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| Title of object: |
differential entropy |
| Canonical Name: |
DifferentialEntropy |
| Type: |
Definition |
| Created on: |
2002-02-13 01:49:03 |
| Modified on: |
2002-02-13 03:31:43 |
Revision comment (for changes between this and next version):
| Changes for correction #2460 ('classification?'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
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%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newcommand{\mv}[1]{\mathbf{#1}} % matrix or vector
\newcommand{\cov}{\mathrm{cov}}
\newcommand{\mvt}[1]{\mv{#1}^{\mathrm{T}}}
\newcommand{\mvi}[1]{\mv{#1}^{-1}}
\newcommand{\mpderiv}[1]{\frac{\partial}{\partial {#1}}} |
Content:
Let $(X, \mathfrak{B}, \mu)$ be a probability space, and let $f \in L^p(X, \mathfrak{B}, \mu)$, $||f||_{p} = 1$ be a function. The \emph{differential entropy} $h(f)$ is defined as
\begin{equation}
h(f) \equiv -\int_{X} |f|^p \log |f|^p\ d\mu
\end{equation}
Differential entropy is the continuous version of the Shannon entropy, $H[\mv{p}] = -\sum_{i} p_i \log p_i$. Consider first $u_a$, the uniform 1-dimensional distribution on $(0,a)$. The differential entropy is
\begin{equation}
h(u_a) = -\int_{0}^{a} \frac{1}{a} \log \frac{1}{a}\ d\mu = \log a.
\end{equation}
Next consider probability distributions such as the function
\begin{equation}
g = \frac{1}{2 \pi \sigma}e^{-\frac{(t-\mu)^2}{2 \sigma^2},
\end{equation}
the 1-dimensional gaussian. This pdf has differential entropy
\begin{equation}
h(g) = -\int_{\mathbb{R}} g \log g\ dt = \frac{1}{2} \log 2 \pi e \sigma^2.
\end{equation}
For a general $n$-dimensional gaussian $\mathcal{N}_{n}(\mv{\mu},\mv{K})$ with mean vector $\mv{\mu}$ and covariance matrix $\mv{K}$, $K_{ij} = \cov(x_i, x_j)$, we have
\begin{equation}
h(\mathcal{N}_{n}(\mv{\mu},\mv{K})) = \frac{1}{2} \log (2 \pi e)^n |\mv{K}|
\end{equation}
where $|\mv{K}| = \det{\mv{K}}$. |
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