Bayesian Probing on Graphs

2026-06-08

AI Summary

This work addresses the Bayesian exploration problem where items exhibit graph-structured dependencies, aiming to adaptively probe edges under a knapsack constraint to maximize expected total reward. Vertices are independently activated with known probabilities, and probing an edge yields an immediate reward while revealing the states of its endpoints, thereby triggering Bayesian updates. The paper presents the first efficient approximation algorithm for this class of problems and extends it to a practical family of prior distributions commonly used in Bayesian active search. By integrating graphical models, Bayesian inference, and stochastic optimization techniques, the proposed method achieves a provable approximation guarantee while significantly improving empirical performance, effectively alleviating the computational bottleneck inherent in Bayesian active search.

Abstract

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.

Problem

• Bayesian Probing
• Stochastic Probing
• Graph Correlations
• Bayesian Active Search
• Knapsack Constraint

Innovation

• Bayesian Probing
• graph correlations
• adaptive probing
• approximation algorithms
• Bayesian Active Search