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Upper Entropy for 2-Monotone Lower Probabilities

Tuan-Anh Vu, Sébastien Destercke, Frédéric Pichon

Abstract

Uncertainty quantification is a key aspect in many tasks such as model selection/regularization, or quantifying prediction uncertainties to perform active learning or OOD detection. Within credal approaches that consider modeling uncertainty as probability sets, upper entropy plays a central role as an uncertainty measure. This paper is devoted to the computational aspect of upper entropies, providing an exhaustive algorithmic and complexity analysis of the problem. In particular, we show that the problem has a strongly polynomial solution, and propose many significant improvements over past algorithms proposed for 2-monotone lower probabilities and their specific cases.

Upper Entropy for 2-Monotone Lower Probabilities

Abstract

Uncertainty quantification is a key aspect in many tasks such as model selection/regularization, or quantifying prediction uncertainties to perform active learning or OOD detection. Within credal approaches that consider modeling uncertainty as probability sets, upper entropy plays a central role as an uncertainty measure. This paper is devoted to the computational aspect of upper entropies, providing an exhaustive algorithmic and complexity analysis of the problem. In particular, we show that the problem has a strongly polynomial solution, and propose many significant improvements over past algorithms proposed for 2-monotone lower probabilities and their specific cases.
Paper Structure (22 sections, 9 theorems, 16 equations, 3 figures, 3 tables, 4 algorithms)

This paper contains 22 sections, 9 theorems, 16 equations, 3 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. 2-monotone lower probability
  4. Belief Function
  5. Possibility Distribution
  6. Probability Intervals
  7. Upper Entropy
  8. Strong Polynomiality of computing Upper Entropy
  9. Computing upper entropy in special cases
  10. Belief function
  11. Possibility distribution
  12. Probability intervals
  13. Computing upper entropy via convex optimization
  14. Experiments
  15. 2-monotonicity
  16. ...and 7 more sections

Key Result

Lemma 1

Finding $A\in \mathop{\mathrm{arg\,max}}\limits_{\emptyset \neq B \subseteq \Omega} \frac{\mu(B)}{|B|}$ and $|A|$ is maximum amounts to solving $O(n)$ SFM.

Figures (3)

  • Figure 1: Flow network in Example \ref{['example:flow']}.
  • Figure 2: Illustration of Example \ref{['example:poss distribution']}
  • Figure 3: Plot of $f(x)$ with $[l_1, u_1]=[0.1, 0.4], [l_2, u_2]=[0.4, 0.5]$ and $[l_3, u_3]=[0.2, 0.6]$.

Theorems & Definitions (23)

  • Definition 1
  • Remark 1
  • Lemma 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 2
  • Example 1
  • Proposition 3
  • ...and 13 more