Elicitation Complexity of Statistical Properties

TL;DR

This paper introduces the concept of elicitation complexity, measuring the dimensions needed to indirectly elicit statistical properties, and provides tight bounds for Bayes risks and other properties.

Contribution

It develops a general theory of elicitation complexity, establishing bounds and analyzing standard properties like variance and risk measures.

Findings

Tight complexity bounds for Bayes risks.

Elicitation complexity varies across different properties.

Framework applicable to financial risk measures.

Abstract

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.