Most machine learning classifiers are designed to output posterior probabilities for the classes given the input sample. These probabilities may be used to make the categorical decision on the class of the sample; provided as input to a downstream system; or provided to a human for interpretation. Evaluating the quality of the posteriors generated by these system is an essential problem which was addressed decades ago with the invention of proper scoring rules (PSRs). Unfortunately, much of the recent machine learning literature uses calibration metrics -- most commonly, the expected calibration error (ECE) -- as a proxy to assess posterior performance. The problem with this approach is that calibration metrics reflect only one aspect of the quality of the posteriors, ignoring the discrimination performance. For this reason, we argue that calibration metrics should play no role in the assessment of posterior quality. Expected PSRs should instead be used for this job, preferably normalized for ease of interpretation. In this work, we first give a brief review of PSRs from a practical perspective, motivating their definition using Bayes decision theory. We discuss why expected PSRs provide a principled measure of the quality of a system's posteriors and why calibration metrics are not the right tool for this job. We argue that calibration metrics, while not useful for performance assessment, may be used as diagnostic tools during system development. With this purpose in mind, we discuss a simple and practical calibration metric, called calibration loss, derived from a decomposition of expected PSRs. We compare this metric with the ECE and with the expected score divergence calibration metric from the PSR literature and argue, using theoretical and empirical evidence, that calibration loss is superior to these two metrics.

本文针对当前机器学习分类器后验概率评估的不足，提出采用期望适当评分规则（PSRs）作为后验质量的主要评估指标，而非常用的校准度量（如期望校准误差，ECE）。研究发现，校准度量只能反映后验质量的一个方面，忽视了区分性能，而期望PSRs能够提供更全面的质量评估，具有重要的理论和实践意义。

评估后验概率：决策理论、适当评分规则与校准