# Large-Sample Asymptotic Approximations for the Sampling and Posterior Distributions of Differential Entropy for Multivariate Normal Distributions

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## Abstract

**:**

## 1. Introduction

## 2. General Result

#### 2.1. Cumulant-Generating Function, Cumulants, and Central Moments of U

**Cumulant-generating function**Let U be the function defined in the introduction, i.e.,

**Cumulants**By construction, the nth cumulant of U is given by ${\kappa}_{n}={g}_{U}^{\left(n\right)}\left(0\right)$. In the present case, ${g}_{U}^{\left(n\right)}\left(t\right)$ can be obtained by direct derivation, yielding for the cumulants

**Central moments**Cumulants and central moments are related as follows: If we denote by μ, ${\sigma}^{2}$, γ and ${\gamma}_{2}$ the mean, variance, skewness and excess kurtosis of U, respectively, we have $\mu ={\kappa}_{1}$, ${\sigma}^{2}={\kappa}_{2}$, ${\gamma}_{1}={\kappa}_{3}/{\kappa}_{2}^{3/2}$, and ${\gamma}_{2}={\kappa}_{4}/{\kappa}_{2}^{2}$. Note that, by definition, μ is equal to the expression of Equation (7) and ${\sigma}^{2}$ to that of Equation (8) with $n=2$.

## 2.2. Asymptotic Expansion

## 2.3. Asymptotic Distribution of U

## 3. Application to Differential Entropy

#### 3.1. Sampling Distribution

#### 3.2. Posterior Distribution

**A**, $ln|{A}^{-1}{|=ln|A|}^{-1}=-ln\left|A\right|$, we have that $h(\mathsf{{\rm Y}}/n)-h\left({S}^{-1}\right)$ is equal to $h\left(S\right)-h(n\Sigma )$ or, equivalently, to $h(S/n)-h(\Sigma )$. As a consequence,

## 4. Application to Mutual Information and Multiinformation

#### 4.1. Sampling Mean

#### 4.2. Posterior Mean

## 5. Simulation Study

## 6. Discussion

**Figure 1.**Error on the mean (top row) and variance (bottom row) of sample entropy for various values of D and ν when using the first-order approximation (circles), the second-order approximation (squares), or the sampling scheme (diamonds). The error was calculated as the absolute value of the difference between the approximation and the true value. For the sampling scheme are represented the median as well as the symmetrical 90% probability interval of the error. Scale on y axis is logarithmic.

## Acknowledgements

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## Appendix

**Results Regarding the Cumulants**

#### 1. Results for ${\kappa}_{1}$

**Result 1:**${f}_{D}\left(\nu \right)$

**is a decreasing function of ν.**Derivation of ${f}_{D}\left(\nu \right)$ with respect to ν leads to

**Result 2: ${f}_{D}\left(\nu \right)$ is an increasing function of D.**We have

**Result 3: ${f}_{D}\left(\nu \right)$ is positive.**${f}_{D}\left(\nu \right)$ is the sum of terms that are strictly positive (cf previous paragraph); it is thus strictly positive.

**Result 4: ${f}_{D}\left(\nu \right)$ tends to infinity as D increases.**From the proof of Result 2, we have

**Result 5: ${f}_{D}\left(\nu \right)$ tends to 0 as ν increases.**We use the following inequality [20]

#### 2. Results for ${\kappa}_{n}$, $n\ge 2$

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**MDPI and ACS Style**

Marrelec, G.; Benali, H. Large-Sample Asymptotic Approximations for the Sampling and Posterior Distributions of Differential Entropy for Multivariate Normal Distributions. *Entropy* **2011**, *13*, 805-819.
https://doi.org/10.3390/e13040805

**AMA Style**

Marrelec G, Benali H. Large-Sample Asymptotic Approximations for the Sampling and Posterior Distributions of Differential Entropy for Multivariate Normal Distributions. *Entropy*. 2011; 13(4):805-819.
https://doi.org/10.3390/e13040805

**Chicago/Turabian Style**

Marrelec, Guillaume, and Habib Benali. 2011. "Large-Sample Asymptotic Approximations for the Sampling and Posterior Distributions of Differential Entropy for Multivariate Normal Distributions" *Entropy* 13, no. 4: 805-819.
https://doi.org/10.3390/e13040805