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We introduce a new setting, optimize-and-estimate structured bandits. Here, a policy must select a batch of arms, each characterized by its own context, that would allow it to both maximize reward and maintain an accurate (ideally unbiased) population estimate of the reward. This setting is inherent to many public and private sector applications and often requires handling delayed feedback, small data, and distribution shifts. We demonstrate its importance on real data from the United States Internal Revenue Service (IRS). The IRS performs yearly audits of the tax base. Two of its most important objectives are to identify suspected misreporting and to estimate the "tax gap" -- the global difference between the amount paid and true amount owed. Based on a unique collaboration with the IRS, we cast these two processes as a unified optimize-and-estimate structured bandit. We analyze optimize-and-estimate approaches to the IRS problem and propose a novel mechanism for unbiased population estimation that achieves rewards comparable to baseline approaches. This approach has the potential to improve audit efficacy, while maintaining policy-relevant estimates of the tax gap. This has important social consequences given that the current tax gap is estimated at nearly half a trillion dollars. We suggest that this problem setting is fertile ground for further research and we highlight its interesting challenges. The results of this and related research are currently being incorporated into the continual improvement of the IRS audit selection methods.