY~2, adaptive enrichment graduated nine of 23 arms to Phase~III. These case studies motivate a methodological contribution: exact backward induction for two-arm binary trials, where Beta-Binomial conjugacy yields closed-form transitions on the integer lattice of success counts with no quadrature. A Pólya-Gamma augmentation bridges this to covariate-adjusted logistic regression. Simulation reveals a fundamental tension: the optimal Bayesian design reduces expected sample sizes to 14--26 per arm (versus 42--100 for alternatives) but with substantially lower power. A calibrated variant embedding the declaration threshold in the terminal utility improves power while maintaining sample-size savings; varying the per-stage cost traces a power frontier for selecting the preferred operating point, with suitability highest in patient-sparing contexts such as rare diseases and paediatrics. The Pólya-Gamma Laplace approximation is validated against exact calculations (mean absolute error below 0.01). We discuss implications for the 2026 FDA draft guidance on Bayesian methodology.

翻译：我们通过一个统一的贝叶斯框架——连接先验设定、序贯适应与决策论优化——检视了三项里程碑式的临床试验：ECMO、CALGB~49907 和 I-SPY~2。对于 ECMO 试验，治疗优势的后验概率在所考察的先验分布范围内均表现稳健。对于 CALGB 试验，预测概率监测在入组 633 名患者时（而非原计划的 1800 名）即停止了招募。对于 I-SPY~2 试验，适应性富集设计使 23 个治疗组中的 9 个成功晋级至 III 期。这些案例研究引出了一项方法论贡献：针对双臂二分类试验的精确逆向归纳法，其中 Beta-二项共轭性在成功计数的整数格上产生了无需数值积分的闭式转移。通过 Pólya-Gamma 数据扩增法，可将其与协变量调整的逻辑回归模型相衔接。仿真揭示了一个根本性矛盾：最优贝叶斯设计将每臂预期样本量降至 14–26（替代方案为 42–100），但代价是检验效能显著降低。一种将声明阈值嵌入终端效用的校准变体在保持样本量节约的同时提升了效能；通过调整每阶段成本可绘制效能边界，以选择最优操作点，该方法尤其适用于需节约患者的场景（如罕见病与儿科研究）。Pólya-Gamma 拉普拉斯近似法通过精确计算验证（平均绝对误差低于 0.01）。我们进一步探讨了其对 2026 年 FDA 贝叶斯方法学草案指南的启示。