  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.
