  Uncertainty estimation (UE), as an effective means of quantifying predictive
uncertainty, is crucial for safe and reliable decision-making, especially in
high-risk scenarios. Existing UE schemes usually assume that there are
completely-labeled samples to support fully-supervised learning. In practice,
however, many UE tasks often have no sufficiently-labeled data to use, such as
the Multiple Instance Learning (MIL) with only weak instance annotations. To
bridge this gap, this paper, for the first time, addresses the
weakly-supervised issue of Multi-Instance UE (MIUE) and proposes a new baseline
scheme, Multi-Instance Residual Evidential Learning (MIREL). Particularly, at
the fine-grained instance UE with only weak supervision, we derive a
multi-instance residual operator through the Fundamental Theorem of Symmetric
Functions. On this operator derivation, we further propose MIREL to jointly
model the high-order predictive distribution at bag and instance levels for
MIUE. Extensive experiments empirically demonstrate that our MIREL not only
could often make existing MIL networks perform better in MIUE, but also could
surpass representative UE methods by large margins, especially in
instance-level UE tasks. Our source code is available at
https://github.com/liupei101/MIREL.
