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The Kullback-Leibler divergence

Let $p$ and $q$ describe two measure of probabilities, the Kullback-Leibler divergence is given by:

\begin{displaymath}D_{KL}(p\Vert q)=\int p(x) \ln \left( \frac{p(x)}{q(x)} \right) \; dx\end{displaymath}

In our case we have modelled the probabilities $\tilde{P}$ by discrete values on a known alphabet $\Sigma$, so:

\begin{displaymath}D\left(\tilde{P}(\cdot\vert\sigma s)\Vert\tilde{P}(\cdot\vert...
...lde{P}(\sigma'\vert\sigma s)}{\tilde{P}(\sigma'\vert s)}\right)\end{displaymath}

Thus:

\begin{displaymath}Err(\sigma s,s)=\sum_{\sigma'\in \Sigma}\tilde{P}(\sigma s \s...
...')\tilde{P}(s)}{\tilde{P}(\sigma s)\tilde{P}(s \sigma')}\right)\end{displaymath}

This measure of a distance between probabilities has been used in [81] together with the Laplace's law of succession in order to correct corrupted texts using a variable length Markov model.



franck 2006-10-01