Is there any known numerical approach to directly compute the log of the incomplete beta function? I would like to be able to compute $$ \log\left( \int_0^u x^{a-1} (1-x)^{b-1} dx \right) $$ accurately. The usual methods for computing the incomplete beta function are not sufficiently accurate in cases where $a$ and $b$ are large (on the order of thousands) and $u$ is far from $\frac{a}{a+b}$.



Write the incomplete Beta function as $$ B(a,b,u) = \frac{u^a (1-u)^b}{a} CF(a,b,u) $$ where $CF(a,b,u)$ is a continued fraction (see e.g. Wolfram function site) which has moderate values even for your parameter range (e.g. using IEEE double I can evaluate $CF(1000,2000,0.5) = 3.9881064361$). With this value, it is a simple exercise to evaluate the log of the product.

  • $\begingroup$ I need to compute the logarithm of the general incomplete beta function $\mathbb{B}(x,y,a,b)$ (lower limit of integration is $x\ge 0$ instead of zero). Any ideas? $\endgroup$ – becko Jun 4 '14 at 12:27
  • $\begingroup$ @becko: Not really, the only hint I can give is, that you may have a look at the Wolfram function site for functions.wolfram.com/GammaBetaErf/Beta4 $\endgroup$ – gammatester Jun 4 '14 at 12:32

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