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Given:

p = $a^l$ * ($\frac{c}{a^l + b^l}$ + $\frac{b}{a^l + c^l}$)

Where a, b, c, p are known and are probabilities.

Solve for l. (1 equation and 1 unknown)

Does a closed form solution to this exist? I can't see how to solve using algebra.

If there is no closed form solution, what is the best way to solve this? I have looked into using Newton's method (but the derivative of this is very ugly) and the secant method (I am concerned about using bad starting values) - these aren't very "elegant".

Edit

Working on this problem, I have found out the following constraints for solutions:

  • $a$ >= $b$ >= $c$
  • $a$ + $b$ + $c$ = $1$
  • $p$ <= $(1 - a)$

I believe that the upper and lower limits of $l$ (that I could use as inputs into the secant method) for the given $a$, $b$, $c$, $p$ are defined by:

  • lower: $p$ >= 0
  • upper: $p$ <= $(1 - a)$
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  • $\begingroup$ What is unknown the power l? $\endgroup$ – Michael R. Chernick Dec 14 '16 at 13:38
  • $\begingroup$ Want to show us how far you've gotten? Perhaps by taking logs and rearranging? $\endgroup$ – ilanman Dec 14 '16 at 13:39
  • $\begingroup$ Mathematica can't solve it, just FYI. $\endgroup$ – user1566 Dec 14 '16 at 16:14
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    $\begingroup$ FWIW, It is a generalization of a logistic expression. For generic $a,b,c$ it has no closed form solution. For many combinations of $a,b,c,p$ Newton's method will work well, but it will not always work. In particular, there may be multiple solutions. Take, for instance, $a=0.454, b=0.375, c=0.7, p=0.537$: there will be solutions $l$ near $0$ and $\pm 4$. Could you please explain the connection to statistics and machine learning? $\endgroup$ – whuber Dec 14 '16 at 16:35

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