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Extension of Yager's negation of probability distribution based on uncertainty measures

Santosh Kumar Chaudhary, Pradeep Kumar Sahu, Nitin Gupta

TL;DR

The paper investigates how negating a discrete probability distribution affects information-theoretic measures, specifically Shannon entropy $H(P)$, varentropy $VH(P)$, and varextropy $VJ(P)$. Using Yager's negation framework with $\bar{p}_i = \frac{1-p_i}{n-1}$, it derives expressions for the negated measures and proves that $H(\bar{P}) \ge H(P)$, $VH(\bar{P}) \ge VH(P)$, and $VJ(\bar{P}) \ge VJ(P)$, with maxima attained at the uniform distribution (and equality in the two-item case). The work employs Lagrangian optimization to show that the uniform distribution maximizes these quantities under negation, offering a rigorous account of how negation amplifies uncertainty metrics. These results have implications for uncertainty modelling and decision-making in contexts where complementary or opposite outcomes are analyzed through entropy-based measures.

Abstract

Existing research on negations primarily focuses on entropy and extropy. Recently, new functions such as varentropy and varextropy have been developed, which can be considered as extensions of entropy and extropy. However, the impact of negation on these extended measures, particularly varentropy and varextropy, has not been extensively explored. To address this gap, this paper investigates the effect of negation on Shannon entropy, varentropy, and varextropy. We explore how the negation of a probability distribution influences these measures, showing that the negated distribution consistently leads to higher values of Shannon entropy, varentropy, and varextropy compared to the original distribution. Additionally, we prove that the negation of a probability distribution maximizes these measures during the process. The paper provides theoretical proofs and a detailed analysis of the behaviour of these measures, contributing to a better understanding of the interplay between probability distributions, negation, and information-theoretic quantities.

Extension of Yager's negation of probability distribution based on uncertainty measures

TL;DR

The paper investigates how negating a discrete probability distribution affects information-theoretic measures, specifically Shannon entropy , varentropy , and varextropy . Using Yager's negation framework with , it derives expressions for the negated measures and proves that , , and , with maxima attained at the uniform distribution (and equality in the two-item case). The work employs Lagrangian optimization to show that the uniform distribution maximizes these quantities under negation, offering a rigorous account of how negation amplifies uncertainty metrics. These results have implications for uncertainty modelling and decision-making in contexts where complementary or opposite outcomes are analyzed through entropy-based measures.

Abstract

Existing research on negations primarily focuses on entropy and extropy. Recently, new functions such as varentropy and varextropy have been developed, which can be considered as extensions of entropy and extropy. However, the impact of negation on these extended measures, particularly varentropy and varextropy, has not been extensively explored. To address this gap, this paper investigates the effect of negation on Shannon entropy, varentropy, and varextropy. We explore how the negation of a probability distribution influences these measures, showing that the negated distribution consistently leads to higher values of Shannon entropy, varentropy, and varextropy compared to the original distribution. Additionally, we prove that the negation of a probability distribution maximizes these measures during the process. The paper provides theoretical proofs and a detailed analysis of the behaviour of these measures, contributing to a better understanding of the interplay between probability distributions, negation, and information-theoretic quantities.

Paper Structure

This paper contains 12 sections, 9 theorems, 74 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Shannon Entropy
  3. Varentropy
  4. Extropy
  5. Varextropy
  6. Negation of a discrete probability distribution
  7. Negation method
  8. Negation with Shannon Entropy
  9. Negation with varentropy
  10. Negation with Varextropy
  11. Examples
  12. Conclusion

Key Result

Theorem 1

Assume the event space $X = \{ x_1, x_2, \dots x_n \}$ and the probability distribution $P = \{ p_1, p_2, \dots, p_n \}$, $\bar{P}$ represents the inverse of P, then $H( \bar{P})\geq H(P).$

Figures (8)

  • Figure 1: Change of $H(P)$ and $H(\bar{P})$
  • Figure 2: Change of $VH(P)$ and $VH(\bar{P})$
  • Figure 3: Change of $VJ(P)$ and $VJ(\bar{P})$
  • Figure 4: Change of $H(P)$ and $H(\bar{P})$ for different $n$
  • Figure 5: Change of $VH(P)$ and $VH(\bar{P})$ for different $n$
  • ...and 3 more figures

Theorems & Definitions (18)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Example 8
  • Example 9
  • Theorem 1
  • ...and 8 more