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I am currently working through an example of a smoothing technique for information. I have finished applying the technique to my data and now want to figure out the probability of the item after it’s been smoothed. I have the computed values of $H$ but am stuck at how to compute $P$ in the equation:

$$H = - P \log_2 P.$$

For one item it has a smoothed information score of $.00324$ bits and by trial and error I have figured out the $P$, where $0<P<1$, should be $.000274$ as $-(.000274)(-11.834) = .00324$ but I haven’t been able to figure out an exact solution. I am not sure if I need use a numerical approximation for this problem or not. Any help would be appreciated as it’s been awhile since I have done any work in information theory.

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  • $\begingroup$ The function is not injective. $\endgroup$ Commented Oct 23 at 21:05
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    $\begingroup$ It's not clear what you mean by am "exact" solution...one can solve it via the Lambert W function, but of course that will still need to be computed numerically. Note that there are two solutions, not just the one you found. $\endgroup$ Commented Oct 23 at 21:06
  • $\begingroup$ For some basic information about writing mathematics at this site see, e.g., here, here, here and here. $\endgroup$ Commented Oct 23 at 21:06
  • $\begingroup$ The solution will always deepend on the Lambert $W$-function, which is non-elementary. $\endgroup$ Commented Oct 23 at 21:16
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    $\begingroup$ A little unexpected. Shouldn’t you have $H=-P\log P - (1-P)\log(1-P)$? $\endgroup$ Commented Oct 23 at 23:02

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