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I'm working on a Wisdom of Crowds solution to be used in conjunction with a Genetic Algorithm for a class. The paper I am reading makes mention of an inverse regularized beta function, given by $I_{a_{ij}}^{-1}(b_1,b_2)$ I've never encountered this function before, and it's not explained in the paper. I tried searching for it, but could not find anything clear on it. I understand $a_{ij}$ in regards to what I am doing, but the $I$ and the $(b_1,b_2)$ are unclear to me. Can someone give me an explanation? Or direct me where I can find an explanation?

John T
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2 Answers2

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The normalized (or regularized) incomplete beta function $I_x(a,b)$ for $a>0,$ $b>0,$ and $0\le x \le 1$ is defined as $$ I_x(a,b) = \frac{B_x(a,b)}{B(a,b)}, \qquad % B_x(a,b)= \int_0^x t^{a-1}(1-t)^{b-1}\,dt. $$ The inverse function $z := I_{a_{ij}}^{-1}(b_1,b_2)$ is the solution of the equation $$I_z(b_1,b_2)=a_{ij}.$$ Fore properties of this function see e.g. http://functions.wolfram.com/GammaBetaErf/InverseBetaRegularized/. I do not know from where your $b_1, b_2$ come from, but a standard application of this function is to compute the quantiles from a beta distribution or the CDF of the binomial distribution. Note that $I_x^{-1}(a,b)$ cannot be expressed with standard functions, normally it's values are computed as numerical solution of the above equation (e.g. with Newton's or Halley's method). If you are interesting in code implementations see e.g. the last two sections of the Boost documentation.

gammatester
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If random variable p is distributed according to a Beta distribution, the Inverse Regularized Beta gives the value for wich the Cumulative Distribution is equal to x. For instance, to calculate the median for a given a and b, you use x=0.5 to give the value of the variable corresponding to the median. Check https://en.wikipedia.org/wiki/Beta_distribution

ferrabras
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