# Empirical Estimation of Information Measures: A Literature Guide

## Abstract

**:**

## 1. Introduction

**Entropy:**$H\left(P\right)$ of a probability mass function P on a discrete set $\mathcal{A}$:$$\begin{array}{c}\hfill H\left(P\right)=\sum _{a\in \mathcal{A}}P\left(a\right)log\frac{1}{P\left(a\right)}.\end{array}$$**Relative Entropy:**$D(P\parallel Q)$ of a pair of probability measures $(P,Q)$ defined on the same measurable space (P and Q are known as the dominated and reference probability measures, respectively; $X\sim P$ indicates $\mathbb{P}[X\in B]=P\left(B\right)$, for any event B):$$\begin{array}{c}\hfill D(P\parallel Q)=\mathbb{E}\left(\right)open="["\; close="]">log\frac{\mathrm{d}P}{\mathrm{d}Q}\left(X\right),\phantom{\rule{1.em}{0ex}}X\sim P.\end{array}$$**Mutual Information:**$I(X;Y)$ of a joint probability measure ${P}_{XY}$:$$\begin{array}{c}\hfill I(X;Y)=D({P}_{XY}\parallel {P}_{X}\times {P}_{Y}),\end{array}$$

## 2. Entropy: Memoryless Sources

## 3. Entropy: Sources with Memory

## 4. Differential Entropy: Memoryless Sources

## 5. Relative Entropy: Memoryless Sources

**Finite alphabet.**In the discrete case, we can base a relative entropy estimator on the decomposition$$\begin{array}{c}\hfill D({P}_{X}\parallel {Q}_{X})=-H\left(X\right)+\mathbb{E}\left(\right)open="["\; close="]">{\u0131}_{{Q}_{X}}\left(X\right),\phantom{\rule{1.em}{0ex}}X\sim {P}_{X}\end{array}$$In the memoryless case, several of the algorithms reviewed in Section 2 for entropy estimation (e.g., [40,47]) find natural generalizations for the estimation of relative entropy. As for entropy estimation, the straightforward ratio of empirical counts can be used in the plug-in approach if $\left|\mathcal{A}\right|$ is negligible with respect to the number of observations. Otherwise, sample complexity can be lowered by a logarithmic factor by distorting the plug-in function; an estimator is proposed in [101], which is optimal in the minimax mean-square sense when the likelihood ratio is upper bounded by a constant that may depend on $\left|\mathcal{A}\right|$, although the algorithm can operate without prior knowledge of either the upper bound or $\left|\mathcal{A}\right|$. Another nice feature of that algorithm is that it can be modified to estimate other distance measures such as ${\chi}^{2}$-divergence and Hellinger distance. The asymptotic (in the alphabet size) minimax mean-square error is analyzed in [102] (see also [101]) when the likelihood ratio is bounded by a function of the alphabet size, and the number of observations is also allowed to grow with $\left|\mathcal{A}\right|$.**Continuous alphabet**. By the relative entropy data processing theorem,$$\begin{array}{c}\hfill D({P}_{X}\parallel {Q}_{X})\ge D({P}_{\phi \left(X\right)}\parallel {Q}_{\phi \left(Y\right)})\end{array}$$$$\begin{array}{c}\hfill D({P}_{\phi \left(X\right)}\parallel {Q}_{\phi \left(Y\right)})=log\left|\mathcal{B}\right|-H\left({P}_{\phi \left(X\right)}\right).\end{array}$$For multidimensional densities, relative entropy estimation via k-nearest-neighbor distances [104] is more attractive than the data-dependent partition methods. This has been extended to the estimation of Rényi divergence in [105]. Earlier, Hero et al. [106] considered the estimation of Rényi divergence when one of the measures is known, using minimum spanning trees.As shown in [107], it is possible to design consistent empirical relative entropy estimators based on non-consistent density estimates.The empirical estimation of the minimum relative entropy between the unknown probability measure that generates an observed independent sequence and a given exponential family is considered in [108] with a local likelihood modeling algorithm.M-estimators for the empirical estimation of f-divergence (according to Equation (16), r-divergence with $r\left(t\right)$ in Equation (15) is the relative entropy)$$\begin{array}{c}\hfill {D}_{f}(P\parallel Q)=\mathbb{E}\left(\right)open="["\; close="]">f\left(\right)open="("\; close=")">\frac{\mathrm{d}P}{\mathrm{d}Q}\left(Y\right)& ,\phantom{\rule{1.em}{0ex}}Y\sim Q;\end{array}$$A recent open-source toolbox for the empirical estimation of relative entropy (as well as many other information measures) for analog random variables can be found in [112]. Software estimating mutual information in independent component analysis can be found in [113]. Experimental results contrasting various methods can be found in [114].

## 6. Relative Entropy: Discrete Sources with Memory

## 7. Mutual Information: Memoryless Sources

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Generation of an estimate for entropy where the middle block is the function $log\frac{1}{t}$, $t\in (0,1]$.

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**MDPI and ACS Style**

Verdú, S.
Empirical Estimation of Information Measures: A Literature Guide. *Entropy* **2019**, *21*, 720.
https://doi.org/10.3390/e21080720

**AMA Style**

Verdú S.
Empirical Estimation of Information Measures: A Literature Guide. *Entropy*. 2019; 21(8):720.
https://doi.org/10.3390/e21080720

**Chicago/Turabian Style**

Verdú, Sergio.
2019. "Empirical Estimation of Information Measures: A Literature Guide" *Entropy* 21, no. 8: 720.
https://doi.org/10.3390/e21080720