A Geometric Proof of Calibration

TL;DR

This paper presents a concise geometric proof of the existence of calibrated forecasters for any finite outcome set, emphasizing the link between approachability theory and calibration.

Contribution

It offers a new, simple proof applicable to multiple outcomes, illustrating the connection between Blackwell's approachability and calibration.

Findings

Proof valid for any finite number of outcomes

Highlights the link between approachability and calibration

Simplifies existing proofs using geometric approach

Abstract

We provide yet another proof of the existence of calibrated forecasters; it has two merits. First, it is valid for an arbitrary finite number of outcomes. Second, it is short and simple and it follows from a direct application of Blackwell's approachability theorem to carefully chosen vector-valued payoff function and convex target set. Our proof captures the essence of existing proofs based on approachability (e.g., the proof by Foster, 1999 in case of binary outcomes) and highlights the intrinsic connection between approachability and calibration.