The Identifiable Elicitation Complexity of the Mode is Infinite

A property is a real- or vector-valued function on a set of probability measures. Common examples of properties include summary statistics such as the mean, mode, variance, or $\alpha$-quantile. Some properties are directly elicitable, meaning they minimize the expectation of a loss function. For a property which is not directly elicitable, it is interesting to consider its elicitation complexity, defined as the smallest dimension of an elicitable vector-valued property from which one can recover the given property. Heinrich showed that the mode is not elicitable, raising the question of its elicitation complexity. We show this complexity to be infinite with respect to identifiable properties.