Entropy of Uniformly Quantized Laplace and Half-Laplace Distributions

This post is about entropy of discrete stochastic variables that are derived by quantizing continuous stochastic variables. A good introduction to the concept of entropy in Information Theory is in [1]. This post is in continuation to the previous post in [2].

Let $x \in \mathbb{R}$ be a continuous stochastic variable with $\text{Laplace}(0, b)$ distribution and let $\hat{x} \in \hat{X}$ such that $\hat{X} \subset \mathbb{R}$ and $0 \in \hat{X}$ be its uniformly quantized version with step size $\Delta$. We would like to find the entropy of the discrete stochastic variable $\hat{x}$, denoted by $H_\hat{x}$, which provides us an estimate of the average number of bits required to encode $\hat{x}$.

HALF-LAPLACE DISTRIBUTION

Half-Laplace distribution refers to the distribution obtained by folding the zero-mean distribution function along the center. The new distribution is equivalent to the distribution of the stochastic variable $y = |x|$, i.e. the absolute value of the original Laplace distributed variable.

Let $\varphi(x)$ and $\Phi(x)$ denote the probability distribution function and cumulative distribution function of the Laplace variable $x$. These are defined as follows.

$$\begin{align}\varphi(x) &= \frac{1}{2b} e^{-\frac{|x|}{b}} \\ \Phi(x) &= \begin{cases} \frac{1}{2} e^{\frac{x}{b}} & \text{if $x < 0$} \\ 1 - \frac{1}{2} e^{-\frac{x}{b}} & \text{if $x \geq 0$}\end{cases}\end{align}$$

The half-Laplace cumulative distribution function of $y$, denoted by $\breve{\Phi}(y)$, is then given as follows.

$$\begin{align} \breve{\Phi}(y) &= \begin{cases} 0 & \text{if $y = 0$} \\ \Phi(y) - \Phi(-y) & \text{if $y > 0$}\end{cases} \\ &= 1 - e^{-\frac{y}{b}} \label{eqn:le} \end{align}$$

The expression in equation ($\ref{eqn:le}$) may be directly recognized as the cumulative distribution function of $\text{Exponential}(1/b)$. Therefore, the entropy of half-Laplace distribution may be found according to the expressions in [2] with $\lambda = 1/b$. 

LAPLACE DISTRIBUTION

Let $f(\hat{x})$ denote the probability mass function of the quantized variable $\hat{x}$. It is related to the probability density function $\varphi(x)$ as follows.

$$\begin{align} f(\hat{x}) = \int_{\hat{x} - \frac{\Delta}{2}}^{\hat{x} - \frac{\Delta}{2}} \varphi(x) dx \label{eqn:lq} \end{align}$$

Further, let $\hat{X}{}^+ \subset \hat{X}$ such that it contains the positive quantized values. The negative entropy $-H_\hat{x}$ is given (by definition) as follows. Note that the simplification is possible due to the symmetry of Laplace distribution.

$$\begin{align} -H_\hat{x} &= \sum_{\hat{x} \in \hat{X}} f(\hat{x}) \log f(\hat{x}) \\ &= f(0) \log f(0) + 2 \sum_{\hat{x} \in \hat{X}{}^+} f(\hat{x}) \log f(\hat{x}) \end{align}$$

By using the definition from equation ($\ref{eqn:lq}$) we have the following:

$$\begin{align} -H_\hat{x} &= \left(1 - e^{-\frac{\Delta}{2b}}\right) \log \left[1 - e^{-\frac{\Delta}{2b}} \right] \nonumber\\ &\quad +  \left(e^{\frac{\Delta}{2b}} - e^{-\frac{\Delta}{2b}}\right) \log \left[ \frac{1}{2} \left(e^{\frac{\Delta}{2b}} - e^{-\frac{\Delta}{2b}}\right) \right] v_1(b) \nonumber\\ &\quad + \left(e^{\frac{\Delta}{2b}} - e^{-\frac{\Delta}{2b}}\right) v_2(b) \end{align}$$

With the functions $v_{1,2}(b)$ given by:

$$\begin{align} v_1(b) &= \sum_{\hat{x} \in \hat{X}{}^+} e^{-\frac{\hat{x}}{b}} \\ v_2(b) &= \sum_{\hat{x} \in \hat{X}{}^+} e^{-\frac{\hat{x}}{b}} \log e^{-\frac{\hat{x}}{b}} \end{align}$$

EXACT RESULT

The analytical expressions for $v_{1,2}(b)$ may be computed starting from the following identities.

$$\begin{align} 1 &= \int_{-\infty}^{\infty} \varphi(x) dx = 2 \int_{0}^{\frac{\Delta}{2}} \varphi(x) dx + 2 \sum_{\hat{x} \in \hat{X}{}^+} \int_{\hat{x} - \frac{\Delta}{2}}^{\hat{x} + \frac{\Delta}{2}} \varphi(x) dx \\ -h_x &= \int_{-\infty}^{\infty} \varphi(x) \log \varphi(x) dx  \nonumber\\ &= 2 \int_{0}^{\frac{\Delta}{2}} \varphi(x) \log \varphi(x) dx + 2 \sum_{\hat{x} \in \hat{X}{}^+} \int_{\hat{x} - \frac{\Delta}{2}}^{\hat{x} + \frac{\Delta}{2}} \varphi(x) \log \varphi(x) dx \end{align}$$

Where $h_x$ is the differential entropy. Upon simplification, we obtain the following results.

$$\begin{align} v_1(b) &= \frac{e^{-\frac{\Delta}{2b}}}{e^{\frac{\Delta}{2b}} - e^{-\frac{\Delta}{2b}}} \\ v_2(b) &= \frac{-h_x - c_1 - \frac{\Delta}{2b} \left[ e^{\frac{\Delta}{2b}} + e^{-\frac{\Delta}{2b}} \right] v_1(b)}{e^{\frac{\Delta}{2b}} - e^{-\frac{\Delta}{2b}}}\end{align}$$

Where $c_1 = log\left(\frac{1}{2b}\right) - 1 + \frac{\Delta}{2b}e^{-\frac{\Delta}{2b}}$.

We see that the expressions are quite similar to the ones obtained for Exponential distribution in [2]. This is expected as Laplace and Exponential distributions are very closely related.

References:

[1] Thomas M. Cover, Joy A. Thomas, Elements of Information Theory, Second Edition, Wiley 2006.

[2] Aravindh Krishnamoorthy, Entropy of Uniformly Quantized Exponential Distribution, Applied Mathematics and Engineering Blog, October 2015.