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The Tsallis Entropy calculator is a tool developed to compute and compare several versions of conditional Tsallis entropies existing in the literature.

Tsallis entropy is a Introduced in Information Theory is a generalization of Boltzmann-Gibbs theory and has been used to prove nonextensive statistical mechanics. Several alternative definitions for conditional Tsallis entropy were proposed but non of them is consensual.

The Tsallis entropy associated with a random variable $X$ is,
$$T_\alpha(X)
= {1\over \alpha-1}\left(1-\sum_x P(x)^\alpha\right),
\text{(for $\alpha> 0$, $\alpha\neq 1$)}.
$$

We consider three definitions of the general conditional Tsallis entropy that exist in the literature and introduce a new proposal of definition; by “general we mean that it can be applied to the Tsallis entropy with any positive parameter~$\alpha$.

Definition 1 (Proposed in [3])

For probability distributions $X$ and $Y$ define the Tsallis Conditional entropy by

\begin{eqnarray}
T_\alpha(Y|X) &=& \sum_x P(x)^\alpha T_\alpha(Y|x) \\
&= &{1\over \alpha-1} \sum_x P(x)^\alpha \left(1-\sum_y P(y|x)^\alpha\right).
\end{eqnarray}

Definition 2 (Proposed in [4])

For probability distributions $X$ and $Y$ define the Tsallis Conditional entropy by

\begin{eqnarray}
\label{re-cond}
S_\alpha(Y|X) &=& \hspace{-2mm} \sum_x P(x) T_\alpha(Y|x) \\
&=& \sum_x P(x) {1\over \alpha-1} \left(1- \sum_y P(y|x)^\alpha\right) \\
&=& {1\over \alpha-1} \sum_x P(x) \left(1-\sum_y P(y|x)^\alpha\right).
\end{eqnarray}

 

Definition 3 (Proposed in [4])

For probability distributions $X$ and $Y$ define the Tsallis Conditional entropy by

\begin{equation}
\label{RWdefA}
S’_\alpha(Y|X) =
{1\over \alpha-1}\left(1-{\sum_{x,y} P(x,y)^\alpha \over \sum_x P(x)^\alpha}\right).
\end{equation}

Definition 4 (new Proposal in [2])

For probability distributions $X$ and $Y$ define the Tsallis Conditional entropy by

\begin{equation}
T’_\alpha(Y|X) =
{1\over \alpha-1} \max_x \left(1-\sum_y P(y|x)^\alpha\right).
\end{equation}


Team

Andreia Teixeira
Luís Filipe Antunes

 


Publications

      1. A. Teixeira, A. Matos, L. Antunes, Conditional Rényi entropies, IEEE Transactions on Information Theory 58 (7), 4273-4277, 2012
      2. A. Teixeira, A. Souto, L. Antunes, Conditional Tsalles entropies, to be submitted soon.
      3. S. Furuichi, Information theoretical properties of Tsallis entropies, Journal of Mathematical Physics 47, 023302 (2006)
      4. S. Manije, M. Gholamreza and A. Mohammad,  Conditional Tsallis Entropy, Cybernetics and Information Technologies 13, No 2, pp:37-42, 2013.