使用嵌套池进行分组测试
Group testing with nested pools
摘要：
为了确定一个群体中的受感染个体，他们的样本被分成大小相等的组，称为样本池，每个样本池都进行一次实验室检测。样本属于检测阴性池的个体被宣布为健康，而每个检测阳性的池被分成较小的、大小相等的池，在下一阶段进行检测。在第$(k+1)$阶段，对所有剩余样品进行试验。如果是$p<1-3^{-1/3}$，我们最小化每个个体的预期测试次数，该次数是阶段数$k+1$和第一个$k$阶段中池大小的函数。我们表明，对于每个$p\in (0, 1-3^{-1/3})$的最佳选择是四个可能的计划，这是明确描述之一。我们推测，对于每个$p$，最佳选择是池大小的两个序列之一$(3^k\text{ or }3^{k-1}4,3^{k-1},\dots,3^2,3 )$，与$p$的范围的精确描述，其中每个是最佳的。这个猜想得到了$p>2^{-51}$的大量数值证据的支持。我们还表明，最好的计划$(3^k,\dots,3)$的成本是为了$O\big(p\log(1/p)\big)$，相当于信息理论下限$p\log_2(1/p)+(1-p)\log_2(1/(1-p))$，伯努利$(p)$随机变量的熵。
英文摘要：
In order to identify the infected individuals of a population, their samples are divided in equally sized groups called pools and a single laboratory test is applied to each pool. Individuals whose samples belong to pools that test negative are declared healthy, while each pool that tests positive is divided into smaller, equally sized pools which are tested in the next stage. In the $(k+1)$-th stage all remaining samples are tested. If $p<1-3^{-1/3}$, we minimize the expected number of tests per individual as a function of the number $k+1$ of stages, and of the pool sizes in the first $k$ stages. We show that for each $p\in (0, 1-3^{-1/3})$ the optimal choice is one of four possible schemes, which are explicitly described. We conjecture that for each $p$, the optimal choice is one of the two sequences of pool sizes $(3^k\text{ or }3^{k-1}4,3^{k-1},\dots,3^2,3 )$, with a precise description of the range of $p$'s where each is optimal. The conjecture is supported by overwhelming numerical evidence for $p>2^{-51}$. We also show that the cost of the best among the schemes $(3^k,\dots,3)$ is of order $O\big(p\log(1/p)\big)$, comparable to the information theoretical lower bound $p\log_2(1/p)+(1-p)\log_2(1/(1-p))$, the entropy of a Bernoulli$(p)$ random variable.
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