We study the problem of actively learning a non-parametric choice model based on consumers' decisions. We present a negative result showing that such choice models may not be identifiable. To overcome the identifiability problem, we introduce a directed acyclic graph (DAG) representation of the choice model. This representation provably encodes all the information about the choice model which can be inferred from the available data, in the sense that it permits computing all choice probabilities. We establish that given exact choice probabilities for a collection of item sets, one can reconstruct the DAG. However, attempting to extend this methodology to estimate the DAG from noisy choice frequency data obtained during an active learning process leads to inaccuracies. To address this challenge, we present an inclusion-exclusion approach that effectively manages error propagation across DAG levels, leading to a more accurate estimate of the DAG. Utilizing this technique, our algorithm estimates the DAG representation of an underlying non-parametric choice model. The algorithm operates efficiently (in polynomial time) when the set of frequent rankings is drawn uniformly at random. It learns the distribution over the most popular items among frequent preference types by actively and repeatedly offering assortments of items and observing the chosen item. We demonstrate that our algorithm more effectively recovers a set of frequent preferences on both synthetic and publicly available datasets on consumers' preferences, compared to corresponding non-active learning estimation algorithms. These findings underscore the value of our algorithm and the broader applicability of active-learning approaches in modeling consumer behavior.

翻译：我们研究了基于消费者决策主动学习非参数选择模型的问题。我们提出了一项负面结果，表明此类选择模型可能无法被识别。为克服可识别性问题，我们引入了选择模型的有向无环图（DAG）表示。该表示可被证明编码了所有能够从可用数据中推断出的关于选择模型的信息，从某种意义上说，它允许计算所有选择概率。我们证明，给定一组商品集合的精确选择概率，就可以重构DAG。然而，试图将这种方法扩展到从主动学习过程中获得的含噪声选择频率数据来估计DAG时，会导致不准确性。为应对这一挑战，我们提出了一种包含-排除方法，该方法有效管理了跨DAG层级的误差传播，从而实现了对DAG更准确的估计。利用该技术，我们的算法估计了底层非参数选择模型的DAG表示。当频繁排名集合均匀随机抽取时，该算法高效运行（多项式时间内）。通过主动且重复地提供商品组合并观察被选中的商品，算法学习了频繁偏好类型中最受欢迎商品的分布。我们在合成数据集和公开的消费者偏好数据集上证明，与相应的非主动学习估计算法相比，我们的算法能更有效地恢复一组频繁偏好。这些发现凸显了我们的算法以及主动学习方法在建模消费者行为方面的广泛适用性。