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Entropies of the Poisson distribution as functions of intensity: "normal" and "anomalous" behavior

Dmitri Finkelshtein, Anatoliy Malyarenko, Yuliya Mishura, Kostiantyn Ralchenko

Abstract

The paper extends the analysis of the entropies of the Poisson distribution with parameter $λ$. It demonstrates that the Tsallis and Sharma-Mittal entropies exhibit monotonic behavior with respect to $λ$, whereas two generalized forms of the Rényi entropy may exhibit "anomalous" (non-monotonic) behavior. Additionally, we examine the asymptotic behavior of the entropies as $λ\to \infty$ and provide both lower and upper bounds for them.

Entropies of the Poisson distribution as functions of intensity: "normal" and "anomalous" behavior

Abstract

The paper extends the analysis of the entropies of the Poisson distribution with parameter . It demonstrates that the Tsallis and Sharma-Mittal entropies exhibit monotonic behavior with respect to , whereas two generalized forms of the Rényi entropy may exhibit "anomalous" (non-monotonic) behavior. Additionally, we examine the asymptotic behavior of the entropies as and provide both lower and upper bounds for them.

Paper Structure

This paper contains 12 sections, 15 theorems, 104 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Entropies of the Poisson distribution
  4. Lower and upper bounds for entropies of the Poisson distribution
  5. Monotonicity of entropies of Poisson distribution as the function of intensity parameter $\lambda$
  6. "Normal" behavior
  7. "Anomalous" behavior of generalized Rényi entropies
  8. The generalized Rényi entropy $\mathbf H_{GR}(\alpha,\lambda)$
  9. The generalized Rényi entropy $\mathbf H_{GR}(\alpha,\beta,\lambda)$
  10. Appendix
  11. Auxiliary results
  12. Some properties of the function $\psi(\alpha, \lambda)$

Key Result

Proposition 2.3

All entropies eq:SH--eq:SM take strictly positive values for all admissible values of the parameters. More precisely, let then

Figures (12)

  • Figure 1: A comparison of the Shannon entropy $\mathbf H_{SH}(\lambda)$ with the lower estimate $L(\lambda)$ given by \ref{['eq:lowerboundEnt']}, the upper estimate $U_{SH}(\lambda)$ given by \ref{['eq:sh-bound']}, and the asymptotic function $A_{SH}(\lambda)$ defined by the right hand side of \ref{['eq:asySH']}.
  • Figure 2: A comparison of the Rényi entropy $\mathbf H_{R}(\alpha,\lambda)$ with the lower estimate $L(\lambda)$ given by \ref{['eq:lowerboundEnt']}, the upper estimate $U_{R}(\alpha,\lambda,\gamma)$ given by \ref{['eq:estR-up1']}--\ref{['eq:estR-up2']} (with optimal values of $\gamma$ for $\lambda\in[1,20])$, and the asymptotic function $A_{R}(\alpha, \lambda)$ defined by the right hand side of \ref{['eq:asyR']}.
  • Figure 3: The Tsallis entropy $\mathbf H_T(\alpha,\lambda)$
  • Figure 4: The Sharma--Mittal entropy for $\alpha = 2$
  • Figure 5: The Sharma--Mittal entropy for $\beta = 2$
  • ...and 7 more figures

Theorems & Definitions (41)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Proposition 3.3
  • ...and 31 more