deletions | additions       diff --git a/subsection_minimising_the_Jensen_Shannon__.tex b/subsection_minimising_the_Jensen_Shannon__.tex  index e568ea7..1f87c17 100644  --- a/subsection_minimising_the_Jensen_Shannon__.tex  +++ b/subsection_minimising_the_Jensen_Shannon__.tex 
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In order to utilise this procedure we must estimate the probability density functions $\phi(dP)$, $\phi(\rho c dU)$, $\phi(dP_+)$ and $\phi(dP_-)$ where $dP_\pm$ are calculated for the assumed value of $c$ as described above. Given $\phi(dP_\pm)$, the probability density functions for their sum and difference are  \[  \phi(dP_+ + dP_-) = \phi(dP_+) \convolution \otimes  \phi(dP_-) \]  \[  \phi(dP_+ - dP_-) = \phi(dP_+) \convolution \otimes  \phi(-dP_-) \]  The problem of estimating the probability density function (pdf) from a sample distribution has received considerable attention over the years. The simplest approach, generally called the maximum likelyhood estimator, is to estimate the pdf from the normalised histogram of the sampled distribution. For example, given the distribution of $dP$