We introduce a stochastic probing problem with correlated items. In our model, which we call Bayesian Probing, the correlations are modeled by an underlying graph $G$. Each vertex is independently active with a known probability. Each item corresponds to an edge in the graph. Probing an edge has some cost, gives some reward if both endpoints are active, and also reveals the state of its endpoints. Hence a probe induces a Bayesian update on the remaining edges. The goal is to adaptively probe items/edges subject to a knapsack constraint to maximize the expected total reward obtained from the probed edges. Bayesian Probing generalizes stochastic knapsack and stochastic probing by allowing correlations between items. Moreover, it gives a tractable model for the Bayesian Active Search problem, a popular problem considered in the machine learning community. In Bayesian Active Search, the goal is to find items in a particular class by adaptively probing at most, say $k$, items. Given a prior distribution over items, we want to compute a Bayesian policy to maximize the number of such items found. For this general problem with arbitrary priors, there are strong lower bounds on efficiently computing good policies. In this paper, we design efficient approximation algorithms for Bayesian Probing. These results give the first efficient approximation algorithms for Bayesian Active Search, for a class of practically-relevant prior distributions.

翻译：我们提出一种包含相关项的随机探测问题。在我们的模型（称为贝叶斯概率探测）中，相关性由底层图$G$建模。每个顶点以已知概率独立激活，每个项对应图中的一条边。探测一条边会产生一定成本，若其两个端点均处于激活状态则获得奖励，同时也会揭示这两个端点的状态。因此，一次探测会触发对剩余边的贝叶斯更新。目标是在背包约束下自适应地探测项/边，以最大化从已探测边中获得的期望总奖励。贝叶斯概率探测通过允许项之间存在相关性，推广了随机背包和随机探测问题。此外，它为机器学习社区关注的热点问题——贝叶斯主动搜索——提供了可处理的模型。在贝叶斯主动搜索中，目标是通过自适应地探测至多$k$个项来寻找特定类别的项。给定项的先验分布，我们希望计算一个贝叶斯策略以最大化找到的此类项的数量。对于具有任意先验的此类一般性问题，高效计算优质策略存在强下界。本文为贝叶斯概率探测设计了高效的近似算法。这些结果首次为实际相关的一类先验分布下的贝叶斯主动搜索问题提供了高效的近似算法。