Elicitation Complexity of Statistical Properties
This work addresses a foundational issue in statistics and machine learning for researchers and practitioners, offering a more nuanced understanding of elicitation beyond mere existence, though it is incremental in building on prior elicitation studies.
The paper tackles the problem of determining the elicitation complexity of statistical properties, which measures the number of dimensions needed to indirectly elicit a property, and provides tight complexity bounds for Bayes risks and applications to properties like variance and entropy.
A property, or statistical functional, is said to be elicitable if it minimizes expected loss for some loss function. The study of which properties are elicitable sheds light on the capabilities and limitations of point estimation and empirical risk minimization. While recent work asks which properties are elicitable, we instead advocate for a more nuanced question: how many dimensions are required to indirectly elicit a given property? This number is called the elicitation complexity of the property. We lay the foundation for a general theory of elicitation complexity, including several basic results about how elicitation complexity behaves, and the complexity of standard properties of interest. Building on this foundation, our main result gives tight complexity bounds for the broad class of Bayes risks. We apply these results to several properties of interest, including variance, entropy, norms, and several classes of financial risk measures. We conclude with discussion and open directions.