针对未知基准的可交换检验
Exchangeable Testing Against an Unknown Benchmark
浏览论文内容
中文总结 AI 辅助
本研究针对未知基准的序贯检验，提出基于组合秩的更新机制，建立贝叶斯框架下可显式计算的预测概率，分析了相关马尔可夫链的渐近行为。
中文摘要 AI 辅助
我们通过将数据点与潜在基准进行序贯比较，生成无限二元可交换序列。假设基准秩\b{R_0}在未观测组内存在先验分布，我们建立贝叶斯机制，以纯组合方式确定运行秩\b{R_n}的后验分布。这得到了可显式计算的、针对战胜基准的预测概率。归一化运行秩收敛于具有多项式密度的潜在强度变量\b{X}，该密度可能经过Beta倾斜处理。一些极小极大锦标赛产生了特别简单的乘法公式，适用于与推广Topp-Leone分布的先验相关的预测概率；对于该类公式，我们分析了相关固定\b{n}的上下马尔可夫链的渐近行为。极限扩散具有经典的Wright-Fisher方差，但非线性漂移通过基准的先验密度显式表达。Beta密度的混合是可交换序列理论中的经典对象。本研究的贡献在于基于组合秩的更新机制，以及针对未知基准序贯检验的显式预测法则。
英文摘要
We generate infinite binary exchangeable sequences by sequential comparison of data points against a latent benchmark. Assuming a prior distribution of the benchmark rank \(R_0\) within an unobserved group, we set up the Bayesian machinery that determines the posterior distribution of the running rank \(R_n\) in purely combinatorial terms. This yields an explicitly computable predictive probability of winning against the benchmark. The normalised running rank converges to a latent strength variable \(X\) with polynomial density, possibly Beta-tilted. Some min-max tournaments lead to particularly simple multiplicative formulae for predictive probabilities related to priors that generalise the Topp--Leone distribution; for that class we analyse the asymptotics of the associated fixed-\(n\) up-down Markov chains. The limiting diffusion has the classical Wright--Fisher variance but a nonlinear drift expressed explicitly via the prior density of the benchmark. Mixtures of Beta densities are classical objects in the theory of exchangeable sequences. The contribution of the present work is the combinatorial rank-based updating mechanism and the resulting explicit predictive laws for sequential testing against an unknown benchmark.
