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Updated: Nov 27, 2025

Setting Limits on Supersymmetry Using Simplified Models
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On the Impossibility of Learning the Missing Mass.

Elchanan Mossel1, Mesrob I Ohannessian2

  • 1Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02142, USA.

Entropy (Basel, Switzerland)
|December 3, 2020
PubMed
Summary

Learning the probability of rare events, known as missing mass, is impossible without structural assumptions. This impossibility extends to tail estimation, highlighting the need for heavy-tailed distributions to predict rare events.

Keywords:
Good–Turingheavy tailslight tailsmissing massno free lunchrare events

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Area of Science:

  • Machine Learning
  • Probability Theory
  • Statistical Inference

Background:

  • Accurate estimation of rare event probabilities is crucial in various fields.
  • Current methods often rely on strong assumptions about data distribution.
  • The challenge lies in learning probabilities outside the observed data range.

Purpose of the Study:

  • To investigate the fundamental learnability of rare event probabilities.
  • To determine if "missing mass" can be learned without distribution-specific assumptions.
  • To analyze the implications for tail estimation and predictive modeling.

Main Methods:

  • Developed a theoretical framework to analyze learnability.
  • Employed a semi-constructive proof technique.
  • Utilized a coupling argument with a dithered geometric distribution.
  • Established a reduction from missing mass to tail estimation.

Main Results:

  • Demonstrated that missing mass is not distribution-free learnable in relative error.
  • Proved that this impossibility extends to both discrete and continuous tail estimation.
  • Showcased that predicting rare events necessitates assumptions like heavy tails.

Conclusions:

  • Fundamental limits exist in learning rare event probabilities without structural priors.
  • The study formalizes the intuition that heavy-tailed distributions are essential for robust rare event prediction.
  • Results have implications for risk assessment, anomaly detection, and statistical modeling.